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Experimental and model-based analyses of physiological determinants of force variability

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Experimental and model-based analyses of physiological determinants of force variability

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Content EXPERIMENTAL AND MODEL-BASED ANALYSES OF PHYSIOLOGICAL DETERMINANTS OF FORCE VARIABILITY By Akira Nagamori A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR Of PHILOSOPHY Biokinesiology August 2020 Copyright 2020 Akira Nagamori Acknowledgements First and foremost, I would like to thank Dr. Valero-Cuevas, my primary advisor, for his continuous support throughout my PhD work. He always encouraged me to explore dierent elds of study and taught me how to look at a problem from a broad and unique perspective. The freedom he allowed me helped me grow as an independent scientist. I also would like to thank all of the rest of my committee members { Drs. Gerald Loeb, James Finley, Lori Michener, Sook-Lei Liew. Dr. Loeb provided constant support, intelligent advices and intellectual challenges that allowed me to bridge computational modeling, sensorimotor physiology and human behavior. Dr. Finley always asked me critical questions that helped me achieve scientic research with the highest standard that I could possibly do. Drs. Michener and Liew encouraged me to think deeply about clinical applications of my research and added an important dimension in my research. There are many lab members that I have interacted and collaborated with throughout my Master's and PhD work. Especially, I am grateful to Dr. Christopher Laine, a former postdoctoral fellow, for his continuous guidance for every aspect of scientic research. He was always there to help me with a wealth of his knowledge and unique ideas whenever I had a problem with my research. I am similarly grateful to all the other unnamed lab members who oered assistance. I enjoyed discussing with each one of them because of their unique background and expertise. Especially, their advice was critical for me to develop a series of computational models of human sensorimotor systems. I have interacted with many faculty members and students in Division of Biokinesiology and Physical Therapy. All of those interactions helped me have condence in my thinking and expressing my opinions. Especially, Computational Motor Control and Learning journal club organized by Drs. Schweighofer, Finley, Liew and Leech give me a special opportunity to learn how to read and interpret scientic studies and how to ask questions through interactions with many bright scientists rst hand. Lastly, this dissertation would not have been completed without the support from both my ii immediate and extended family in Japan and US. Especially, my wife, Hanaka, was always there patiently waiting for me to complete this work without any complaints and she helped me grow as a person outside my research and school. The distance made it dicult to communicate in persons with many of my family members, but I always felt supported and enjoyed the moments when we got to see each other. I am and will always be grateful to have such a supportive family. iii Contents Acknowledgements ii List of Tables ix List of Figures x Abstract xxvi Introduction 1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 What is force variability? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Variability vs. Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Force variability and motor noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Theoretical applications of motor noise . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Signicance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Chapter 1: Aerented Muscle Model 13 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Material and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Closed-Loop Simulation of Aerented Muscle Model . . . . . . . . . . . . . . . . . . 16 Primary Simulation Protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Secondary Motor Unit Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Simulation 1: Closed-Loop Control of Musculotendon . . . . . . . . . . . . . . . . . 27 Simulation 2: Ia Aerent Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 iv Simulation 3: Fusimotor Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Simulation 4: Ib Aerent Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Chapter 2: Force variability is not motor noise: theoretical implications for motor control 49 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Material and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Conversion of spike trains into motor unit force using Fuglevand model . . . . . . . . 52 New model of a motor unit pool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Experimental Design and Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 70 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 New model of a population of motor units improves the simulated behavior of motor units compared to the Fuglevand model. . . . . . . . . . . . . . . . . . . . . . 72 Both `onion-skin' and `reverse onion-skin' patterns emerge from our model. . . . . . 75 Unfused tetanic contraction is not the cause of motor noise. . . . . . . . . . . . . . . 77 The Fuglevand model overestimates the contribution of motor unit properties to motor noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Musculotendon mechanics are essential for realistic simulation of the spectral char- acteristics of motor noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Motor noise cannot fully account for the experimentally observed amplitude of force variability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Motor unit functional organization is not a primary contributor to signal-dependent noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Physiological Mechanisms of Motor Noise . . . . . . . . . . . . . . . . . . . . . . . . 85 Alternative sources of force variability: uctuating synaptic input due to feedback- driven control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 v Theoretical and clinical implications . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Chapter 3: Synaptic noise is not a sucient explanation for force variability. 90 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Material and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Model of a population of motor units . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Motor unit model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Motoneuron model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Model of synaptic inputs to a population of motoneurons . . . . . . . . . . . . . . . 101 Model of stochasticity in motoneuron discharges . . . . . . . . . . . . . . . . . . . . 102 Hysteresis in motoneuron channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Simulation protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Data and statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Setting the parameters for the electrical properties of motoneurons . . . . . . . . . . 114 Synaptic noise does not follow the pattern of signal-dependent noise . . . . . . . . . 115 Synaptic noise does not generate signal dependent noise . . . . . . . . . . . . . . . . 117 Intrinsic properties of motoneurons facilitate generation of stable force output . . . . 117 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Physiological mechanisms of noise in synaptic input . . . . . . . . . . . . . . . . . . 120 Alternative sources of force variability . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Implication for eects of neuromodulatory inputs on force variability . . . . . . . . . 125 Theoretical and clinical implications . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Chapter 4: Physiologically-Realistic Sensorimotor Systems Model of Human Force Control 129 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 vi Model of a population of motor units . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Model of hierarchical sensorimotor control systems . . . . . . . . . . . . . . . . . . . 135 Simulation protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Comparisons to experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Eects of visual feedback sensitivity on force variability . . . . . . . . . . . . . . . . 152 Estimation and motor uncertainties in visuomotor error correction are necessary to replicate the experimental observed force variability below 2 Hz. . . . . . . . 154 Monosynaptic Ia excitation aects force variability across the entire frequencies. . . 156 Synchronization among motor units re ects changes in the frequency content of shared synaptic input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Eects of disynaptic Ib inhibition and Renshaw inhibition on force variability are limited. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 The combined eects of proprioceptive feedback on force variability are not a simple summation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Origin of low-frequency force variability below 3 Hz and its modulation by monosy- naptic Ia excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Physiological mechanisms for high-frequency force uctuations above 3 Hz . . . . . . 173 Clinical implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Summary and Future Work 178 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Appendix 184 Flexor-bias of the 10 Hz shared drive to synergistic muscles . . . . . . . . . . . . . . . . . 184 vii Bibliography 187 viii List of Tables 1 Model parameters for muscle based on architectural parameters of tibialis anterior muscle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Model parameters for muscle based on architectural parameters of exor carpi radialis muscle(Lieber et al., 1990; Loren and Lieber, 1995) . . . . . . . . . 66 3 Model parameters for the motoneuron model: . . . . . . . . . . . . . . . . . . 95 4 Model parameters for the sensorimotor systems: . . . . . . . . . . . . . . . . 146 ix List of Figures 1 Primary closed-loop simulations of arented muscle model. A musculoten- don unit receives a neural drive (ND) derived from three input sources and signal dependent noise (SDN). The muscle spindle generates excitatory inputs through group Ia and II aerents at a given fusimotor drive level. The Golgi tendon organ (GTO) sends an inhibitory input through the group Ib aerent pathway. The in- put contributions of Ia and Ib aerents are controlled by presynaptic control (PC). The tracking controller provides a control signal (C) based on a fraction of the error between reference force (F Reference ) and actual force output (F Output ). These inputs are integrated at the spinal level and signal dependent noise (SDN) is added. The resulting neural drive (ND), ltered to account for Ca + dynamics, induces contrac- tion of the muscle (length (L), velocity (V), acceleration (A)), taking into account mechanical factors such as the pennation angle (), mass (M), viscosity (B), parallel (PE1 and PE2) and series elastic elements (SE). Each aerent or eerent pathway has associated delays which account for the conduction velocities of each ber, the distance between the spinal cord and the muscle of 0.8 m (Elias et al., 2014) and synaptic delays of 2 ms Kandel et al. (2000) . . . . . . . . . . . . . . . . . . . . . . . 25 2 Example time-series signals and their power spectra obtained from the primary closed-loop simulation of aerented muscle. . . . . . . . . . . . . . . 26 3 Secondary Motor Unit Simulations. We used the neural drive (ND) obtained from the closed-loop simulations of aerented muscle to investigate changes in syn- chronization patterns of motor units in dierent simulated conditions. The simulated neural drive is fed to a simulated motor unit pool as a common input across the pool. The motor unit pool consists of 120 motor units. Individual motor units also receive independent noise (IN). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 x 4 Low-frequency `common drive' originates in musculotendon mechanics but is shaped and amplied by closed-loop control of muscle force. (A) Power spectra of output force from the aerented muscle model using two types of control strategies (open-loop control in red and closed-loop (error correction mech- anism only) control in green). Noise level was adjusted to match the amplitudes of overall force variability between the two control strategies. (B) Motor unit synchro- nization in the common drive range (1-5 Hz) as per the common drive index for the two control strategies. Note that open-loop control in the presence of broad-band noise produces only low-frequency force uctuations (below 5 Hz), which can be attributed to the low-pass ltering nature of musculotendon. Also, note that the closed-loop control only with a tracking controller results in generation of a distinct peak within the common drive range and increased common drive. . . . . . . . . . . 30 5 Modulation of Ia aerent feedback gain (i.e., presynaptic control input) in uences the amplitude of overall force variability through its eects across the force-relevant frequencies. (A) Coecient of variation (mean SE across 20 trials) at each presynaptic control level of Ia aerent feedback. The smaller the value of presynaptic control level, the lesser the contribution from this pathway. (B) Power spectra of force (mean across 20 trials at each level of presynaptic control). Power spectra were divided into two frequency ranges: 1-5 Hz common drive (CD) and 5-12 Hz physiological tremor (PT). Note that overall force variability shows U- shaped response to changes in the gain of Ia aerent feedback. Also, note that those changes in overall force variability occur across the range of frequencies below 12 Hz with distinctive peaks in the common drive and physiological tremor ranges. . . . . . 32 xi 6 Increases in the gain of Ia aerent feedback (i.e., larger presynaptic con- trol level) decrease low-frequency force variability (1-5 Hz) while they increase physiological tremor (5-12 Hz). All of the dependent variables are normalized to their respective mean values at the presynaptic control level of -0.5. (A) Mean force power within the common drive range (mean SE across 20 trials). (B) Mean force power within the physiological tremor range (mean SE across 20 trials). (C) Common drive index (CDI, mean SE of mean CDIs across 20 trials) (D) Coherence (mean SE of mean Fisher z-transformed coherence across 20 trials) in the common drive range. (E) Coherence in the physiological tremor range (mean SE of mean Fisher z-transformed coherence across 20 trials). Note that increases in the gain of in Ia aerent feedback (i.e., decreases in presynaptic inhibition of Ia aerent feedback) result in decreases in low-frequency force variability and increases in physiological tremor. Also, note that changes in motor unit synchronization gen- erally re ect changes in force variability. . . . . . . . . . . . . . . . . . . . . . . . . . 33 7 Frequency response of the closed-loop aerented muscle system with or without Ia aerent feedback. (A) gain of the system quantied as the ratio of the amplitude of output force, normalized to MVC, to the amplitude of input sinusoid. (B) phase delay of output force with respect to input sinusoids. Note that addition of Ia aerent feedback attenuates the amplication of low-frequency inputs and removes phase delays, present in the absence of Ia aerent feedback. . . . . . . . 35 xii 8 Modulation of Ia aerent activity through the fusimotor system aects the amplitude of force variability through its eects across the force-relevant frequencies. The top gure panel depicts the coecient of variation of force and the bottom gure panel shows power spectra of force. (A) Dynamic and static fusimotor drives were co-modulated. (B) dynamic fusimotor drive was varied while static fusimotor drive was kept constant at 70 pps. (C) static fusimotor drive was varied while dynamic fusimotor drive was kept constant at 70 pps. Note that overall amplitude of force variability is less sensitive to modulation of dynamic fusimotor drive, while that of static fusimotor drive has signicant eects on the amplitude of overall force variability. Also, eects of concurrent increases in dynamic and static fusimotor drives are a combination of their respective contributions. Importantly, those changes in overall force variability predominantly occur in the common drive range and physiological tremor ranges. . . . . . . . . . . . . . . . . . . . . . . . . . . 37 9 Modulation of Ia aerent activity through the fusimotor system decreases low-frequency force variability (1-5 Hz) while it increases physiological tremor (5-12 Hz). Three combinations of fusimotor drive modulation are indicated as follows; co-modulation of dynamic and static fusimotor drives (circles with solid line) and modulation of dynamic (triangles with dashed line) or static (squares with long-dashed line) fusimotor drives alone. All of the dependent variables presented here are normalized to their respective mean values at the lowest fusimotor drive for each condition. (A) Mean force power within the common drive range (meanSE across trials). (B) Mean force power within the physiological tremor range (meanSE across 20 trials). (C) Common drive index (CDI, meanSE across 20 trials). (D) Coherence (meanSE across 20 trials) in the common drive range. (E) Coherence (meanSE across 20 trials) in the physiological tremor range. Note that changes in force variability and motor unit synchronization in both frequency ranges can be attributed primarily to eects of static fusimotor drive. . . . . . . . 38 xiii 10 Modulation of presynaptic control of Ib feedback alters the amplitude of overall force variability mainly through its eects on the common drive range. (A) Ib-related changes in coecient of variation (meanSE across 20 trials). The smaller the value of presynaptic control, the smaller the contribution from the pathway. (B) Changes in power spectra of (mean across 20 trials) as a function of presynaptic control levels of Ib aerent feedback. Note the U-shaped response of the amplitude of overall force variability with increases in the strength of Ib aerent feedback. Also note that eects of Ib aerent feedback occurs mainly in the common drive range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 11 Increasing the strength of Ib aerent feedback decreases low-frequency force variability, but can lead to excessive force uctuations in that range. All of the dependent variables are normalized to their respective mean values at the lowest presynaptic control level. (A) Mean force power within the common drive range (meanSE across 20 trials). (B) Mean force power within the physiological tremor range (meanSE across 20 trials). (C) Common drive index (CDI, meanSE across 20 trials). (D) Coherence (meanSE across across 20 trials) in the common drive range. (E) Coherence (meanSE across 20 trials) in the physiological tremor range. Note that increased strength Ib aerent feedback preferentially aects force variability in the common drive range. Also note that CDI and coherence show dierent responses to excessive force uctuations in the common drive range. . . . . 41 xiv 12 Schematic representation of our new model of a motor unit pool. The model consists of three modules. Module 1 converts synaptic input, U eff , into spike trains of individual motor units. Module 2 turns spike trains into motor unit activa- tion,A, through three-stage process shown below. Stage 1 simulates calcium kinetics driven by action potentials (R). The calcium kinetics is described using ve states, [s], [cs], [c], [f] and [cf] with associated rate constants (k 1 ,k 2 ,k 3 and k 4 ) between those states (see text for details). Stage 2 converts [cf] into the intermediate activa- tion, e A, through a non-linear lter, which describes cooperativity and saturation of calcium binding and cross-bridge formation. Stage 3 introduces an additional rst- order dynamics to generate motor unit activation, A, from e A. Module 3 describes the contraction dynamics between muscle and a series elastic element and generates tendon force, F se , as the output. The detail descriptions of each module are given in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 13 Recruitment scheme of our new model. The frequency-synaptic input relation- ship of the selected motor units (n = 10) is shown to illustrate our new recruitment scheme. U r indicates the level of synaptic input at which all motor units are re- cruited. Lower-threshold motor units (red) (below 20% of the maximal synaptic input) show rapid acceleration upon recruitment and saturation of their discharge rates. Higher-threshold units (blue) linearly increase their discharge rates up to the maximal synaptic input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 xv 14 Population characteristics of our default motor unit pool. a) Histogram of peak tetanic force of 120 motor units (blue: slow-twitch, red: fast-twitch). The distribu- tion follows an exponential distribution where a large portion of motor units produce relatively smaller tetanic force. b) Peak tetanic forces as a function of recruitment thresholds. The relationship assumes the size-principle (Henneman, 1957): smaller units get recruited earlier than larger units. (c) Histogram of contraction time 120 motor units. The distribution follows the Rayleigh distribution which spans 20 ms to 110 ms. d) The relationship between peak tetanic force and contraction time. Slow- twitch units (in blue) have slower contraction time and smaller peak tetanic force. Within each ber type, no correlation between peak tetanic force and contraction time was assumed. e) The relationship between contraction time and twitch-tetanus ratio. Within each ber type, twitch-tetanus ratio is positively correlated with con- traction time (correlation coecients, r, are 0.587 and 0.531 for slow-twitch and fast-twitch units, respectively). f) The relationship between contraction time and the frequency at which half the tetanic force is achieved f 0:5 . Consistent with previ- ous experimental data (Botterman et al., 1986; Kernell et al., 1983), those parameters in our default motor unit pool are highly correlated (r = 0.794). . . . . . . . . . . . 69 xvi 15 Our new model improves predictions of force production at the individual motor unit level. (Row a) Output of representative motor units, one slower and one faster, from each model to constant synaptic input to their motoneuron at various frequencies. Note the output of the Fuglevand model is force in arbitrary units, whereas that of our model is motor unit activation (0{1) that is then scaled by peak tetanic force to produce force. (Row b) The output-to-frequency relationship of those same motor units. The shaded area represents the range of simulated discharge rates for those motor unit, which for the Fuglevand model does not correspond to the region of the steepest force-frequency relationship. Also, note that our new model includes the length-dependent output-to-frequency relationship described in (Brown et al., 1999; Song et al., 2008b). (Row c) The degree of fusion as a function of discharge rates. In the Fuglevand model, the increase in fusion is not monotonic and some units do not approach complete fusion. These issues are corrected in our new model. (Row d) The degree of fusion attained as output levels increase compared to the experimental observation from Maceeld et al. (1996) shows that the degree of fusion increases too slowly only to rise abruptly at higher outputs in the Fuglevand model, which is corrected in our new model. The dotted identity line is included for reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 16 Both `onion-skin' and `reverse onion-skin' patterns emerge from our model. (a) The relationship of peak discharge rate vs. recruitment threshold shows a signicant pos- itive correlation (r = 0.450, p<0.01). This demonstrates that higher-threshold units tend to show higher peak discharge rates (reverse-onion scheme features), without us having built that in. (b) The relationship between the discharge rate at 10% maximum input vs. recruitment threshold shows a signicant negative correlation (r = -0.247, p = 0.038). In contrast to (a), higher-threshold units tend to show slower discharge rates at 10% maximum input, demonstrating the onion-skin pattern as the emergent features of rate coding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 xvii 17 Increased discharge variability causes an increase in force variability through its interaction with the muscle force generating dynamics. a) Power spec- trum of a motor unit spike train. Note that increased discharge variability introduces low-frequency (<5 Hz) power. b) Power spectral density of motor unit force. In- creased discharge variability increases low-frequency force uctuations while attenu- ating those associated with unfused tetanic contraction. c) Power spectral density of tendon force. The spatial ltering of motor unit forces selectively attenuates higher- frequency force uctuations associated with unfused tetanic contraction. d) CoV of force with varying degrees of discharge variability. Increases in discharge variability result in increases in the overall amplitude of force variability. . . . . . . . . . . . . . 79 18 The Fuglevand model can overestimate the contribution of motor noise to force variability. Three simulated conditions are presented as follows: Fugle- vand model in gray, the new model without a series-elastic element (SSE) and the new model with SSE. a) Mean force as a function of the synaptic input. All three conditions exhibit a non-linear force-input relationship. b) SD of force normalized to the maximal force, plotted as a function of synaptic input levels. The slope and amplitude SD-input relationship of our new models are substantially dierent from those of Fuglevand model. c) CoV for force as a function of synaptic input levels. Our new model predicts substantially lower contribution of motor noise across all synaptic input levels tested here. Addition of a series-elastic element further reduces such contribution (blue line) at certain synaptic input levels. . . . . . . . . . . . . . 80 19 Viscoelastic properties of the contractile element damps high-frequency oscillations associated with discharge rate of motor units. Power spectra of output force at dierent synaptic input levels (5%, 20%, 40% and 70%) for our new model without a series-elastic element (SSE) in red and with SSE and blue. Note that addition of SSE substantially reduces power at frequencies> 5 Hz (shaded areas). 81 xviii 20 Motor noise cannot fully account for the experimentally observed ampli- tude of force variability. The amplitude of motor noise predicted by our new model is compared to the amplitude of force variability recorded from the rst dor- sal interosseus muscle in 20 participants reported by Moritz and colleagues (2005). Furthermore, our prediction is then compared to assumptions made in previous the- oretical models by (Jones et al., 2002) and (Todorov, 2005). Note that the predicted motor noise is smaller than the experimentally measured force variability (a black dotted line) the entire possible range of synaptic input and our prediction highly de- viates from the nature of motor noise assumed in previous theoretical models (green and magenta lines). Figure adapted from Fig 6. in Moritz et al. (2005) . . . . . . . . 83 21 Signal-dependent noise is not the by-product of the organization of a motor unit pool. The range of peak tetanic force (PR), the range of recruit- ment thresholds (U r ), and the number of motor units in a pool (N) were altered to demonstrate the sensitivity of our results. a) Mean force as a function of synaptic input levels. Note that this relationship is non-linear and highly dependent on those model parameters. b) SD of force (% of maximal force) as a function of synaptic input levels. A solid black line represents the theoretical scaling factor of the SD- input relationship (p = 1 in Eq. 31). Also, this result is robust to important model parameters suggested to in uence the relationship (Jones et al., 2002; Moritz et al., 2005). c) CoV for force as a function of synaptic input levels. Only a change in the number of motor units in a pool (orange line) had an appreciable in uence on CoV for force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 22 Equivalent electrical circuit of the motoneuron model. A motoneuron con- sists of two-compartments: soma and dendrite. The membrane potentials of each compartment,V s andV d , respectively, are described by the system of coupled dier- ential equations in Eq. 33. Synaptic inputs, I syn , are injected through the dendritic compartment. Model parameters are described in texts and in Table 3. . . . . . . . . 96 xix 23 Input-output relationships of a motor unit population. a) Discharge rates of ten motor units (ve slow and ve fast-twitch units) as a function of synaptic input levels. b) Force output of a motor unit population as a function of synaptic input levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 24 Short-term synchronization to quantify the degree of intrinsic noise in our model of motoneurons. a) Cross-correlation histograms between spike trains of two motoneurons (top) and their cumulative sum of the normalized histograms (bot- tom). The solid line indicates the average response across 10 trials and shaded lines indicate individual trials. b) Comparison of the values of CIS between our model and the data from (Binder and Powers, 2001). The error bars indicate standard deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 25 F measurement. The top panel shows the force output of the motor unit popula- tion in response to a triangular current with peak amplitude of 15% of the maximum synaptic current. The smoothed instantaneous discharge rates of a lower-threshold control unit and higher-threshold test unit are shown in the bottom panel. F was calculated as a dierence in discharge rates of the control unit at recruitment and derecruitment of the test unit as indicated in dotted, vertical lines. . . . . . . . . . . 113 26 Electrical properties of motoneurons. a) Distribution of input resistance. b) Distribution of membrane time constant. c) Distribution of recruitment threshold normalized to the maximal eective synaptic current required to recruit all motor units at their peak discharge rates. d) Distribution of AHP duration. e) Distribu- tion of AHP magnitude. f) Distribution of AHP half-decay time. g) Relationship between input resistance and AHP duration. Note that these parameters are pos- itively correlated (r = 0.55), consistent with a previous experimental observation (Kernell, 1966). h) Relationship between input resistance and frequency-current (f- I) slope. Note that there is no association between these parameters, consistent with previous experimental observations (Kernell, 1979; Schwindt, 1973). . . . . . . . . . 116 xx 27 Statistical properties of synaptic noise. a) The relationship between mean synaptic current and its standard deviation (SD) generated by asynchronous, random synaptic inputs. Note that this relationship is characterized by SD/ mean 0:5 . b) Discharge rates of a motoneuron as a function of synaptic currents. c) Discharge variability of a motoneuron as a function of synaptic currents. Note that increasing the amount of synaptic current decreases discharge variability. d) Autocorrelation functions of simulated noise generated by asynchronous, random synaptic inputs and of synthetic noise used in the following simulations. . . . . . . . . . . . . . . . . . . . 118 28 Synaptic noise does not generate signal dependent noise. Standard deviation of force variability is plotted as a function of mean force levels. The predicted motor noise is compared to experimental data by Moritz et al. (2005) and the theoretical relationship assumed in two previous models by Jones et al. (2002) and Todorov (2005). Note that the predicted motor noise by our model is lower than the exper- imentally observed force variability (a dotted line) at higher force levels (>40%). Also note that our model prediction deviates greatly from the amplitude of motor noise used in previous theoretical models (green and magenta lines). . . . . . . . . . 119 29 Hysteresis in motoneuron channels prevents sporadic discharges and re- duces discharge variability. a) Raster plots of all active motor units at 5% of the maximal synaptic input. The top and bottom panels represent example motor unit responses to noisy synaptic inputs without hysteresis and with hysteresis, re- spectively. Note that in the absence of hysteresis, some motor units display sporadic discharges. b) Coecient of variation (CoV) of inter-spike intervals (ISIs) with and without hysteresis. Note that a higher degree of discharge variability in the absence of hysteresis, which is conned within the 8-30% range in the presence of hysteresis. 121 xxi 30 Intrinsic properties of motoneurons facilitate generation of smooth force output. a) The dierence in control unit discharge rates at recruitment and dere- cruitment of test units (i.e., F measure) as a function of PIC amplitudes. Note that higher PIC amplitudes induce stronger hysteresis in motor unit discharge patterns. b) CoV of force at varying degree of PIC amplitudes compared to experimental data by Moritz et al. (2005). Note that the greater extent of hysteresis reduces force variability at low forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 31 Schematic representation of the model of our sensorimotor system. An visual feedback controller at the brain level sends a descending command signal based on the error perceived between the output force and the desired constant force. Synaptic inputs from spinal segmental feedback pathways (i.e., homologous monosynaptic Ia excitation, Ib inhibition, and Renshaw cell inhibition) are pre- synaptically added to the descending command at the spinal cord to generate the eective synaptic input to the population of 120 motor units. The population of motor units converts this eective synaptic input into output force. . . . . . . . . . . 133 32 Detailed descriptions of the model of our sensorimotor system. Visuomotor controller is represented as an proportional controller with its gain, K F , acting on the dierence (i.e., error) between the delayed information regarding output tendon force and reference force. Spinal segmental feedback pathways (i.e., monosynaptic Ia excitation, Ib inhibition, and Renshaw inhibition) are added to the descending com- mand signal from the visuomotor controller at the spinal cord with their respective delays and gains. Muscle spindle receives dynamic and static fusimotor drive via -motoneurons. Motoneurons convert synaptic inputs into spikes trains, which are then transformed into motor unit forces with the motor unit population model. The resulting muscle force (the sum of individual of motor unit forces) is then converted into tendon force via a model of musculotendon dynamics. . . . . . . . . . . . . . . . 136 xxii 33 Example time-series of each component of our sensorimotor systems model and their respective power spectral density. Panels on the left column show from the top to bottom time-series of Ia aerent ring rate, Ib aerent ring rate, a single Renshaw cell activity, command signal from the error-driven feedback control, eective synaptic input and force output. Panels on the right show their respective power spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 34 Experimental setup. a) The experimental setup. Participants were asked to produce and hold a given magnitude of isometric force against a force transducer using their wrist exor muscles (while keeping their ngers as relaxed as possible). The arm was restrained by supports at the forearm (not shown) and wrist level. b) The last 10-sec of a representative 15-sec force tracking trial targeting 5% MVC. c) High sensitivity visual feedback condition. The full range of the computer display was set to3% MVC from the target, thus even small variability in force were seen as large visual deviations from the target force. d) Low sensitivity visual feedback condition. The full range of the computer display was set to60% MVC from the target. Note that a give cursor deviation from the target force (i.e., tracking error) in (b) is seen 20x larger in the high sensitivity condition (d) compared to the low sensitivity condition (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 35 High sensitivity visual feedback aects force variability across frequen- cies. a) Comparisons of the amplitude of force variability (raw and de-trended force signals) between high and low sensitivity visual feedback conditions. Note that CoV of force for raw force signals was signicantly lower in high sensitivity visual feedback compared to low sensitivity visual feedback condition (p< 0:01). No signicant dif- ference was found for de-trended force signals (p = 0:40). b) Comparisons of power spectral density (PSD) in the low-frequency component below 3 Hz between the two conditions. c) Comparisons of power spectral density (PSD) in the high-frequency (3-15 Hz) component between the two conditions. Note that high sensitivity visual feedback increases power in the 1-2 Hz and 4-5 Hz ranges (green-shaded area) while reducing power in the 0-0.5 Hz range (red shaded area). . . . . . . . . . . . . . . . . 153 xxiii 36 Error-driven feedback with estimation and execution errors is necessary to replicate experimentally observed amplitude and spectral features of force variability. a) CoV of force compared across three simulated conditions: open-loop control, closed-loop control with and without estimation and motor un- certainties. CoV of force was computed with and without de-trending. b) Average CoV of ISIs of motor units in those three simulated conditions. denotes statistically signicant dierence (p < 0:05). c) Power spectrum of force as proportion of total power compared between open-loop condition in simulation and low visual feedback sensitivity condition in experiment. Note that our simulated force output under- estimates relative power in the very low-frequency range and overestimates that in higher frequencies. d) Power spectrum of force simulated for two closed-loop con- ditions compared against high visual feedback sensitivity condition in experiment. Note that addition of estimation and execution uncertainties dramatically improve the predicted power spectrum by our simulations. . . . . . . . . . . . . . . . . . . . . 157 37 Frequency-specic eects of monosynaptic Ia excitation on force variabil- ity. Its gain was modulated from low to high (K Ia = 0.00006 and 0.0001, respec- tively). Control condition refers to a condition in which all proprioceptive feedback pathways are removed (i.e., K Ia = 0). We ran 20 trials for each condition. a) The low-frequency (0-3 Hz) component of force variability. b) The high-frequency (3-15 Hz) component of force variability. c) CoV of raw and de-trended force. denotes signicant dierences at = 0.05. d) CoV of ISIs. . . . . . . . . . . . . . . . . . . . 159 38 Frequency-specic eects of spindle sensitivity modulation by the fusimo- tor system. Eects of the fusimotor systems on the low and high frequency com- ponents of force variability are shown as follows. a-b) Dynamic and static fusimotor drives are modulated simultaneously and tested at two levels: 0 Hz and 50 Hz. c-d) Dynamic fusimotor drive is modulated from 0 to 50 Hz while static fusimotor drive is kept constant at 25 Hz. e-f) Static fusimotor drive is modulated from 0 to 50 Hz while dynamic fusimotor drive is kept constant at 25 Hz. . . . . . . . . . . . . . . . 162 xxiv 39 Synchronization among motor units re ects changes in the frequency con- tent of shared synaptic input. Z-score coherence between two groups of cumu- lative spike trains is compared between a) control and low gain conditions in Fig. 37, b) low gain and high gain conditions in Fig. 37 and c) low and high dynamic fusimotor drive in Fig. 38. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 40 Ib aerent feedback has limited eects on force variability. Its gain was modulated from low to high (K Ib = 0.00006 and 0.0001, respectively). The control condition refers to a condition in which all proprioceptive feedback pathways are re- moved (i.e.,K Ib = 0). a) The low-frequency (0-5 Hz) component of force variability. b) The high-frequency (3-15 Hz) component of force variability. c) CoV of raw and de-trended force. d) CoV of ISIs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 41 Renshaw inhibition reduces 10 Hz force uctuations. Its gain was modulated from low to high (K RC = 0.0125 and 0.05, respectively). Control condition refers to a condition in which all proprioceptive feedback pathways are removed (i.e.,K RC = 0). a) The low-frequency (0-5 Hz) component of force variability. b) The high-frequency (3-15 Hz) component of force variability. c) CoV of raw and de-trended force. d) CoV of ISIs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 42 Combined eects of proprioceptive feedback on force variability are not a simple summation. Combined eects of all proprioceptive feedback pathways are compared to those of Ia excitation only. The gains of individual feedback pathways are set to their respective lowest values used in previous simulations. a) The low- frequency (0-5 Hz) component of force variability. b) The high-frequency (3-15 Hz) component of force variability. c) CoV of raw and de-trended force. d) CoV of ISIs. 168 43 Flexor-bias of the 10 Hz shared drive to synergistic muscles. a) Comparisons of coherence spectra between wrist exors and extensors. Note that the highly synchronized activity in the 8-15 Hz between wrist exors, which is almost absent between extensors. b) Comparisons of coherence in the 8-15 Hz range between the two muscle pairs. Note that all participants show higher coherence in wrist exors in this frequency range compared to wrist extensors (p< 0:01, paired t-test). . . . . 186 xxv Abstract This dissertation systematically investigates physiological determinants of force variability during constant isometric force production in humans. I use an integrative approach between computa- tional modeling of human sensorimotor systems and experimental observations. Despite the long history of research on this topic, there are still continuing debates about the fundamental sources of force variability and their relative importance. Here, we attempt to resolve an apparent dichotomy in the current literature: force variability is either a xed and unavoidable consequence of inherent noise in sensorimotor systems vs. a malleable consequence of control strategies embodied through hierarchically distributed sensorimotor systems. To this end, we present a series of physiologically realistic models of sensorimotor systems that allows us to perform 1) critical evaluations of previous assumptions and the underlying physiology and 2) direct quantication of the relative importance of individual sources of force variability and their interactions to explain experimentally observed features of force variability. We do so by comparing our model predictions against published and our own new experimental observations. We show that force variability is mostly not the result of constant, low-level, random noise as assumed in today's prevailing theories, but rather it contains a rich source of information about the consequences and contributions of our hierarchical and dis- tributed sensorimotor system to intended and unintended variability in voluntary action in health and disease. Furthermore, we highlighted the integrative approach between computational model- ing and experimental observations is critical to establish the causal relationship between a proposed mechanism and an observed phenomenon. To this end, detailed spectral analysis of synaptic input and the resulting force variability likely provide a tenable method to characterize the underlying physiological mechanisms for altered sensorimotor control in health and disease. xxvi Introduction Background Variability is an ubiquitous feature of biological sensorimotor systems. Variability can be observed at many levels in biological systems from neural activity to behavior. Ever since the early observa- tions on experienced blacksmiths by Nicholas Bernstein (Whiting, 1983), variability in kinematic and kinetics of human behavior has attracted considerable attention to understand principles of human sensorimotor control, and their disruption in aging and disease. He observed that the tra- jectories of a hammer and the joints of the arm vary considerably from movement to movement despite consistently hitting a chisel with the hammer (Song, 2008; Whiting, 1983). This observation suggests that variability directly associated with task performance is structured and that the ability to regulate/minimize it is a critical determinant that separates highly skilled professional/Olympic athletes from others. Furthermore, variability is a feature of human sensorimotor behavior that any theory of human sensorimotor control must include and explain. As such, it has informed theoretical models about the mechanisms used by the central nervous system to learn and produce motor behaviors (Dhawale et al., 2017; Diedrichsen et al., 2010; Harris and Wolpert, 1998; Scott, 2004; Seethapathi and Srinivasan, 2019; Sternad, 2018; Todorov and Jordan, 2002; van Beers et al., 2004; Wu et al., 2014). Previous studies have proposed that observed kinematic variability originates from two main sources: 1) variability in planning of movement (Churchland et al., 2006; Gordon et al., 1994a) and 2) variability in execution of the planned movement (Hamilton and Wolpert, 2002; Harris and Wolpert, 1998; van Beers et al., 2004). The relative importance of these two sources of variability is still debated, yet popular theoretical models seem to place the relatively stronger importance on the second mechanism as reviewed below. This variability in movement execution has been postulated to emerge from variability in muscle force output, or `force variability,' inherent in our motor systems. 1 What is force variability? To quantify the extent of force variability generated by our motor systems, previous studies used uctuations in force output measured during an isometric force production (Jones et al., 2002; Schmidt, 1988). Thus, throughout this dissertation, force variability refers to uctuations in force output when participants attempt to maintain a constant isometric force at a given target level for a short time (usually 1-30 sec). The amplitude of force variability is expressed as a variance over time such as the standard deviation (SD) and/or coecient of variation (CoV) (SD divided by mean). There exists an important methodological dierence across studies that is worth mentioning. While some studies compute SD or CoV from the raw force signal, others remove any slow linear trend from the raw force signal (or de-trend it) before computing SD or CoV. De-trending is per- formed when a signicant amount of drift in a raw force signal is observed which is common in the absence of visual (e.g., (Smith et al., 2018)) or auditory ((Matthews, 1996)) feedback. In such cases, CoV is calculated as SD of de-trended force divided by the mean of the raw force signal. It is important to note that de-trending tends to reduce the magnitude of SD and CoV (Tracy, 2007) by removing the very low-frequency component of force variability (usually less than 0.5 Hz). This eect depends on the duration of a force signal in which de-trending is performed. Thus, considera- tion of this methodological dierence is critical to compare the results across studies. Accordingly, we use or don't use de-trending depending on whether it was used in the experimental/simulated data we compare against. A common observation for the amplitude of force variability is that SD of force increases mono- tonically with the magnitude of the force (de C. Hamilton et al., 2004; Dideriksen et al., 2010, 2012; Enoka et al., 2003; Hu et al., 2014; Jesunathadas et al., 2012; Jones et al., 2002; Moritz et al., 2005; Petersen and Rostalski, 2019; Slifkin and Newell, 2000; Taylor et al., 2003; Yao et al., 2000; Zhou and Rymer, 2004), referred to as signal-dependent noise. This relationship between SD and mean force levels is assumed to be linear (Jones et al., 2002). In other words, the relative variance, CoV, is constant across mean force levels. Although some studies report the relationship is best t with an exponential relationship (Slifkin and Newell, 1998, 2000), we assume a linear relationship for 2 subsequent discussions throughout our discussion because this assumption is used as the theoretical underpinning of implemented noise in current theoretical models as discussed below. Given the limitations of variance (SD or CoV) as the simplest measure of force variability discussed throughout this dissertation, we also use spectral analysis to quantify force variability at dierent frequency ranges. We focus our attention on force variability below 15 Hz as those frequencies are most relevant to activities of daily living and clinical studies, and because there is little power at higher frequencies due to the low-pass ltering properties of musculotendon units (Baldissera et al., 1998; Bawa et al., 1976; Krylow and Rymer, 1997; Mannard and Stein, 1973) that limits the contribution of force variability above this frequency to the overall amplitude. The low-frequency component below 5 Hz is the dominant source of the experimentally measured force variability (e.g., Baweja et al. (2010, 2009); Slifkin and Newell (2000); Taylor et al. (2003)). Another component of force variability that has been studied extensively is force variability in the 6-12 Hz range, often called physiological tremor (e.g., Christakos et al. (2006); Laine et al. (2013, 2014); Lippold (1970); McAuley (2000)). We do not present others metrics used in previous studies to characterize force variability such as the amount of information transmission (mean/SD) (e.g., Slifkin et al. (2000); Vaillancourt et al. (2001a)) and the temporal structure of force variability (e.g., randomness) using approximate en- tropy (e.g., Slifkin and Newell (1998); Slifkin et al. (2000)) and detrended uctuation analysis (R acz and Valero-Cuevas, 2013) because there is no additional information provided by these phenomeno- logical metrics that is critical to improving our understanding of the neurophysiological mechanisms that produce force variability. Variability vs. Noise Variability and noise are often used interchangeably, yet there is a clear, important distinction to be made. Variability refers to variations in the quantity of a given variable. For example, an endpoint location after reaching movement or a joint trajectory during reach varies across trials. Or, a force output varies continuously across time. Importantly, variability does not specify its source/mechanism (Faisal et al., 2008; Song et al., 2008a), but rather it is a description of an 3 observed phenomenon. On the other hand, `noise' is a random disturbance that corrupts a signal (Faisal et al., 2008; Song et al., 2008a). For example, a force signal recorded from a transducer may contain random uctuations due to high-frequency electrical measurement noise. In biological systems, there are many sources of noise at various levels due to the stochastic nature of biochemical processes such as neurotransmitter release and regulation of ion channels, which aects higher-level neuronal processing (Faisal et al., 2008). Importantly, noise is an inherent aspect of biological sensorimotor systems that cannot be eliminated|but the eect of noise on a particular behavior can at times be mitigated or shaped by adequate control strategies (Loeb and Marks, 1985). Importantly, variability can arises from uctuations/variations in the signal itself. In other words, variability can contains both the signal and noise components. For example, sensory signals vary over time due to the ever-changing nature of the sensory information they encode. At a much higher level, a biological systems may purposefully increase trial-to-trial variability to search for better solutions (purposeful exploration, active sensing, persistence of excitation) or may decrease it to maximize rewards (exploitation) (Dhawale et al., 2019, 2017; Uehara et al., 2019). Variability from changes in the signal must, therefore, be clearly separated from those arising from random noise since they have clearly distinct origins, and consequences to sensorimotor control. As such, an attempt to explain observed variability requires careful examinations of each contribution. Force variability and motor noise The popular view in the current literature is that the amplitude of force variability and its signal de- pendence arise predominately, if not exclusively, as an unavoidable consequence of `signal-dependent motor noise:' random variations in muscle force output that increase with the magnitude of the force (Diedrichsen et al., 2010; Harris and Wolpert, 1998; Scott, 2004; Seethapathi and Srinivasan, 2019; Sternad, 2018; Todorov and Jordan, 2002; van Beers et al., 2004; Wu et al., 2014). Previous studies have proposed two main physiological mechanisms and sources for this signal-dependent mo- 4 tor noise: unfused tetanic contractions of motor units and stochasticity in motoneuron discharges (Faisal et al., 2008; Harris and Wolpert, 1998; Jones et al., 2002; Slifkin and Newell, 2000). Unfused tetanic contraction of motor units refer to uctuations in motor unit force due to Ca 2+ kinetics and cross-bridge turnover (Kandel et al., 2000). When an action potential in a motoneuron reaches a bundle of muscle bers that it innervates, it allows for the release of Ca 2+ from the sarcoplasmic reticulum into myolaments. The Ca 2+ diuses into myolaments and binds to troponin, which exposes actin-binding sites and allows cross-bridges to form. Those cross-bridges produce contractile force. This process does not occur instantaneously, but may take 20{50 ms for Ca 2+ to diuse and activate actin-binding sites and for cross-bridges to form. Simultaneously, the Ca 2+ in myolaments are actively removed, causing decreases in the concentration of Ca 2+ and thereby the number of cross bridges to be formed. This reuptake process results in a decrease in contractile force, which is slower (80{200 ms) than its increase due to Ca 2+ diusion. These two competing processes result in the initial rise and subsequent (slower) fall in contractile force as described as twitch contraction. When a subsequent action potential triggers the release of Ca 2+ before it is completely removed from the myolaments, it allows more cross-bridges to form and develop greater contractile force. This is because a single action potential does not result in the release of Ca 2+ enough to activate all of the actin bindings sites. If the ring rate of a motoneuron is low, then reuptake of Ca 2+ occurs between two successive action potentials. This process results in unfused tetanic contraction where contractile force shows ripples around a stable mean level. Once the ring rate of a motoneuron becomes high enough such that all actin binding sites have been activated, contractile force becomes smoother (fused tetanic contraction). However, complete fused tetanic contraction (i.e., truly constant force) requires a ring rate of a motoneuron higher than physiological recruitment (Kandel et al., 2000). Therefore, force ripples associated with unfused tetanic contractions of motor units at physiological motoneuron ring rates inevitably result in variability in force output. Another mechanism for force variability is that discharge patterns of motor units recorded during voluntary contraction in humans display variability in their successive inter-spike intervals (e.g., Clamann (1969); Kranz and Baumgartner (1974)). The degree of stochasticity is often quantied as the coecient of variation (CoV) of inter-spike intervals (ISIs), which typically ranges from 8 5 {30 % of CoV of ISIs (e.g., Clamann (1969); Matthews (1996); Nordstrom et al. (1992); Nordstrom and Miles (1991)). The sources of such stochastic discharge patterns of motor units can be divided into intrinsic and extrinsic factors. The intrinsic factors is related to the stochastic nature of biochemical processes that in uences the spike-generating mechanisms of a motoneuron such as uctuations in the voltage threshold and time-course of afterhyperpolarization (AHP) of the action potential (Diba, 2004; Mainen and Sejnowski, 1995). The extrinsic factors concern with apparently random membrane voltage uctuations in motoneurons caused by synaptic bombardment by many asynchronous, stochastic excitatory and inhibitory dendritic and somatic inputs (i.e., synaptic noise) (Calvin and Stevens, 1968). This discharge variability seems to t the above denition of noise if we consider the mean ring rate of a neuron as the signal, which is the putative primary determinant of a stable mean level of force output. The importance of motor noise arising from these two physiological mechanisms is supported by many simulations based upon the seminal computational model of motor unit recruitment and rate coding originally proposed by Andrew Fuglevand (Fuglevand et al., 1993). The Fuglevand model converts spike trains (binary signals with the value of 1 indicating the presence of an action potential) of motor units, with explicitly imposed stochasticity, into motor unit force output. The motor unit force generating mechanism was modeled as an impulse response of a second-order critically damped system whose time constant was adjusted to simulate contraction time of a twitch response of individual motor units. Since the model was originally proposed, many studies repurposed and modied it to quantify the contribution of motor noise to the experimentally observed force variability (de C. Hamilton et al., 2004; Dideriksen et al., 2010, 2012; Enoka et al., 2003; Hu et al., 2014; Jesunathadas et al., 2012; Jones et al., 2002; Moritz et al., 2005; Petersen and Rostalski, 2019; Taylor et al., 2003; Yao et al., 2000; Zhou and Rymer, 2004). Remarkably, all of these studies were able to successfully replicate signal-dependent noise, thanks to the model's wide range of parameters that can be used to t experimentally recorded discharge statistics of motor units. Furthermore, Moritz et al. (2005) demonstrated that the amplitude of simulated force variability (as per CoV of force) can be closely matched to the experimental counterpart from rst dorsal interosseous muscle by appropriately adjusting model parameters. In spite of some studies questioning the extrapolation of the Fuglevand model to functional domains beyond its 6 original intent, the numerous studies using it have led many to believe that the amplitude of force variability is determined entirely by motor noise and its signal dependence is "a natural by-product of the organization of the motor unit pool" (Jones et al., 2002). Theoretical applications of motor noise Many theories for human sensorimotor behavior (e.g., minimum variance theory and optimal control theory) invoke this signal-dependent motor noise to explain observed kinematic variability (Dhawale et al., 2017; Diedrichsen et al., 2010; Harris and Wolpert, 1998; Scott, 2004; Seethapathi and Srinivasan, 2019; Sternad, 2018; Todorov and Jordan, 2002; van Beers et al., 2004; Wu et al., 2014). For example, the seminal work by Hamilton and Wolpert (2002); Harris and Wolpert (1998) proposed the minimum-variance theory built based on the TOPS model (task optimization in the presence of signal-dependent noise) that attempts to minimize variability at the end point of movement in the presence of signal-dependent noise whose amplitude increase proportionally to the mean of control input. This model accurately replicated many stereotypical features of human eye and arm movements despite their drastically dierent dynamics that the central nervous system needs to contend with. Those features include the well known bell-shaped velocity prole of the hand during a reach (e.g., Flash and Hogan (1985)) and the velocity prole of saccade (e.g., Collewijn et al. (1988)) and its dependence on the amplitude of movement (Fig. 2 and 3 in (Harris and Wolpert, 1998)). Furthermore, they successfully replicated the Fitts' law during reaching movements where movement duration, T , is related to movement amplitude, A, and target width, W , as follows: T =a +blog 2 ( 2A W ); (1) where a and b are constants tted to experimental data (Fitts, 1954). The authors argued the TOPS theory provides a governing principle for human sensorimotor behavior as the model can describe many kinematic features observed from systems with drastically dierent dynamics requires a simple, common control scheme with a single assumption of motor noise. 7 Signal-dependent noise also plays an integral part in another in uential theory in human senso- rimotor control: optimal feedback control theory proposed originally by He et al. (1991); Todorov (2005); Todorov and Jordan (2002). This theory shares the basis with the minimum variance the- ory described above in that it describes human behaviors as a result of optimization of a certain cost function. Optimal feedback control theory extends this idea by incorporating a feedback com- ponent and suggests that humans not only execute but also adjust their control strategies based on the optimality principle in the face of uncertainty due motor and sensory noise. This theory was able to replicate other human behaviors such as reductions of variability directly linked to task performance independently of variability unrelated to task performance and task-dependent feedback responses, which previous models could not account for (Diedrichsen et al., 2010; Scott, 2004). Under the framework of these theories, observed kinematics and its variability re ect the perfor- mance limitation imposed by the motor system (Harris and Wolpert, 1998; van Beers et al., 2004) attempting to minimize the deleterious eects of `motor noise' on behavior (Todorov, 2005; Todorov and Jordan, 2002). As a consequence of these in uential studies, it become common practice to include signal-dependent noise in the motor command in models built to explain experimental ob- servations (Seethapathi and Srinivasan (2019); van Beers et al. (2004)). Furthermore, many clinical expression of increased kinematic/kinetic variability has been interpreted simply as a consequence of increased motor noise interfering with optimal control strategies implemented by the central nervous system (Chu and Sanger, 2009; Lunardini et al., 2015; McCrea and Eng, 2005; Therrien et al., 2016). Statement of the Problem Although the apparent success in describing experimentally observed force variability and its ap- plications to many in uential theories of sensorimotor control discussed above, the current concept of motor noise has a fundamental aws. This concept does not account for many other exper- imental observations that suggest the amplitude and spectral content of force variability re ect control strategies and properties inherent to distributed and hierarchical sensorimotor systems that 8 malleable and state-dependent. First, the amplitude and spectral content of force variability are shown to be modiable by experimental manipulations of other aspects of sensorimotor control. Manipulation of the charac- teristics of visual feedback such as feedback frequency of output force and sensitivity of displayed force can alter not only the amplitude of force variability (Baweja et al., 2010; Keenan et al., 2017; Slifkin et al., 2000), but also its frequency content (Baweja et al., 2010; Keenan et al., 2017; Laine et al., 2013, 2014). Moreover, prolonged vibration of muscle spindle has been shown to aect the amplitude of force variability (Shinohara et al., 2005; Yoshitake et al., 2004). Finally, the am- plitude of force variability during power grip is much smaller in patients with spinal cord injury than healthy controls, which was attributed to the increased level of physical activity with with the unimpaired upper limb (Nakanishi et al., 2019). These observations are powerful evidence that contradict considering force variability as an immutable performance limitation imposed by motor noise. Second, the assumption that participants can maintain a constant level of motor command is likely false. Without veridical visual or auditory feedback, motor output (i.e., force or discharge rate of a motor unit) tends to drift (e.g., Ambike et al. (2016); Smith et al. (2018)). As a con- sequence, maintaining output at a constant level requires closed-loop control via sensory feedback (often visual). Control of force output through a visuomotor loop introduces low-frequency (1-2 Hz) uctuations in synaptic drive and resulting muscle force (Baweja et al., 2010, 2009; Keenan et al., 2017; Slifkin and Newell, 2000; Smith et al., 2018). Some studies attempt to minimize the ef- fect of variability introduced by such a control mechanism by removing visual feedback and instead de-trending the force signal before computing the amplitude of force variability (de C. Hamilton et al., 2004; Jones et al., 2002). However, some involvement of a (perhaps haptic) feedback control mechanism is certainly necessary even in the absence of visual feedback since participants are still required to somehow maintain a constant level of motor command. Even if the eects of such a higher-order control mechanism were minimized, proprioceptive feedback constantly modulates synaptic inputs in the 5-12 Hz range due to its interactions with musculotendon mechanics (Nag- amori et al., 2018). Furthermore, descending inputs may contain oscillations in the frequency range above 8 Hz (Brown et al., 1998; de Vries et al., 2016; Gross et al., 2002; Grosse and Brown, 2003; 9 Soteropoulos and Baker, 2006)). These oscillations by themselves may not contribute signicantly to the overall amplitude of force variability because of the low-pass ltering properties of a mus- culotendon unit (Baldissera et al., 1998; Bawa et al., 1976; Krylow and Rymer, 1997; Mannard and Stein, 1973), but they can still be de-modulated to the low-frequency (< 5Hz) component of synaptic drive/force indirectly aecting the overall amplitude (Watanabe and Kohn, 2015). An additional consequence of an inevitably uctuating level of synaptic input is that these uc- tuations are often shared across the pool of motor units and can cause synchronization among them (Farina and Negro, 2015). The eects of synchronization would be largest below 10 Hz (Farina and Negro, 2015; Farina et al., 2014; Negro et al., 2016) due, again, to the low-pass ltering properties of motor units (Fig. 7, cf. (Mannard and Stein, 1973)). Accordingly, recent studies have found the high degree of motor unit synchronization below 5 Hz, which is correlated with the amplitude of force variability (Castronovo et al., 2018; Negro et al., 2009; Pereira et al., 2019). On the other hand, many previous models based on the Fuglevand model assumed no synchronization among motor units and, if included, the estimated force variability greatly exceeds that found experimen- tally (Taylor et al., 2003; Yao et al., 2000). This is because correlated motor unit discharges are expected to have disproportionately large eects on force variability compared to asynchronous motor unit discharges. These observations imply that the contribution of motor noise might have been overestimated in previous simulation studies. Finally, certain clinical manifestations of increased force variability are frequency-specic. These observations are not limited to many forms of pathological tremor that appear at well-dened frequencies (Hallett, 2008) and extends to many other clinical conditions without overt pathological tremor. For example, increased force variability due to aging (Castronovo et al., 2018; Pereira et al., 2019) and neurological conditions such as stroke (Lodha and Christou, 2017) and essential tremor (Neely et al., 2015) have been associated with increases in the low-frequency component of force variability, not necessarily a broad band increase in power as predicted by the popular concept of noise. To date, this apparent con ict in the current literature has not be resolved. Many experimental and computational studies discussed so far have focused on a certain aspect of force variability 10 and provided additional experimental/computational data to provide piecemeal support of one argument over the other, but they have failed to present a comprehensive view that encompasses the potential sources and mechanisms of force variability, and their interactions, to account for all aspects of force variability. To address this gap in the current literature, this dissertation presents a series of computational models in which we perform critical evaluations of the physiological validity of previous assumptions and quantify the contribution of specic physiological mechanisms to force variability at much deeper levels than simply observing behavior. To test our model predictions, we compare them against experimental observations both in the literature and in the present study. Signicance Although computational models are critical to derive theories of sensorimotor control from human behavior as discussed above, those theories risk being misleading if they are based on misconcep- tions about the underlying physiology. Our work will provide a much needed comprehensive and critical evaluation of the underlying physiological mechanisms of force variability. Thus, our study provides a critical path toward developing theories and models of sensorimotor control that provide a physiologically valid and clinically useful understanding of healthy and pathologic force variability. Furthermore, the integrative approach we take between computational modeling and experimen- tal studies provides a tenable method to characterize the underlying physiological mechanisms for sensorimotor control in health and disease. Thesis outline Chapter 2 presents a closed-loop simulation of aerented muscle model through which we investigate eects of closed-loop interactions among musculotendon dynamics, proprioceptive feedback and error-driven feedback on the amplitude and spectral structure of force variability. This has been published as Nagamori et al. (2018). Chapter 3 presents a new model of motor unit force generation to quantify the extent of motor 11 noise contribution to force variability. This model evaluates critical limitations and assumptions in previous models of a motor unit population and improves the extend of predicted motor noise by incorporating physiologically important features of motor unit force generation. The work in this chapter is currently in review (Nagmaori et al. 2020). Chapter 4 extends the motor unit population model presented in Chapter 3 to include a model of motoneuron spike generating dynamics. Addition of this critical component allows us to sim- ulate dynamic interactions between non-stationary, uctuating synaptic input and properties of motoneurons that determine how noise in synaptic input in uences the output action potential discharge patterns, and the resulting force variability. Some element of the work in this chapter was included in the manuscript in review (Nagmaori et al. 2020). Chapter 5 introduces a sensorimotor systems model that combines all models presented in Chapter 2, 3, and 4. This model enables, for the rst time, quantication of emergent behaviors that arise from interactions between distributed sensorimotor systems and realistic noise. Simulation results are compared to experimental data both in the literature and in the current study to fully characterize eects of emergent properties of distributed sensorimotor systems on force variability. This work will be published shortly. Chapter 6 summarizes the research presented from Chapter 2 to 5 and discusses future work. Unpublished experimental data critical for discussion in Chapter 5 is included in Chapter 7 as an Appendix. 12 Chapter 1: Aerented Muscle Model Abstract Involuntary force variability below 15 Hz arises from, and is in uenced by, many factors includ- ing descending neural drive, proprioceptive feedback, and mechanical properties of muscles and tendons. However, their potential interactions that give rise to the well-structured spectrum of involuntary force variability are not well understood due to a lack of experimental techniques. Here, we investigated the generation, modulation, and interactions among dierent sources of force variability using a physiologically-grounded closed-loop simulation of an aerented muscle model. The closed-loop simulation included a musculotendon model, muscle spindle, Golgi tendon organ (GTO), and a tracking controller which enabled target-guided force tracking. We demonstrate that closed-loop control of an aerented musculotendon suces to replicate and explain surpris- ingly many cardinal features of involuntary force variability. Specically, we present 1) a potential origin of low-frequency force variability associated with co-modulation of motor unit ring rates (i.e.,`common drive'), 2) an in-depth characterization of how proprioceptive feedback pathways suf- ce to generate 5-12 Hz physiological tremor, and 3) evidence that modulation of those feedback pathways (i.e., presynaptic inhibition of Ia and Ib aerents, and spindle sensitivity via fusimotor drive) in uence the full spectrum of force variability. These results highlight the previously under- estimated importance of closed-loop neuromechanical interactions in explaining involuntary force variability during voluntary `isometric' force control. Furthermore, these results provide the basis for a unifying theory that relates spinal circuitry to various manifestations of altered involuntary force variability in fatigue, aging and neurological disease. Introduction Involuntary uctuations in muscle force are inherent to human motor control. Evidence suggests that this apparent `noise' is functionally signicant for movement execution and learning (Cohen and Sternad, 2009; Enoka et al., 2003; Faisal et al., 2008; Harris and Wolpert, 1998; Stein et al., 2005). Furthermore, amplication of force variability or distortion of its frequency content is an 13 almost universal phenomenon whenever neuromuscular control is altered, for example by aging Enoka et al. (2003); Tracy et al. (2005), fatigue (Cresswell and Loscher, 2000; Hunter et al., 2002), and neurological diseases (Chu and Sanger, 2009; Lafargue et al., 2003; Lodha et al., 2010; Rothwell et al., 1982; Vaillancourt et al., 2001a). However, whether such phenomenon is caused by common or distinct factors is not known because the sources of involuntary force variability and their potential interactions are not well understood. By some descriptions, involuntary force variability is a manifestation of broad-band neural noise (Faisal et al., 2008; Harris and Wolpert, 1998; Stein et al., 2005). However, neural drive to muscles is known to have a highly structured frequency spectrum (Farina et al., 2016). Accordingly, dierent neural sources of involuntary force variability, such as descending drive (Llin as, 2009; Lodha and Christou, 2017; Watanabe and Kohn, 2015; Williams et al., 2010) and proprioceptive feedback (Christakos et al., 2006; Lidierth and Wall, 1996; Lippold, 1970; Stein and O^ guzt oreli, 1976), are often described specically in terms of their frequency content. Frequency-specic force variability can also stem from mechanical sources (e.g. mechanical resonance), even if the neural drive itself contains no distinct oscillatory components (Lakie et al., 2012; Vernooij et al., 2013). Attempts to understand the relative contribution that each `source' of involuntary force variability makes to the total have been dicult, given that they all act concurrently during muscle activation, and are dicult to experimentally isolate and manipulate. While dierent sources of involuntary force variability may be distinct, they are not likely to be independent. For example, there is recent evidence suggesting an inverse relationship between low (1-5 Hz)- and high-frequency (5-12 Hz) neural drive to muscles (De Luca et al., 2009; Laine et al., 2013, 2014). The high-frequency drive may originate from stretch-re ex circuitry (Christakos et al., 2006; Lippold, 1970). The low-frequency drive (the so-called `common drive') does not have a known origin, but appears to be negatively in uenced by Ia aerent feedback, since it is strongest in muscles which have low spindle densities(De Luca et al., 2009). Further, experimental conditions which increase high-frequency neural drive and H-re ex amplitudes also decrease low frequency neural drive (Laine et al., 2013, 2014). Together, the clear implication is that Ia aerent feedback oppositely aects high and low frequency neural drive (and thus force variability), but the mechanistic details are not yet understood. 14 In this study, we establish how neural and mechanical sources of force variability interact to produce the structured force spectrum observed experimentally using a physiologically-grounded model of aerented muscle. Our simulation of an aerented musculotendon set inside of a closed- loop control scheme allowed us probe the mechanistic interactions that exist among an error cor- rection mechanism for muscle force, proprioceptive feedback, and mechanical properties of muscle. Further, we describe these interactions in terms of their eects on involuntary force variability and on the behavior of a simulated pool of motor units. Our hypotheses were 1) neuromechanical interactions inherent to the closed-loop control of viscoelastic musculotendon would suce to produce low-frequency force variability, 2) tuning of proprioceptive feedback (i.e., known modulation of fusimotor drive or presynaptic gains) would impact the entire frequency spectrum of force variability, and 3) those changes in force variability would be re ected in motor unit synchronization. Our ndings not only support these predictions, but (i) emphasize the importance of neurome- chanical interactions to levels not previously recognized, and (ii) they describe how isolated changes in each proprioceptive pathway gain in uences the full spectrum of involuntary force variability. This novel demonstration lls a critical gap in our understanding of how error correction mech- anisms, proprioceptive feedback, noise, and musculotendon mechanics are interrelated, and our results emphasize the critical importance of investigating involuntary force variability within the context of closed-loop control. Our results are an important step towards a unifying theory that relates spinal circuitry to various manifestations of altered involuntary force variability in functional performance (Enoka et al., 2003), aging (Enoka et al., 2003; Tracy et al., 2005), fatigue (Cresswell and Loscher, 2000; Hunter et al., 2002) and neurological disease (Chu and Sanger, 2009; Lafargue et al., 2003; Lodha et al., 2010; Rothwell et al., 1982; Vaillancourt et al., 2001a). 15 Material and Methods Closed-Loop Simulation of Aerented Muscle Model We used a closed-loop simulation of an aerented muscle model, which is an extension of a previously published model (Laine et al., 2016), to identify the sources of frequency-specic force variability and to characterize interactions among them. The schematic diagram of this model is provided in Fig. 1. The aerented muscle model is comprised of a musculotendon unit (Brown et al., 1996; Cheng et al., 2000; Song et al., 2008a,b), muscle spindle (Mileusnic et al., 2006), and Golgi tendon organ (GTO) (Elias et al., 2014), which is controlled by a tracking controller (Laine et al., 2016). The model was implemented in the MATLAB environment (The MathWorks Inc., Natick). Full details of each model are given in the corresponding references and only brief descriptions are provided here. All model parameters were taken from the corresponding references except for musculotendon architecture as described below. Musculotendon Model The musculotendon model incorporates realistic physiological properties of muscle force production and resulting contraction dynamics (Brown et al., 1996; Cheng et al., 2000; Song et al., 2008a,b). Those properties include non-linear properties of passive elements of muscle and tendon, force-length and force-velocity relationships of contractile elements, and activation-force dynamics such as calcium kinetics, sag, yield, and activation-frequency relationship (Brown et al., 1999; Brown and Loeb, 2000). Architectural parameters of the musculotendon model was adjusted according to those for the tibialis anterior muscle (Table 1) (Arnold et al., 2010; Elias et al., 2014). The tibialis anterior was chosen because it is a large supercial muscle often chosen for motor unit analysis (De Luca et al., 2009; Laine et al., 2014). Closed-loop Control System A closed-loop controller enabled the entire system to perform a force tracking task at a constant force level as previously described (Laine et al., 2016). The control signal (C) from a tracking controller and proprioceptive feedback signals were delayed and integrated at the spinal level (the box in Fig. 1) to generate the neural drive (ND) to the muscle. The neural drive was then passed through a sigmoid transfer function, which describes the non-linear input-output behavior of a pool of motoneurons (Raphael et al., 2010; Tsianos et al., 2014). In this study, signal 16 Table 1: Model parameters for muscle based on architectural parameters of tibialis anterior muscle Mass (g) 150 Optimal ber length (cm) 6.8 Tendon length (cm) 27.5 Pennation angle (deg) 9.6 Ia aerent conduction velocity (m/s) 64.5 II aerent conduction velocity (m/s) 32.5 Ib aerent conduction velocity (m/s) 59 -motoneuron conduction velocity (m/s) 56 Synaptic delay (ms) 2 17 dependent noise (SDN) was added to the neural drive such that the magnitude of output force variability closely matched experimental literature ( 2-10 % coecient of variation)(Galganski et al., 1993). The SDN was modeled as zero-mean, white noise signal low-pass ltered at 100 Hz (4th order Butterworth lter) with variance of 0.3. The sum of ND and SDN was delayed by 16ms to account for conduction delay along-motoneuron and passed through the activation lter, which accounts for rst-order calcium dynamics of muscle (Song et al., 2008a). The resulting activation signal (MA) induces contraction of muscle and associated dynamics within the musculotendon. Delays along each aerent or eerent pathway accounted for the conduction velocities of each ber, the distance between the spinal cord and the muscle of 0.8 m (Elias et al., 2014) and synaptic delays of 2 ms (Kandel et al., 2000) (Table 1). Tracking Controller: The tracking controller was implemented to ensure successful force tracking at a given contraction level as in (Laine et al., 2016). It continuously adjusts the level of the control signal (C) based on a fraction of the error between actual force output (F Output ) and reference force (F Reference ) as shown in Fig. 1 and in the equation below, C =G (F Reference F Output ) (2) where G refers to the gain of the tracking controller, which was set at 0.00035. The gain was empiri- cally determined such that coecient of variation of force is around the upper end of experimentally observed range (10 %) in the absence of proprioceptive feedback. This controller is not designed to re ect a specic neural circuit, but rather it is meant to describe general error correction mechanisms used during various motor tasks as done in our previous study (Laine et al., 2016). Accordingly, the latency (total feedback loop delay of 66-ms) was set to approximately match the fastest EMG responses measured during tracking tasks (Schmidt, 1988), which is thought to involve the supraspinal circuits (Pruszynski and Scott, 2012). Muscle Spindle: The muscle spindle model was adopted from (Mileusnic et al., 2006). This model generates Ia and II outputs as functions of muscle ber length (L), velocity (V) and accel- eration (A) through three types of intrafusal bers (i.e., bag 1 , bag 2 , and chain bers). Ia and II 18 pathways provide monosynaptic and dysynaptic excitation of -motoneuron, respectively. Also, each of intrafusal ber receives ber-type-specic fusimotor drive (dynamic and static) through -motoneuron. Golgi Tendon Organ: The Golgi tendon organ (GTO) model was adopted from (Elias et al., 2014). This model describes population behavior of GTOs and resulting Ib aerent activity. The GTO model converts tendon force into Ib ber output using the transfer function described in (Elias et al., 2014). The Ib aerent feedback was modeled as a disynaptic inhibitory pathway to -motoneuron (Elias et al., 2014). Although some evidence suggests convergence of Ia aerent feedback onto Ib interneurons (Wolpaw, 2010) and reversal to positive feedback loops (McCrea, 1986), we simplied the model as done previously (Elias et al., 2014; Laine et al., 2016; Raphael et al., 2010; Tsianos et al., 2014). Gain control of proprioceptive feedback: We used gain control mechanisms of propriocep- tive feedback pathways similar to ones used in the previous literature (Raphael et al., 2010; Tsianos et al., 2014). Activity of individual aerents (Ia, II, and Ib) were rst normalized to a value between 0 and 1. Each proprioceptive feedback pathway except for II aerents received presynaptic control input (PC Ia and PC Ib in Fig. 1), whose value could range from -1 to 1. The sum of the nor- malized aerent activity and presynaptic control input was then passed through a sigmoid transfer function, which bounds the output to be a value between 0 and 1, to account for nonlinearity in the input-output relationship of a given cell population as described previously (Raphael et al., 2010; Tsianos et al., 2014). This presynaptic control mechanism is a salient feature of gain control of pro- priocetive feedback systems at the spinal cord level, which can be controlled by various descending pathways as well as interneuronal circuits within the spinal cord (Fink et al., 2014; Rudomin et al., 1983; Rudomin and Schmidt, 1999; Seki et al., 2003). We did not consider presynaptic control of group II aerent feedback in this study because the connectivity of group II interneurons is not well understood (Loeb and Tsianos, 2015). However, we included weak excitatory input from II aerent feedback ( 2% of total input), as their weak contributions to stretch re ex have been suggested (Windhorst, 2007). Another gain control mechanism for Ia aerent feedback is fusimotor drive through -motoneurons. 19 dynamic and static fusimotor drives (fusimotor drive in Fig. 1) alter responses of intrafusal bers to muscle stretch and therefore the behavior of Ia and II aerent feedback from muscle spindles (Mileusnic et al., 2006). Dynamic fusimotor drive determines the dynamic response of muscle spin- dles, while static fusimotor drive determines the baseline level of Ia aerent activity and negatively aects the dynamic response (Mileusnic et al., 2006). Therefore, when dynamic and static fusimo- tor drives are concurrently increased, Ia aerents increase their static bias while their dynamic sensitivity is maintained (Mileusnic et al., 2006). Primary Simulation Protocols We simulated a force tracking task using a closed-loop simulation of the aerented muscle model. Each trial consisted of 1-s zero input phase, 2-s ramp-up and 32-s hold at 20% MVC. The last 30 s of each trial were used for further analysis. We simulated 20 trials for each condition described below. In the rst set of simulations (Simulation 1), we examined the frequency spectrum of output force variability arising from interactions among musculotendon, broad-band neural noise, and error cor- rection mechanism. First, we simulated the force tracking task without the tracking controller and proprioceptive feedbacks (i.e., open-loop control) to characterize the interaction between mechan- ical properties of musculotendon and broad-band neural noise (Simulation 1.1). In this open-loop control condition, the neural drive consisted of open-loop input (i.e., 1-sec zero input , 2-sec ramp- up and 32-sec hold at 20% MVC) and signal dependent noise. The amplitudes of the open-loop input and noise were adjusted such that the mean force level was at 20% MVC and coecient of variation of force equaled to that from the closed-loop condition described below. We made these adjustments so that 1) the same force/input level is used in both conditions, thus facilitating comparisons of force variability and motor unit synchronization, and 2) because normalizing the total force variability across conditions makes comparison of their spectral characteristics more straightforward. Then, we replaced the open-loop input with the tracking controller (i.e., closed-loop control) to investigate how an error correction mechanism interacts with force variability arising from mechan- 20 ical properties of musculotendon (Simulation 1.2). In this closed-loop control condition, proprio- ceptive feedback was removed by setting presynaptic control inputs to the value of -0.5 for each pathway. In Simulation 2-4, we examined eects of proprioceptive feedback on force variability. To do so, we ran a set of simulations varying one of the proprioceptive pathway gains (i.e., presynaptic control level of Ia aerent feedback (Simulation 2), dynamic and static fusimotor drives (Simulation 3), and presynaptic control level of Ib aerent feedback (Simulation 4)) from its minimum to maximum values, while keeping the other gains constant. The minimum value corresponded to the value at which the contribution of that particular proprioceptive feedback is completely removed. The maximal value was determined empirically by the presence of non-physiological high-frequency force variability. The minimum and maximum values of fusimotor drive were set at 10 and 250 pulse per second (pps), respectively. At each parameter value, we run 20 trials. Following Simulation 2, we ran two additional sets of simulations to investigate the mechanism through which increases in Ia aerent feedback gain lead to reductions in low-frequency force variability. In the rst set of simulations (Simulation 2.1), we tested whether reductions in the relative strength of the tracking controller in response to increased excitatory input from Ia aerent feedback could explain that observation. To do so, we ran a set of simulations (20 trials) where the presynaptic control level of Ia aerent feedback was set at -0.5 and we added a constant excitatory input whose amplitude corresponded to the average input contribution from Ia aerent feedback at its presynaptic control level of -0.1. This presynaptic control value was chosen because we observed the smallest amplitude of low-frequency force variability. All the other gain parameters were held at 70 pps for dynamic and static fusimotor drives and -0.3 for presynaptic control level of Ib aerent feedback. In the second set of simulations (Simulation 2.2), we quantied the frequency response of the closed-loop aerented muscle to investigate how addition of Ia aerent feedback changes the dy- namics of this closed-loop system. To obtain the frequency response, we ran two sets of simulations using presynaptic control levels of Ia aerent of -0.5 and -0.1, while presynaptic control level of Ib aerent, dynamic and static fusimotor drives were kept constant at -0.3, 70, and 70, respectively. 21 In these simulations, we removed the signal dependent noise and injected a set of sinusoids (0.5 to 15 Hz in steps of 0.5 Hz) with amplitude of 1% of the maximum neural drive 5 sec after the initiation of simulations. We quantied gain and phase of output force in response to an input sinusoid. The gain was computed as the ratio of the amplitude of output force to the amplitude of the input sinusoid. Phase was calculated as a phase dierence between mean phase of the output force and input sinusoid during each trial. Secondary Motor Unit Simulations We performed secondary simulations for all conditions tested (Simulations 1-4) to investigate whether or not changes in the involuntary force variability produced in the closed-loop simula- tions can be detected as synchronization of motor unit activities within a pool. To do so, we used our simulated neural drive as a common input to a simulated motor unit pool obtained from (Fugl- evand et al., 1993). While this model is simplistic by comparison with more biophysically-nuanced compartment-based models (Dideriksen et al., 2015; Elias et al., 2014; Kim and Heckman, 2014), it is nonetheless entirely sucient to describe the basic phenomenon of motor unit entrainment by common input (Feeney et al., 2018; Yao et al., 2000) for the following reason. The `eective' (i.e., force-generating) neural drive is common to all motor units within a pool (Farina et al., 2016). Thus, the population activity is a linear transformation of the common input, even though indi- vidual motor unit responses to that input are nonlinear (Negro et al., 2009). As a result, specic non-linearities present in each motor neuron response such as plateau potentials, adaptation, res- onance, accommodation, etc., which can be modeled by those more complex models, are not an important consideration for our present application. This motor unit model describes the orderly recruitment of motor units and rate coding in response to a common excitatory input. The motor unit pool consisted of 120 motoneurons, whose recruitment threshold to excitatory input had exponential distribution with a greater number of motoneurons with low thresholds and a small fraction of motoneurons with high thresholds as described previously (Fuglevand et al., 1993). The range of recruitment thresholds was set such that the highest value was 30-fold that of the lowest. Firing rate of a motoneurons was linearly 22 scaled to excitatory input with a constant minimum ring rate of 8 imp/s. All of these properties were same as those in the original study (Fuglevand et al., 1993). We added discharge rate variability (5% CoV) to spike trains of individual motor units, indicated as IN in Fig. 3, to simulate the eects of independent noise. In this simulation, we also included a previously modeled intrinsic property of motor unit known as persistent inward current (Revill and Fuglevand, 2011). Data Analysis I. Force The amplitude of overall force variability obtained from close-loop simulations of aerented muscle was quantied by coecient of variation (CoV) of force. The force variability was further analyzed in the frequency domain. Power spectrum of the force was obtained using pwelch function in MATLAB with 5-s Gaussian window, 50% overlap, and the frequency resolution of 0.1Hz. Statistical analysis was performed to identify dependence of force power at each frequency (dependent variable) on the level of proprioceptive feedback gains (independent variable). We used the heteroscedastic one-way ANOVA for means (Wilcox, 2011) from "WRS2" package in the R environment for statistical computing (The R Foundation for Statistical Computing, Vienna, Austria). Signicance level was set to p< 0.05. Furthermore, force variability was analyzed in two frequency ranges (i.e., common drive (1-5 Hz) and physiological tremor (5-12 Hz)). Mean force power within each frequency range was calculated. II. Motor Unit Synchronization Common Drive Index: Common drive index (CDI) is a time-domain measure of synchronization between motor units in the 1-5 Hz range (de Luca et al., 1982). This value ranges from -1 to 1, 1 being perfect positive linear correlation and 0 being no correlation between motor units. From the simulated motor unit pool, we randomly sampled 50 pairs of concurrently active motor units. Spike trains of each motor unit were smoothed with a 400-ms Hann window and high-pass ltered at 0.75Hz (Contessa et al., 2009; de Luca et al., 1982; Semmler and Nordstrom, 1998). We calculated cross-correlations of pairs of the processed spike trains. CDI values were obtained by nding the peak correlation coecient value within 100-ms time lag from the cross-correlogram 23 (Contessa et al., 2009; de Luca et al., 1982). The correlation coecient was transformed using Fisher's z-transform (hyperbolic arctangent) before averaging (Silver and Dunlap, 1987). For each iteration, the mean Fisher z-transformed CDI value across 50 pairs of motor units was calculated and transformed back to correlation coecients using the inverse of the Fisher z-transformation (Silver and Dunlap, 1987). This provided 20 CDI values for each condition. Coherence: Similarly, synchronization between pairs of motor units was quantied in the frequency domain using coherence analysis (Farina et al., 2016; Laine et al., 2013, 2014; Myers et al., 2004). Coherence at each frequency ranges from 0 to 1, 1 being high synchronization at that frequency. Magnitude squared coherence between each randomly selected, unique pair of motor units (same as used in the common drive analysis) was calculated using 5-s Gaussian windows with 50% overlap and 0.1 Hz frequency resolution. Then, Fisher z-transformed coherence spectra were averaged across 50 pairs of motor units to obtain a single coherence spectrum for each condition. Mean Fisher z-transformed coherence values within the common drive (1-5 Hz) and physiological tremor (5-12) ranges were extracted from the averaged coherence spectrum for each trial. Results We used a closed-loop simulation of an aerented muscle model, which is an extension of a previously published model (Laine et al., 2016) (Details are given in Materials and Methods). With this model, we identied the sources of frequency-specic force variability, and characterized interactions among them. We simulated an isometric force tracking task at 20% of maximal voluntary contraction (MVC). The schematic diagram of the model is provided in Fig 1. The aerented muscle model is comprised of a musculotendon unit (Brown et al., 1996; Cheng et al., 2000; Song et al., 2008a,b), muscle spindle (Mileusnic and Loeb, 2006), and Golgi tendon organ (GTO) (Elias et al., 2014), which is driven by a tracking controller to ensure successful force tracking tasks Laine et al. (2016). Fig 2 shows sample time-series signals of each element from the closed-loop simulation of the aerented muscle model, along with their respective power spectra. We also performed sets of secondary simulations to investigate whether or not changes in involun- tary force variability characterized in the closed-loop simulations can be detected as synchronization 24 Figure 1: Primary closed-loop simulations of arented muscle model. A musculotendon unit receives a neural drive (ND) derived from three input sources and signal dependent noise (SDN). The muscle spindle generates excitatory inputs through group Ia and II aerents at a given fusimotor drive level. The Golgi tendon organ (GTO) sends an inhibitory input through the group Ib aerent pathway. The input contributions of Ia and Ib aerents are controlled by presynaptic control (PC). The tracking controller provides a control signal (C) based on a fraction of the error between reference force (F Reference ) and actual force output (F Output ). These inputs are integrated at the spinal level and signal dependent noise (SDN) is added. The resulting neural drive (ND), ltered to account for Ca + dynamics, induces contraction of the muscle (length (L), velocity (V), acceleration (A)), taking into account mechanical factors such as the pennation angle (), mass (M), viscosity (B), parallel (PE1 and PE2) and series elastic elements (SE). Each aerent or eerent pathway has associated delays which account for the conduction velocities of each ber, the distance between the spinal cord and the muscle of 0.8 m (Elias et al., 2014) and synaptic delays of 2 ms Kandel et al. (2000) 25 Figure 2: Example time-series signals and their power spectra obtained from the pri- mary closed-loop simulation of aerented muscle. 26 of motor unit activities within a pool. We considered this to be an important link to the literature because experimental studies consider synchronization across motor units as a measure of `eective' (i.e., force-generating) neural drive to muscle (Farina et al., 2016). Accordingly, synchronization across motor units has been used extensively to characterize the neural drive that generates involuntary force variability (De Luca et al., 2009; de Luca et al., 1982; Farmer, 2002, 1998; Laine et al., 2013, 2014). In this set of secondary simulations, we took the neural drive generated by the primary closed-loop simulations (Fig 3) and fed it into a simulated pool of 120 motor units (Fuglevand et al., 1993) as a common input. The resulting total discharge rate variability arising from independent as well as common synaptic input ranged from 17 { 33% of coecient of variation, which is compatible with that observed experimentally (Moritz et al., 2005). The synchronization between randomly chosen pairs of motor units was quantied in both time and frequency domain using (i) common drive index (de Luca et al., 1982) and (ii) coherence (Farmer et al., 1993; Laine et al., 2013, 2014). Simulation 1: Closed-Loop Control of Musculotendon First, we investigated the interactions between mechanical properties of the musculotendon and broad-band neural noise using an open-loop input without any feedback (Simulation 1.1). For this simulation, our control input was simply the target trajectory (i.e., 1-sec zero input, 2-sec ramp-up and 32-sec hold at 20% MVC), with added signal-dependent noise. The coecient of variation of force was 8.73%. This open-loop control resulted in force variability which fell almost entirely below 5 Hz, within the `common drive' range (red line in Fig 4 A). It is worth noting that there was no distinct peak within this frequency range (i.e., 1-5 Hz). Accordingly, the neural drive produced in this simulation also caused a small degree of common drive, as measured by the `common drive index' (red boxplot in Fig 4 B). A similar result was observed using motor unit coherence analysis. It is also important to note that high-frequency force variability (5-12 Hz) did not arise from the interaction between mechanical properties of musculotendon and broad-band noise. 27 Figure 3: Secondary Motor Unit Simulations. We used the neural drive (ND) obtained from the closed-loop simulations of aerented muscle to investigate changes in synchronization patterns of motor units in dierent simulated conditions. The simulated neural drive is fed to a simulated motor unit pool as a common input across the pool. The motor unit pool consists of 120 motor units. Individual motor units also receive independent noise (IN). 28 We then ran the simulation in closed-loop condition using only the error correction mechanism (i.e., tracking controller) (Simulation 1.2). The amplitude of overall force variability was 8.39%, which was not signicantly dierent from the open-loop condition (independent sample t-test using Yuen's method, p = 0.19). This addition of an operational tracking controller resulted in the generation of a peak at 1.8 Hz in the power spectrum of muscle force (green line in Fig 4 A). Also, the degree of motor unit synchronization in this range increased accordingly (green boxplot in Fig 4 B). These results altogether suggest that low-frequency force variability and common drive are primarily an emergent property of a close-loop control of muscle force. Also, these results show that high-frequency force variability does not emerge in the absence of proprioceptive feedback. Simulation 2: Ia Aerent Feedback Gain control of Ia aerent feedback at the spinal cord, often experimentally quantied by H-re ex amplitude, plays an important role in human motor control and learning to achieve a variety of movements (Ludvig et al., 2007; Wolpaw, 2010). Here, we examined how changes in the gain of Ia aerent feedback, modeled as presynaptic control input, in uence force variability. We systemat- ically altered the level of this presynaptic control input from the value of -0.5 to 0 while keeping the other gain parameters constant (70 pps for dynamic and static fusimotor drives and -0.3 for presynaptic control level of Ib aerent feedback). This range was set such that the mean input contribution of Ia aerent feedback to the neural drive spanned a range from 0 (i.e., no contribution from Ia aerent feedback) to 30% of the maximum neural drive. The amplitude of force variability decreases as the presynaptic input level is increased and becomes minimal at the value of -0.15 (Fig 5A). Further increases negatively aect the amplitude of force variability (the presynaptic control level of -0.05 and 0 in Fig 5A). Analyses of force variability in the frequency domain show the change in force variability amplitude occurred across the frequency range (p< 0.01 at all the frequencies between 1 and 12 Hz), but prominent peaks exist in the two distinct frequency ranges, namely the common drive range (1-5 Hz) and physiological tremor range (5-12 Hz) (shown as blue and red bands for common drive range and physiological tremor range, respectively, in Fig 5B). These observations demonstrate that modulation of the 29 Figure 4: Low-frequency `common drive' originates in musculotendon mechanics but is shaped and amplied by closed-loop control of muscle force. (A) Power spectra of output force from the aerented muscle model using two types of control strategies (open-loop control in red and closed-loop (error correction mechanism only) control in green). Noise level was adjusted to match the amplitudes of overall force variability between the two control strategies. (B) Motor unit synchronization in the common drive range (1-5 Hz) as per the common drive index for the two control strategies. Note that open-loop control in the presence of broad-band noise produces only low-frequency force uctuations (below 5 Hz), which can be attributed to the low-pass ltering nature of musculotendon. Also, note that the closed-loop control only with a tracking controller results in generation of a distinct peak within the common drive range and increased common drive. 30 strength of Ia aerent feedback is an important factor that in uences overall force variability during `isometric' force production. Further analyses on frequency-specic eects of Ia aerent feedback show increasing the gain of Ia aerent feedback reduces force variability within the common drive range (Fig 6A) while it increases the amplitude of physiological tremor (Figure 6B). Excessive Ia gain led to excessive phys- iological tremor as suggested in previous studies (Fink et al., 2014; Stein and O^ guzt oreli, 1976). As Ia aerent feedback increases, common drive decreases more than physiological tremor increases, after which physiological tremor dominates the spectrum and a monotonic increase in total force variability is observed. These observations suggest that the U shaped response comes from the rela- tive contribution of common drive and physiological tremor to total force variability. As Ia aerent feedback increases, common drive decreases more than physiological tremor increases, after which physiological tremor dominates the spectrum and a monotonic increase in total force variability is observed. Importantly, these concurrent changes in the common drive and physiological tremor are consistent with previous speculations (De Luca et al., 2009; Laine et al., 2013, 2014). These observations suggest that relatively faster excitation cycles of Ia aerent feedback can function as a negative feedback (i.e., withdrawal of Ia aerent input during muscle shortening and its excitation during muscle stretch), thereby interrupting the development of low-frequency force uctuations, characteristic of a close-loop control of muscle force. Changes in motor unit synchronization in the common drive and physiological tremor ranges are shown in (Fig 6C-E). Stronger Ia aerent feedback reduces the degree of common drive (Fig 6 C,D). In contrast, it induces a higher degree of synchronization in the physiological tremor range (Figure 6 E). These results further conrms a previously suggested relationship between the strength of Ia aerent feedback and motor unit synchronization in the common drive and physiological tremor ranges (De Luca et al., 2009; Laine et al., 2013, 2014). Additional Simulation 2.1: Contribution of Tracking Controller to Reduction in Low-Frequency Force Variability: To determine the specic mechanism through which increases in the strength of Ia aerent feedback reduce low-frequency force variability, we ran two additional sets of simu- lations. The rst (Simulation 2.1) aimed to address the question of whether changing the relative 31 Figure 5: Modulation of Ia aerent feedback gain (i.e., presynaptic control input) in- uences the amplitude of overall force variability through its eects across the force- relevant frequencies. (A) Coecient of variation (mean SE across 20 trials) at each presynaptic control level of Ia aerent feedback. The smaller the value of presynaptic control level, the lesser the contribution from this pathway. (B) Power spectra of force (mean across 20 trials at each level of presynaptic control). Power spectra were divided into two frequency ranges: 1-5 Hz common drive (CD) and 5-12 Hz physiological tremor (PT). Note that overall force variability shows U-shaped response to changes in the gain of Ia aerent feedback. Also, note that those changes in overall force variability occur across the range of frequencies below 12 Hz with distinctive peaks in the common drive and physiological tremor ranges. 32 Figure 6: Increases in the gain of Ia aerent feedback (i.e., larger presynaptic control level) decrease low-frequency force variability (1-5 Hz) while they increase physiolog- ical tremor (5-12 Hz). All of the dependent variables are normalized to their respective mean values at the presynaptic control level of -0.5. (A) Mean force power within the common drive range (mean SE across 20 trials). (B) Mean force power within the physiological tremor range (mean SE across 20 trials). (C) Common drive index (CDI, mean SE of mean CDIs across 20 trials) (D) Coherence (mean SE of mean Fisher z-transformed coherence across 20 trials) in the common drive range. (E) Coherence in the physiological tremor range (mean SE of mean Fisher z-transformed coherence across 20 trials). Note that increases in the gain of in Ia aerent feedback (i.e., decreases in presynaptic inhibition of Ia aerent feedback) result in decreases in low-frequency force variability and increases in physiological tremor. Also, note that changes in motor unit synchronization generally re ect changes in force variability. 33 contribution of the tracking controller with respect to the total input could have resulted in the reduction in low-frequency force variability. We injected an additional constant excitatory input (amplitude of 1.7 % of the maximum neural drive, equivalent to the Ia contribution when the presynaptic control value is -0.1), while the presynaptic control level of Ia aerent feedback was held at -0.5 to remove its contribution. All the other gain parameters were same as in Simulation 2. This additional constant excitatory input, however, did not change force variability in the common drive range (independent sample t-test using Yuen's method, p = 0.27), nor the degree of motor unit synchronization (p = 0.87), compared to the original condition where there was no additional excitatory input (presynaptic control level of -0.5 in Fig 3). This result suggests that the reductions in low-frequency force variability cannot be explained by reduction in the relative strength of the tracking controller alone. Additional Simulation 2.2: Frequency Response of Closed-Loop System In the second set of simulations (Simulation 2.2), we quantied the frequency response of the closed-loop aerented muscle by removing the signal dependent noise and adding sinusoidal inputs of dierent frequencies with equal amplitudes. For each input, we quantied the gain (i.e., ratio of output amplitude to input amplitude) and phase delay (dierence in phase between input and output). The Bode plots in Fig 7 characterize the consequences of altering presynaptic control on the frequency response of the system. Fig 7 shows that increasing Ia aerent feedback attenuates low-frequency inputs and removes associated delays seen in the absence of Ia aerent feedback. These results show that Ia aerent feedback induces a high-pass ltering behavior on the system, which prevents the development of slow force uctuations and enables more rapid corrections. This is a previously unrecognized functional contribution of muscle aerentation. Simulation 3: Fusimotor Drive Understanding the operation of the fusimotor system is hindered by the lack of techniques which can directly measure -motoneuron activities (Loeb and Tsianos, 2015). However, experimental evidence based on human group Ia and II aerent activities has suggested that humans have control over the fusimotor system which is independent of-motoneuron drive, and which can be modulated 34 Figure 7: Frequency response of the closed-loop aerented muscle system with or with- out Ia aerent feedback. (A) gain of the system quantied as the ratio of the amplitude of output force, normalized to MVC, to the amplitude of input sinusoid. (B) phase delay of output force with respect to input sinusoids. Note that addition of Ia aerent feedback attenuates the am- plication of low-frequency inputs and removes phase delays, present in the absence of Ia aerent feedback. 35 by attention and task requirements (Hospod et al., 2007; Ribot-Ciscar et al., 2003, 2009, 2000). Here, we postulate that fusimotor-induced changes in the dynamic sensitivity and static bias of Ia aerent activity will have profound eects on force variability as well. Therefore, we tested three scenarios; 1) co-modulation of dynamic and static fusimotor drives, 2) modulation of dynamic or 3) static fusimotor drive independently while the other is held constant, as done previously (Mileusnic and Loeb, 2006). In this study, we varied them from 10 to 250 pps by increment of 20 pps. When dynamic or static fusimotor drive was varied independently, the other was kept at 70 pps. The presynaptic control levels of Ia and Ib aerent feedback were set at -0.15 and -0.3. Results show that the amplitude of overall force variability depends on the levels of fusimotor drives (Fig 8 A-C top gures). When both dynamic and static fusimotor drives are varied, the amplitude of overall force variability shows a similar response to the presynaptic manipulation of Ia aerent feedback (Fig 8 A top gure). Also, the changes again occur predominantly in the common drive and physiological tremor ranges (p < 0.01 at all the frequencies between 1 and 12 Hz) as indicated by prominent peaks in those ranges (Fig 8 A bottom gure). Independent modulation of the only dynamic fusimotor drive has comparably smaller eects on the amplitude of overall force variability (Fig 8 B top gure). On the contrary, modulation of static fusimotor drive produces eects similar to co-modulation of both fusimotor drives (Fig 8 C top gure). Again, their eects occur in the common drive and physiological tremor ranges (Figure 8 B-C bottom gures). These results show that the fusimotor system, especially static fusimotor drive, has profound eects on force variability in a frequency specic manner similar to presynaptic modulation of Ia aerent gain. This dierential sensitivity to dynamic and static fusimotor drives might speak to dierences in their functional signicance during isometric force production. Also, it is important to note that too high levels of static fusimotor drives can lead to greater overall force variability accompanied by excessive physiological tremor, which might be similar to eects of fatigue (Biro et al., 2007; Cresswell and Loscher, 2000) (see Discussion). Further analyses in the two frequency ranges show greater fusimotor drives are associated with smaller force variability in the common drive range and larger physiological tremor (Fig 9A,B). The eects of static fusimotor drive are substantially larger than those of dynamic fusimotor drive in both frequency ranges and the combination of those eects is illustrated in the case of 36 Figure 8: Modulation of Ia aerent activity through the fusimotor system aects the amplitude of force variability through its eects across the force-relevant frequencies. The top gure panel depicts the coecient of variation of force and the bottom gure panel shows power spectra of force. (A) Dynamic and static fusimotor drives were co-modulated. (B) dynamic fusimotor drive was varied while static fusimotor drive was kept constant at 70 pps. (C) static fusimotor drive was varied while dynamic fusimotor drive was kept constant at 70 pps. Note that overall amplitude of force variability is less sensitive to modulation of dynamic fusimotor drive, while that of static fusimotor drive has signicant eects on the amplitude of overall force variability. Also, eects of concurrent increases in dynamic and static fusimotor drives are a combination of their respective contributions. Importantly, those changes in overall force variability predominantly occur in the common drive range and physiological tremor ranges. 37 Figure 9: Modulation of Ia aerent activity through the fusimotor system decreases low-frequency force variability (1-5 Hz) while it increases physiological tremor (5-12 Hz). Three combinations of fusimotor drive modulation are indicated as follows; co-modulation of dynamic and static fusimotor drives (circles with solid line) and modulation of dynamic (triangles with dashed line) or static (squares with long-dashed line) fusimotor drives alone. All of the dependent variables presented here are normalized to their respective mean values at the lowest fusimotor drive for each condition. (A) Mean force power within the common drive range (meanSE across trials). (B) Mean force power within the physiological tremor range (meanSE across 20 trials). (C) Common drive index (CDI, meanSE across 20 trials). (D) Coherence (meanSE across 20 trials) in the common drive range. (E) Coherence (meanSE across 20 trials) in the physiological tremor range. Note that changes in force variability and motor unit synchronization in both frequency ranges can be attributed primarily to eects of static fusimotor drive. co-modulation of dynamic and static fusimotor drives. These results are consistent with those from presynaptic Ia aerent feedback gain such that increased bias level (mean input contribution) of Ia aerent feedback, rather than the dynamic sensitivity of Ia aerent feedback, plays a more important role in shaping the power spectrum of force variability and generating physiological tremor Hagbarth et al. (1975). Changes in motor unit synchronization correspond well to changes in force variability, as shown in Fig 9C-E. Greater fusimotor drives result in lower CDI values and low-frequency coherence (Fig 9C,D), as well as higher coherence in the physiological tremor range. These results suggest that modulation of dynamic and static fusimotor drives can also alter the degree of motor unit 38 synchronization across the force-relevant frequencies. Simulation 4: Ib Aerent Feedback Given that Ib aerent feedback in general provides inhibition of-motoneurons as a function of force level, one can easily expect that it helps stabilize force uctuations (Goodwin et al., 2000; Loeb and Tsianos, 2015). However, exactly how such a feedback system in uences either overall amplitude or frequency-specic components of involuntary force variability is unknown. Here, the presynaptic control value of Ib aerent feedback was varied from -0.5 to 0, while the presynaptic control value of Ia aerent feedback was kept at -0.3 and gamma and static fusimotor drives at 70 pps. This range corresponds to a Ib contribution of 0 to 45% of the maximum neural drive, respectively. The upper range of these values would be non-physiological as the Ib input contribution of 45% of the maximal neural drive, for example, means 45% total input is continuously inhibited and it requires other compensatory mechanisms through Ia aerent feedback and a tracking controller to maintain the target force level. Here, we merely try to fully characterize eects of Ib aerent feedback on force variability and thereby highlight dierences between Ia and Ib aerent feedback. As expected, greater inhibition of -motoneurons through Ib aerent feedback reduces the am- plitude of overall force variability (Fig 10 A). However, excessive Ib gain can also lead to increased force variability at4 Hz (Fig 10 A) although it requires non-physiologically large Ib input con- tributions. In the frequency domain, changes in force variability occur across the frequencies (p < 0.01 at all the frequencies between 1 and 12 Hz), but mainly in the common drive as indicated by peaks appearing only in that range (Fig 10 B). The slightly lower frequencies at which the second peak occurs compared to those of Ia aerent feedback might result from the longer loop delay of Ib aerent feedback (Fig 10 B). These results highlight that Ib aerent feedback can regulate force variability much like presynaptic/fusimotor modulation of Ia aerent feedback, but its eects are mostly conned in the common drive range. Increasing the strength of Ib inhibition results in smaller force variability in the common drive range (Fig 11A), but excessive Ib inhibition can lead to excessive force uctuations in this range as shown in (Fig 10A). Its eects on physiological tremor are considerably smaller than presynap- 39 Figure 10: Modulation of presynaptic control of Ib feedback alters the amplitude of overall force variability mainly through its eects on the common drive range. (A) Ib-related changes in coecient of variation (meanSE across 20 trials). The smaller the value of presynaptic control, the smaller the contribution from the pathway. (B) Changes in power spectra of (mean across 20 trials) as a function of presynaptic control levels of Ib aerent feedback. Note the U-shaped response of the amplitude of overall force variability with increases in the strength of Ib aerent feedback. Also note that eects of Ib aerent feedback occurs mainly in the common drive range. 40 Figure 11: Increasing the strength of Ib aerent feedback decreases low-frequency force variability, but can lead to excessive force uctuations in that range. All of the dependent variables are normalized to their respective mean values at the lowest presynaptic control level. (A) Mean force power within the common drive range (meanSE across 20 trials). (B) Mean force power within the physiological tremor range (meanSE across 20 trials). (C) Common drive index (CDI, meanSE across 20 trials). (D) Coherence (meanSE across across 20 trials) in the common drive range. (E) Coherence (meanSE across 20 trials) in the physiological tremor range. Note that increased strength Ib aerent feedback preferentially aects force variability in the common drive range. Also note that CDI and coherence show dierent responses to excessive force uctuations in the common drive range. tic/fusimotor modulation of Ia aerent feedback (Fig 11B). These results highlight the dierences in cross-frequency interactions between Ia and Ib aerent feedback pathways, which has not been reported previously. As before, the frequency-specic eects of presynaptic Ib modulation on force variability are also re ected in motor unit synchronization (Fig 11C-E). Higher Ib feedback gain is associated with lower synchronization in the common drive range and higher synchronization in the physiological tremor range. Interestingly, CDI and coherence in the common drive range respond dierently to excessive force uctuations at4 Hz seen with excessive Ib inhibition (Fig 11C-D), suggesting that these two measures have diering sensitivity to synchronization at dierent frequencies within 1-5 Hz. 41 Discussion A series of closed-loop simulations of aerented muscle show that many cardinal features of invol- untary force variability emerge from closed-loop neuromechanical interactions. Our results reveal that closed-loop control of a viscoelastic musculotendon unit, combined with the tuning of pro- prioceptive feedback gains, naturally generate both low-frequency (1-5 Hz) force variability and high-frequency oscillations analogous to physiological tremor (5-12 Hz). Moreover, we show that these low- and high-frequency phenomena are in fact mechanistically related to each other|which suggests novel and fruitful directions for future research. This study is, to our knowledge, the rst to directly conrm mechanistic links between low- and high-frequency force variability, as was proposed earlier (De Luca et al., 2009; Laine et al., 2013, 2014). Finally, we also used the emergent time histories of closed-loop net neural drive (`ND' in Fig. 1) to drive the model of a motor unit pool. We nd that these inputs suce to produce motor-unit synchronization compatible with experimental ndings (De Luca et al., 2009; Laine et al., 2013, 2014). Involuntary force variability at low frequencies (1-5 Hz) can arise from various sources, including low-frequency variability in the neural drive to muscle (the so-called `common drive') de Luca et al. (1982). As such, the amplitude of this common drive is a contributor to error during voluntary control of precision forces (de Luca et al., 1982; De Luca and Mambrito, 1987; Farina et al., 2016; Lodha and Christou, 2017). Although common drive has been studied for over 30 years, its origins remain debatable (Lodha and Christou, 2017). Our results are signicant because they suggest that common drive can emerge due to a com- bination of factors inherent to any neuromuscular control loop. Foremost among them is the vis- coelasticity of the musculotendon, which acts as a mechanical low-pass lter that naturally allows the preferential conversion of low frequencies in the neural drive into muscle force as previously shown in (Baldissera et al., 1998; Bawa et al., 1976; Bobet and Stein, 1998). It is this low fre- quency component (1-5 Hz) of muscle force that would be selectively reinforced by any imperfect physiological error correction mechanism. Thus, our results demonstrate that low-frequency force variability emerges naturally when controlling viscoelastic muscles|and do not require the pres- ence of proprioceptive feedback. This is a novel alternative to other peripheral explanations. For 42 example, Watanabe and Kohn suggested that high-frequency neural drive can be demodulated into lower frequencies (Watanabe and Kohn, 2015), which still remains to be tested. In fact, our results are congruent with previous evidence for peripheral mechanisms, such as the fact that common drive persists even after disruption of the cortico-spinal tract, as in capsular stroke (Farmer et al., 1993). Another component of force variability is oscillations in the 5-12 Hz range, often called `phys- iological tremor.' Physiological tremor may arise from multiple factors (McAuley, 2000). One of the earliest and most well-supported mechanisms is cycles of excitation around the stretch re ex loop (Christakos et al., 2006; Lippold, 1970; Stein and O^ guzt oreli, 1976). The rst important im- plication of our results is that, in contrast to common drive, physiological tremor does require the proprioceptive feedback in order to arise as shown experimentally (Christakos et al., 2006; Lippold, 1970) and in computational simulations Stein and O^ guzt oreli (1976). Thus mechanical resonance of musculotendons as proposed by (Lakie et al., 2012; Vernooij et al., 2013), did not suce. In fact, we could not elicit physiological tremor via interactions between broad-band noise and the mechanical properties of musculotendon using an open-loop input which consisted of the target tra- jectory and signal-dependent noise. This result is consistent with previous experimental evidence and simulation (Christakos et al., 2006; Lippold, 1970; Stein and O^ guzt oreli, 1976). Moreover, our simulations allowed us to characterize how physiological tremor amplitude is modulated by proprioceptive pathway gains. Those include both presynaptic control levels of inhibition/disinhibition (`PC Ia ' & `PC Ib ' in Fig. 1) and `descending' fusimotor drive to muscle spindles (`fusimotor drive' in Fig. 1). This detailed characterization was not possible in the previous simulation study by Stein and Oguztoreli (Stein and O^ guzt oreli, 1976) and added a new insight that physiological tremor amplitude is mostly determined by the bias level (i.e., mean input contribution) of Ia aerent feedback, not dynamic sensitivity of muscle spindle. Importantly, excessive Ia aerent gains could produce excessive oscillations primarily in the physiological tremor range in Figs. 5 and 8, similar to what has been shown previously in animal models (Fink et al., 2014). Interestingly, excessive Ib aerent gains could lead to excessive oscillations in the lower frequency range (3-5 Hz) possibly due to the longer delay along this pathway (Fig. 10. These ndings are particularly important to design hypotheses about how peripheral mechanisms interact with descending neural 43 drive to produce physiological and other kinds of tremor in healthy and pathological conditions (Hallett, 2008; Llin as, 2009; Williams and Baker, 2009b; Williams et al., 2010). Although we nd that proprioceptive feedback is not strictly necessary to generate common drive, we do nd that it can in uence its strength. This is compatible with experimental ndings (De Luca et al., 2009; Laine et al., 2013, 2014). Specically, De Luca and colleagues report a negative correlation between the degree of common drive and muscle spindle density (De Luca et al., 2009). Further, Laine and colleagues showed that heightening the perception of task-related errors during a force tracking task led to increases in physiological tremor and H-re ex|while common drive decreased (Laine et al., 2013, 2014). Their interpretation was that the changes in common drive and physiological tremor both stemmed from the tuning of proprioceptive gains due to alterations in psycho-sensory state (Hospod et al., 2007; Pinar et al., 2010; Ribot-Ciscar et al., 2009, 2000; Roche et al., 2011; Taube et al., 2008). These lines of experimental evidence, however, could not test a mechanistic link between common drive and physiological tremor. Here, we show that increasing the strength of proprioceptive feedback (via `PC Ia ' and ` static fusimotor drive') increases physiological tremor but concurrently decreases common drive (Fig 6). Thus, our results demonstrate that peripheral mechanisms suce to reproduce those experimental ndings. This close link between the amplitude of involuntary force variability and proprioceptive pathway gains (in Figs. 5, 8, and 10) may explain many experimental ndings. For example, removing proprioceptive feedback leads to greater overall involuntary force variability (i.e., smaller values of `PC Ia ', `PC Ib ' and ` static fusimotor drive' in Figs. 5, 8, and 10). This is similar to what has been seen in patients with deaerentationLafargue et al. (2003). Moreover, we show that excessive proprioceptive pathway gains result in greater overall force variability and excessive physiological tremor (i.e., larger values of `PC Ia ', `PC Ib ' and ` static fusimotor drive' in Figs. 5, 8, and 10). Interestingly, fatigue can produce similar eects on force variability and physiological tremor (Cresswell and Loscher, 2000; Hunter et al., 2002); however, a precise mechanism for this phenomenon has not been established. The enhancement of physiological tremor in fatigue can be attenuated by blocking Ia aerent feedback (Cresswell and Loscher, 2000). Further, the sensitivity of stretch/tonic vibration re ex responses is enhanced during fatigue (Biro 44 et al., 2007). An emerging picture is that Ia aerent feedback gains are increased during fatigue, but it is not clear how this occurs (i.e., via presynaptic inhibition or fusimotor modulation), and it is not clear why fatigue in uences overall force variability rather than just physiological tremor. Biro and colleagues suggested that augmented Ia aerent feedback during fatigue re ects a fusimotor-dependent compensation for reduced descending drive (Biro et al., 2007). This sugges- tion was based on previous ndings in cat where 1) the activity of fusimotor system is enhance by activation of group III and IV aerents (Jovanovi et al., 1990; Ljubisavljevi c et al., 1992), which respond to an accumulation of metabolites during fatigue (Kniki et al., 1978; Kumazawa and Mizumura, 1977), 2) Ia aerent ring rates increase accordingly during fatigue contractions (Ljubisavljevi c and Anastasijevi c, 1994; Ljubisavljevi c et al., 1992), and 3) group III and IV aer- ents, on the contrary, enhance presynaptic inhibition of Ia aerent feedback (Kalezic et al., 2004). Since presynaptic inhibition would reduce Ia aerent feedback gain, only the increased fusimotor activation seems a plausible compensatory mechanism. Thus it is important to mention that, when we tested the eects of increased fusimotor drive in our simulation, the results of static fusimotor drive (` static fusimotor drive' in Fig. 8) accurately predicted changes in force variability, as might occur during fatigue. Our ndings therefore may provide a mechanistic link between several complementary lines of investigation related to fatigue. As demonstrated in the cases of deaerentation and fatigue, the close link between our results and experimental ndings may represent an important step in developing a unifying theory of human sensorimotor control that further relates spinal circuitry to manifestations of altered involuntary force variability under various neuromuscular conditions such as aging (Baudry et al., 2010; Enoka et al., 2003; Galganski et al., 1993), stroke (Lodha et al., 2010), cerebral palsy (Chu and Sanger, 2009), Parkinson's disease (Vaillancourt et al., 2001a), and essential tremor (Neely et al., 2015). For example, we show that increased Ia aerent feedback gains result in increased force variability below 0.5 Hz (i.e., larger values of `PC Ia ' and co-modulation and ` static fusimotor drive' in Figs. 5 and 8). This might provide a link between increased force variability below 0.5 Hz seen in patients post stroke (Lodha and Christou, 2017; Lodha et al., 2010) and their heightened Ia aerent feedback 45 gains (Thilmann et al., 1991) or lower re ex threshold (Katz and Rymer, 1989). Thus, a unifying principle emerges. Namely, that the task-specic tuning of proprioceptive pathway gains in spinal circuitry|or its disruption|produces characteristic changes in the spectra of neural drive. Importantly, these can be quantied by measuring force variability. Our results highlight the signicance of considering closed-loop control of aerented muscle in the generation and modulation of involuntary force variability in motor control research. Historically, the force uctuations have been considered as manifestation of `neural noise' that is intrinsic to neural drive (Jones et al., 2002). Despite the fact that such noise (e.g., signal dependent noise) is usually not frequency-specic, involuntary force uctuations tend to be highly structured (Farina et al., 2016). Our results now show that neuromechanical interactions impose structure onto noisy neural drive, and thus involuntary force variability and `noise' are not independent, as is often assumed (Lodha and Christou, 2017). This idea may be signicant in formulation of theoretical frameworks in motor control. For example, the ability of the proprioceptive feedback system to regulate the amplitude of overall involuntary force variability provides a neural mechanism to minimize it, as suggested by some (Faisal et al., 2008; Todorov and Jordan, 2002). It is important to discuss how the limitations of our model do not aect our conclusions. Our af- ferented muscle model was not intended to represent the full complexity of the spinal cord circuitry. We used a simplied version of a previously described model of a spinal-like regulator (Raphael et al., 2010; Tsianos et al., 2014) that can replicate experimental behavior. Specically, we did not include Renshaw inhibitory interneurons, which are known to provide recurrent inhibition of -motoneurons and inhibition of Ia inhibitory internuerons (Windhorst, 2007). However, in our simulation of a single muscle, the role of Renshaw inhibitory interneurons would be restricted to recurrent inhibition and therefore have eects similar to that of Ib inhibitory feedback, which we did include. Secondly, our model did not attempt to replicate the exact biophysical structure of-motoneurons and sensory aerents. Rather, we used a single-input/single-output structure to describe the pop- 46 ulation behavior of each system. We believe this simplication is reasonable because 1) the pop- ulation response of an -motoneuron pool is linear with respect to its common/shared synaptic input, since noise and non-linear properties of i0ndividual neurons get canceled out in the overall population behavior (Farina et al., 2016; Scott and Loeb, 1994), and 2) the common input to an -motoneuron pool is the `eective' neural drive, that is, the input that is actually translated into muscle force (Farina et al., 2016). Therefore, it was appropriate for the contractile element in the aerented muscle to be modeled as a single input-output element. Another outcome of using a lumped parameter model of muscle is that force is not generated by the summation of twitches from progressively recruited motor units. However, neither physiological tremor nor `common drive' is thought to relate directly to this aspect of physiological force generation (Taylor et al., 2003). It is also worth noting that since we simulated constant-force contractions, the number of units recruited/derecruited during each trial would have been very small and therefore would have only minor in uence on the overall amplitude of force variability. Similarly, the population behavior of muscle spindles can be appropriately modeled as a single element, as muscle spindles are in general believed to distribute their synaptic inputs widely across a motor unit pool (Mendell and Henneman, 1971). While potential non-uniformity of Ia projections has been suggested (Scutter and T urker, 1999), this remains to be validated, and conrmed across dierent muscles. Thirdly, we did not include modulation of-motoneuron excitability through various neuromod- ulatory inputs arising from the brainstem, which can in uence re ex sensitivity (Heckman et al., 2003). Such neuromodulatory eects would be widespread and more dicult to interpret, while also greatly increasing the complexity of our analyses. Finally, our simulation was limited to that of a single muscle during isometric contraction, which is a valuable and informative experimental paradigm (Chu and Sanger, 2009; Enoka et al., 2003; Laine et al., 2016, 2013, 2014). As in those experimental studies, it is dicult to extrapolate our ndings to complex actions involving movement and coordination among multiple muscles. Still, we believe that our results help establish a strong basis for future study of peripheral and neuromechanical factors in uencing the control of muscle force. Lastly, we demonstrate that the modulation of involuntary force variability via proprioceptive 47 pathway gains gives the nervous system a certain degree of control over involuntary force variabil- ity. Properly regulating those gains is important if disruptive tremor is to be avoided (Fink et al., 2014). Our ability to understand and modify these relationships will be instrumental to providing insights into the neural mechanisms and circuits associated with functional performance(Enoka et al., 2003), aging (Enoka et al., 2003; Tracy et al., 2005), fatigue (Cresswell and Loscher, 2000; Hunter et al., 2002) and neurological disease (Chu and Sanger, 2009; Lafargue et al., 2003; Lodha et al., 2010; Rothwell et al., 1982; Vaillancourt et al., 2001a). Finally, our approach of combining experimental observations with a computational simulation should provide a springboard for future investigation of neuromechanical interactions and task-dependent tuning of sensorimotor integra- tion and proprioceptive mechanisms during voluntary actions in healthy development and aging; and disease. 48 Chapter 2: Force variability is not motor noise: theoretical impli- cations for motor control Abstract Variability in muscle force is a hallmark of healthy and pathological human behavior. Predomi- nant theories of sensorimotor control assume `motor noise' leads to force variability and its `signal dependence' (variability in muscle force whose amplitude increases with intensity of neural drive). Here, we demonstrate that the two proposed mechanisms for motor noise (i.e., the stochastic nature of motoneuron discharge and unfused tetanic contraction) cannot account for the majority of force variability nor for its signal dependence. We do so by considering three previously unmodeled, but physiologically important, features of a population of motor units: 1) calcium kinetics that drives fusion of motor unit twitches, 2) coupling between motoneuron discharge rate, cross-bridge dynamics, and muscle mechanics, and 3) a series-elastic element to account for the aponeurosis and tendon. These results argue strongly against the idea that force variability and the resulting kine- matic variability are generated primarily by `motor noise.' Rather, they underscore the importance of variability arising from properties of control strategies embodied through distributed sensorimo- tor systems. As such, our study provides a critical path toward developing theories and models of sensorimotor control that provide a physiologically valid and clinically useful understanding of healthy and pathologic force variability. Introduction Variability is a hallmark of healthy and pathological human behavior. As such, the structure of kinematic (Whiting, 1983) and kinetic variability (Newell and Carlton, 1988; Sherwood and Schmidt, 1980) is a rich behavioral phenomenon that informs theoretical models about the mech- anisms used by the central nervous system to learn and produce motor behaviors (Dhawale et al., 2017; Diedrichsen et al., 2010; Harris and Wolpert, 1998; Scott, 2004; Seethapathi and Srinivasan, 2019; Sternad, 2018; Todorov and Jordan, 2002; van Beers et al., 2004; Wu et al., 2014). These 49 theories are then used to help identify the mechanisms underlying dysfunction and features of var- ious neurological conditions (Therrien et al., 2016, 2018). Therefore, the physiological validity of these theoretical models is critical to their proper and eective clinical translation. Many theoretical models for human motor behavior (e.g., minimum variance theory and optimal control theory) assume that observed kinematic variability arises predominantly, if not exclusively, from `signal-dependent motor noise;' random variations in muscle force output whose amplitude increases with the input level (Dhawale et al., 2017; Diedrichsen et al., 2010; Harris and Wolpert, 1998; Scott, 2004; Seethapathi and Srinivasan, 2019; Sternad, 2018; Todorov and Jordan, 2002; van Beers et al., 2004; Wu et al., 2014). Under this theoretical framework, observed kinematics and its variability re ect the performance limitation imposed by the motor system (Harris and Wolpert, 1998; van Beers et al., 2004) attempting to minimize the deleterious eects of `motor noise' on behavior (Todorov, 2005; Todorov and Jordan, 2002). In these models, the specic im- plementations of motor noise (i.e., amplitude and its relationship with input levels) are convenient free parameters that determine model performance and allow tting of model output to experimen- tal data. Furthermore, many clinical manifestations of increased force/kinematic variability have been interpreted simply as a consequence of increased motor noise interfering with optimal control strategies implemented by the central nervous system (Lunardini et al., 2015; McCrea and Eng, 2005; Therrien et al., 2016). Despite the success of these theoretical models replicating certain experimental observations, their theoretical framework is incompatible with many other experimental observations. For one, they cannot explain why force/kinematic variability can be modied by various factors such as visual feedback and physical activity (Keenan et al., 2017; Nakanishi et al., 2019), how such variability can be tuned as needed to meet task demands, enhance sensing or exploration (Dhawale et al., 2017), or why the amplitude and spectral structure of force variability is so heavily dependent on closed-loop elements of force control, such as segmental and visiomotor feedback loops (Keenan et al., 2017; Nagamori et al., 2018; Shinohara et al., 2005; Vaillancourt et al., 2003; Yoshitake et al., 2004) Secondly, the physiological basis for `motor noise' is very weak. It has been assumed that motor 50 noise arises from the physiological properties of motor units, in particular, stochastic timing of motor unit discharges and unfused tetanic contraction (Faisal et al., 2008; Harris and Wolpert, 1998; Jones et al., 2002; Slifkin and Newell, 2000). This view is supported by many simulations based upon the seminal computational model of motor unit recruitment and rate modulation originally proposed by Andrew Fuglevand (Fuglevand et al., 1993). Despite many successful applications of the Fuglevand model, its usage to explain the origin of force variability is, in fact, an extrapolation of its original intent and scope. It has been extensively re-purposed and modied to t experimental recordings of force variability and its signal dependence (de C. Hamilton et al., 2004; Dideriksen et al., 2010, 2012; Enoka et al., 2003; Hu et al., 2014; Jesunathadas et al., 2012; Jones et al., 2002; Moritz et al., 2005; Petersen and Rostalski, 2019; Taylor et al., 2003; Yao et al., 2000; Zhou and Rymer, 2004), despite containing various non-physiological motor unit features/assumptions that may render it inappropriate for this purpose. As a result, the signicance of signal-dependent random motor noise in shaping motor performance has been a dominant, but perhaps unwarranted, assumption for decades (Jones et al., 2002). Despite those successful applications of the Fuglevand model, its usage to explain the origin of force variability is, in fact, an extrapolation of its original intent and scope. In particular, the Fuglevand model includes various which may render it inappropriate to study the physiological origins and behavioral consequences of force variability. As a result, some studies may have over- stated the signicance of motor noise in shaping experimentally observed force variability during isometric contractions (Jones et al., 2002). Such overstatement promoted the emergence of a pre- mature consensus in the community that force variability arises mostly or exclusively from motor noise. This interpretation is incompatible with early and recent experimental observations that closed-loop elements of force control (e.g., such as segmental and visuomotor feedback loops) can alter the amplitude and spectral structure of force variability (Keenan et al., 2017; Nagamori et al., 2018; Shinohara et al., 2005; Vaillancourt et al., 2003; Yoshitake et al., 2004). In this study, we systematically examine the physiological validity of the assumptions that under- lie various motor unit models and their implications to force variability. To this end, we developed a new model of a population of motor units that now includes three physiologically important features: 1) calcium kinetics and cross-bridge turnover that drives fusion of motor unit twitches, 51 2) coupling among motoneuron discharge rate, cross-bridge dynamics, and muscle mechanics, and 3) a series-elastic element to account for the aponeurosis and tendon. By exploring a plausible range of parameters for these known physiological processes, we were able to test the following two hypotheses: Hypothesis A: Those model renements signicantly reduce the amplitude of `motor noise' and its signicance in shaping overall force variability. and Hypothesis B: The rened model output directly contradicts the assumption that motor noise should increase continually with force (i.e, is `signal dependent', as typically dened), because the completeness of twitch fusion at higher levels of synaptic input should increase. Our results support the above two hypotheses, demonstrating that experimentally observed amplitude of force variability and its signal dependence cannot be explained by motor noise alone. Therefore, our results argue strongly against the idea that force variability should be modeled as the consequence of `motor noise.' Rather, our results emphasize the importance of alternative sources of force variability arising from control strategies embodied through distributed sensorimotor systems, which are underestimated or ignored in current models of motor behavior. Therefore, our study informs fruitful directions to better understand and interpret force variability in health and disease (Enoka et al., 2003; Lodha and Christou, 2017; Mai et al., 1988; Vaillancourt et al., 2001a). Material and Methods Conversion of spike trains into motor unit force using Fuglevand model The model of rate coding and recruitment of a motor unit pool developed by (Fuglevand et al., 1993) was replicated and tested against known physiological properties of motor units. Only brief descriptions of this model are provided here, as it is fully described in the original paper (Fuglevand et al., 1993). Motor unit force is modeled as the impulse response of a second-order critically damped system as in the following equation (Fuglevand et al., 1993): f i (t) =g i;j P i t T i e 1(t=T i ) ; (3) 52 where g, P , T and t are the gain, peak twitch force, contraction time and time, respectively. The subscripts, i and j, denote the indexes of motor units and of motor unit spike events. The gain, g i;j , was originally introduced to replicate the sigmoidal relationship between the discharge rate of a motor unit and its output force as described in Eq. 17 in (Fuglevand et al., 1993). The peak twitch force,P i , and contraction time,T i , of each motor unit follow a exponential distribution with a specied range between the smallest (slowest) and largest (fastest) motor units. The original model proposed in (Fuglevand et al., 1993) uses 100-fold and 3-fold ranges for the peak twitch force and contraction time, respectively, which were used here. Similarly, the recruitment threshold of motor units follows an exponential distribution such that the range of lowest- and highest-threshold units equals to 17-fold in an excitation unit, E, in the original model. This value corresponds to the recruitment of all motor units at 50% of the maximal excitation, which has been reported for rst dorsal interosseus muscle (de Luca et al., 1982; Kukulka and Clamann, 1981). A motor unit pool consisted of 120 motor units. The minimal discharge rate of all motor units was set to 8 Hz while the peak discharge rate was linearly decreased from 35 to 25 Hz from the lowest-threshold to the highest-threshold units. All of these parameters were the same as proposed in (Fuglevand et al., 1993). New model of a motor unit pool Despite its original purpose for simulation of isometric muscle force and electromyogram of muscle, the Fuglevand model has been repurposed to simulate force variability (de C. Hamilton et al., 2004; Dideriksen et al., 2010, 2012; Enoka et al., 2003; Hu et al., 2014; Jesunathadas et al., 2012; Jones et al., 2002; Moritz et al., 2005; Petersen and Rostalski, 2019; Taylor et al., 2003; Yao et al., 2000; Zhou and Rymer, 2004). However, such use of the Fuglevand model has several critical drawbacks that limit its physiological faithfulness to simulate force variability: 1) The peak tetanic force of motor units, and therefore that of muscle, depend on an arbitrary choice of model parameters such as the peak discharge rate of a given unit (Fig. 15 b). We found this non-physiological because the determinants of the maximal force of a given unit are the number of muscle bers (i.e., innervation ratio), the muscle bers cross-sectional area, and their specic 53 tension (Totosy de Zepetnek et al., 1992). 2) The simulated range of discharge rates for a given motor unit was based on empirical obser- vations from human motor units (Fig. 15 b), but these do not always re ect current theoretical and functional understanding of muscle (see Discussion). 3) The model does not explicitly simulate the fusion of force twitches with increases in discharge rate and the concomitant saturation of calcium binding to troponin. In fact, the model does not always produce the fusion of force twitches as shown in Fig. 15 c&d. 4) The model lacks a series elastic element (i.e., tendon and aponeurosis). Even during the isometric condition it was originally intended to simulate, muscle length uctuates due to this in- series compliance. Such ber length uctuations inevitably aects the viscoelastic properties of a musculotendon unit on output force, which can signicantly alter the frequency content of force variability (Nagamori et al., 2018). To address these issues and re-evaluate the contribution of motor unit properties to force vari- ability, we have developed a new model drawn schematically in Fig. 12 and described in detail below. Muscle Architecture: As in the Fuglevand model, we included 120 motor units (N = 120) in a pool. Architectural parameters of the muscle were chosen based on exor carpi radialis muscle (Lieber et al., 1990; Loren and Lieber, 1995). This muscle was chosen instead of intrinsic hand muscles often used in motor unit simulations (e.g., (Fuglevand et al., 1993)) because the muscle architecture is reasonably compatible with the parallelogram assumption needed to simulate mus- culotendon dynamics in Module 3 (Segal et al., 1991). In comparison, rst dorsal interosseous, for example, is bipennate (Infantolino and Challis, 2010) and does not conform to the assump- tion. Module 1: Conversion of synaptic input into discharge patterns: The motor unit pool is driven by an eective synaptic input,U eff , whose normalized value ranges from 0 and 1. The time course of the eective synaptic input was modeled as a ramp-and-hold input (1-sec zero input phase, 2-sec ramp to a target, and 13-sec hold at the target), which was applied equally to all motor units 54 Figure 12: Schematic representation of our new model of a motor unit pool. The model consists of three modules. Module 1 converts synaptic input, U eff , into spike trains of individ- ual motor units. Module 2 turns spike trains into motor unit activation, A, through three-stage process shown below. Stage 1 simulates calcium kinetics driven by action potentials (R). The cal- cium kinetics is described using ve states, [s], [cs], [c], [f] and [cf] with associated rate constants (k 1 ,k 2 ,k 3 and k 4 ) between those states (see text for details). Stage 2 converts [cf] into the inter- mediate activation, e A, through a non-linear lter, which describes cooperativity and saturation of calcium binding and cross-bridge formation. Stage 3 introduces an additional rst-order dynam- ics to generate motor unit activation, A, from e A. Module 3 describes the contraction dynamics between muscle and a series elastic element and generates tendon force, F se , as the output. The detail descriptions of each module are given in the text. 55 in a pool. It is important to note that some synaptic inputs (e.g., Ia aerent and rubrospinal inputs) may be distributed non-uniformly across motor units (Heckman and Enoka, 2012; Johnson et al., 2017) and such non-uniform distribution can in uence the range of recruitment thresholds and the general frequency-input relationship of a motor unit pool (Heckman and Binder, 1991b, 1993). However, it is dicult to accurately simulate how a non-uniform distribution of synaptic input would aect those parameters due to limited experimental data from humans and potential anatomical dierences across muscles and across species (e.g., potential absence of the rubrospainal tract in humans (Nathan and Smith, 1982)). Instead, the range of recruitment thresholds and the frequency-input relationship were directly manipulated assuming the uniform distribution of synaptic inputs across motor units. Intracellular recordings of cat motoneurons have shown that the frequency-current relationship of a motoneuron (discharge rate vs. injected current to the motoneuron soma) is best described as two linear ranges: primary and secondary (Kernell, 1965a,b, 1979; Schwindt, 1973). Almost all motoneurons can sustain repetitive discharges in the primary range (Gydikov and Kosarov, 1973) and motor unit forces reach 65-95% of their respective peak tetanic force at the transition frequency to the secondary range (Heckman and Binder, 1991b). Thus, we assume in our model that the discharge rate of motoneurons linearly increases up to the rate that corresponds to approximately 90% of peak tetanic force of that motor unit (Fig. 15b). Recruitment scheme: We developed a new recruitment scheme, which resembles experimentally observed motor unit discharge patterns in human (i.e., the rate limiting of low-threshold motor units (Bigland and Lippold, 1954; Binder et al., 1996; de Luca et al., 1982; Fuglevand et al., 2015; Gydikov and Kosarov, 1973; Heckman and Binder, 1991b, 1993; Kanosue et al., 1979; Monster and Chan, 1977; Moritz et al., 2005) presumably due to motoneuron intrinsic mechanisms such as adaptation (Gorman et al., 2005; Granit et al., 1963; Kernell et al., 1999; Powers et al., 1999; Sawczuk et al., 1995; Spielmann et al., 1993) and persistent-inward current (Fuglevand et al., 2015; Heckman and Enoka, 2012; Heckman et al., 2008; Hornby et al., 2002; Lee and Heckman, 1998a,b). We did so modifying and combining previously proposed methods (De Luca and Contessa, 2015; Fuglevand et al., 1993; Song et al., 2008b). Specically, this new scheme demonstrates low-threshold motor units whose recruitment threshold is below 20% of maximal synaptic input show rapid acceleration 56 and saturation of their discharge rates, while the remaining `high-threshold' units linearly increase their discharge rates and reach their peak discharge rates at the maximal synaptic input. These were modeled using the following equation: DR i (t) =g ei [U eff (t)RT i ] +MDR i (4) g ei = PDR i MDRi 1RT i ; (5) where RT i is recruitment threshold of the i-th motor unit in a unit of U eff (0-1), and MDR i and PDR i are the minimal and peak discharge rates of that motor unit, respectively. Note that the gain of the frequency-input relationship, g e , diers across motor units depending on the values of RT i , MDR i and PDR i . The frequency-input relationship of the low-threshold units is described as two linear functions using the following equations: DR i (t) = 8 > > < > > : i k e [U eff (t)RT i ] +MDR i RT i <=U eff (t)<=U ti PDR i k ei [1U eff (t)] U eff (t)>U ti (6) i =100RT i + 21 (7) k ei = f t f 0:5 MDR i + i (PDR i f t f 0:5 ) i (1RT i ) (8) U ti = k ei [PDR i (f t f 0:5 )] k ei : (9) These equations were derived such that the slope of the rst linear region (i.e., when U eff <U ti ) is much steeper than that of the second linear region, whereU ti determines the level of synaptic input at which a given motor unit transitions from the rst linear region into the second. i determines the relative slope of the rst to second linear function, which was modeled to decrease from a value of 20 forRT i = 1 to 1 forRT i = 0:2. Given the value of i , the value ofk ei is then calculated, which determines the rate at which the discharge rate increases with a given increment in synaptic input. The transition frequency,f t describes the frequency at which the slope transitions from the rst to 57 Figure 13: Recruitment scheme of our new model. The frequency-synaptic input relationship of the selected motor units (n = 10) is shown to illustrate our new recruitment scheme. U r indicates the level of synaptic input at which all motor units are recruited. Lower-threshold motor units (red) (below 20% of the maximal synaptic input) show rapid acceleration upon recruitment and saturation of their discharge rates. Higher-threshold units (blue) linearly increase their discharge rates up to the maximal synaptic input. the second linear function. The value of f t was chosen to be 1.2 in the unit of f 0:5 , the discharge rate required to reach half the peak tetanic force of a motor unit. The value of 1.2f 0:5 corresponds to approximately 60% of the maximal motor unit activation and achieves 90% of fusion at muscle length of 1:0L ce . All of these parameters were set to qualitatively mimic discharge patterns of human motor units (Heckman and Binder, 1991b; Kanosue et al., 1979; Monster and Chan, 1977; Moritz et al., 2005) such that lower-threshold units have more rapid acceleration of discharge rate upon recruitment as shown in Fig. 13 c. Peak and minimal discharge rate: Peak (PDR i ) and minimal discharge rates (MDR i ) of indi- vidual motor units were determined based on f 0:5i . The value of f 0:5i for individual motor units was determined empirically based on their respective activation-frequency relationship (see below). As done previously (Song et al., 2008b), PDR i andMDR i were set equal to 2f 0:5i and 0:5f 0:5i , respectively. We chose this method based on empirical evidence that the discharge rate of a mo- toneuron and speed of contraction of muscle bers that it innervates are tightly coupled (Bakels and Kernell, 1993; Botterman et al., 1986; Cooper and Eccles, 1930; Grimby et al., 1979; Kernell, 1979; Kernell et al., 1999, 1983; Sasaki, 1963) and such a tight connection is regulated by both genetic and epigenetic factors (Buller et al., 1960; Jansen and Fladby, 1990; Mendell et al., 1994; 58 Thompson et al., 1984). This diers from previous studies where minimal and peak discharge rates were determined based on empirical ndings in human motor unit recordings (e.g., De Luca and Contessa (2015); Fuglevand et al. (1993); Moritz et al. (2005)). Although observations of motor unit activity during voluntary muscle contraction are certainly important, there are methodological limitations of accurately extracting motor unit discharge rates in humans especially at higher levels of recruitment (Binder et al., 1996) and their use for the present purpose leads to non-physiological discharge behaviors that fundamentally contradict known principles of muscle energetics and func- tion (see Results and Discussion). Recruitment threshold: The distribution of recruitment thresholds of motoneurons in a motor unit pool is determined by intrinsic electrical properties of motoneurons (e.g., input resistance and rheobase) as well as the organization of excitatory, inhibitory and neuromodulatory inputs across the constituent motor units (Binder et al., 1996; Burke, 1979; Fleshman et al., 1981; Heckman and Enoka, 2012; Johnson et al., 2017). The distribution of rheobase (an index of excitability of motoneurons (Fleshman et al., 1981)) across motoneurons in a pool is skewed to the high rheobase (Fleshman et al., 1981; Gustafsson and Pinter, 1984; Powers and Binder, 1985). Similarly, it was found both in human and animal that the distribution of recruitment thresholds in a pool as a fraction of maximal force follows an exponential distribution where a larger proportion of units are recruited at low force levels (i.e., weaker synaptic inputs) (Duchateau and Hainaut, 1990; Henneman et al., 1965b; Milner-Brown et al., 1973; Stephens and Usherwood, 1977). As such, we modeled the distribution of recruitment thresholds by tting an exponential function (t function in MATLAB) that spans the range from the lowest recruitment threshold (U 1 ) to the highest recruitment threshold (U r ) in the unit of eective synaptic drive (0-1). The value of U 1 was always set to 0.01 (1% of maximal excitation). The value of U r was set to 0.5 by default (i.e., all motor units are recruited at 50% of maximal excitation). The order in which individual motor units are recruited largely follows Henneman's size principle where smaller motoneurons are always recruited before larger ones (Binder et al., 1983; Calancie and Bawa, 1985; Carpentier et al., 2001; Clark et al., 1993; Cope and Clark, 1991; Duchateau and Hainaut, 1990; Dum and Kennedy, 1980; Fleshman et al., 1981; Gustafsson and Pinter, 1984; Haftel et al., 2001; Henneman, 1957; Henneman et al., 1965a,b; Milner- Brown et al., 1973; Mishelevich, 1969; Monster and Chan, 1977; Riek and Bawa, 1992; Stephens 59 and Usherwood, 1977; Thomas et al., 1987, 1986; van Zuylen et al., 1988; Zajac and Faden, 1985). This recruitment order tends to be robust regardless of types of synaptic inputs in most muscles (Clark et al., 1993; Cope and Clark, 1991; Haftel et al., 2001) with the possible exception of certain hand muscles (Garnett and Stephens, 1981; Kanda and Desmedt, 1983). Accordingly, we assigned lower recruitment thresholds to motor units with smaller peak tetanic force, which resulted in the relationship shown in Fig. 14d. This positive correlation between the recruitment threshold and peak tetanic force (therefore twitch force) is consistent with previous experimental ndings (Calancie and Bawa, 1985; Garnett and Stephens, 1981; Milner-Brown et al., 1973; Stephens and Usherwood, 1977). The methods we used to determine the peak tetanic force of individual motor units are described below. Variability in motoneuron discharge: Variability in timing of motoneuron discharge was intro- duced using the method described Fuglevand et al. (Fuglevand et al., 1993). The j-th spike time of the i-th motor unit is described in the following equation: t i;j = +cvZ +t i;j1 : (10) Inter-spike interval, , is calculated from discharge rate predicted from the Eqs. 4-6. The value of cv determines the degree of stochasticity in motor unit discharges as per coecient of variation (CoV) for inter-spike intervals (ISIs). For each discharge, the value of Z was randomly drawn from the standard normal distribution whose values range from -3.9 to 3.9 (Fuglevand et al., 1993), which simulates a renewal process where inter-spike intervals between successive spikes are uncorrelated and the distribution of inter-spike intervals follow a Gaussian distribution. The value of cv was set to 0.2 (or 20% of CoV of ISIs) except for the following two sets of simulations: 1) one in which CoV of ISIs were varied from 0 to 20% to quantify how the degree of stochasticity in uence the overall amplitude of force variability (Fig. 17) and 2) the other in which CoV of ISIs were tted to the experimentally observed values in (Moritz et al., 2005) to compare the predicted motor noise from our model to the experimentally measured amplitude of force variability reported in (Moritz et al., 2005) (Fig. 20). In case 2), the value ofcv were adjusted as a function of synaptic input using the equation provided by (Moritz et al., 2005), which was 60 modied as follows: cv = 10 + 20e (U eff RT i )=2:5 : (11) This equation simulates an exponential decline in CoV of ISIs with increasing force levels observed experimentally by (Moritz et al., 2005). Module 2: Conversion of spike trains into muscle activation Module 2 was inspired by the model proposed by Williams et al. (1998). We chose this model over others (e.g., models by (Bobet and Stein, 1998; Kim et al., 2015; Shorten et al., 2007)) for its relative simplicity and its ability to replicate properties of motor unit force generation as described in details in the Results section. This module converts motor unit spike trains into muscle activation (a value between 0 and 1), A, via three-stage processes (Fig. 12). First, a motor unit spike train is converted into a state variable, [cf]; a fraction of cross-bridges bound to calcium that can participate in force generation, based on the model of simplied calcium kinetic described previously (Williams et al., 1998) (Stage 1). Second, the state variable, [cf], is then passed through a non-linear lter that accounts for cooperativity and saturation of calcium binding and cross-bridge formation (Stage 2). Third, the output of the non-linear lter, e A, is converted into muscle activation, A, using the rst-order dynamics (Stage 3). Stage 1: This stage describes calcium kinetics with ve states, [s], [cs], [c], [f] and [cf] with associated rate constants (k 1 ,k 2 ,k 3 and k 4 ) between those states (Stage 1 in Fig. 12). Since full derivations were given previously (Williams et al., 1998), only important equations are described here. The dynamics of free calcium [c] and that of calcium bound to myolaments [cf] can be expressed using the following equations: d[c] dt =k 1 (C [c] [cf])Rk 2 [c]fSC + [c] + [cf]g (k 3 [c]k 4 [cf])(1 [cf]) (12) d[cf] dt = (1 [cf])(k 3 [c]k 4 [cf]): (13) 61 R = n X i=1 (1 + 2A )exp( tt i 1 ) (14) The parameters,C andS, describe the total number of calcium and sarcoplasmic reticulum binding sites relative to the total number of myolament binding sites, F . These parameters have the following relationships: S = S C (15) C = C F; (16) where the value ofF was assumed to be 1 and S and C were determined for each motor unit. We added Eq. 14 to the original model by Williams et al. (1998) based on the methods described by Ding et al. (2000) and Kim et al. (2015) to `activate' calcium kinetics with action potentials. The value of in Eq. (14) was also determined for each motor unit. Stage 2: we introduce cooperativity and saturation of calcium binding and cross-bridge forma- tion using the Hill equation (Bobet and Stein, 1998; SwartzS and Moss, 1992) described below: e A = ([cf]S i Y ) N ([cf]S i Y i ) N +K N : (17) The coecients, N and K, describe non-linearity due to cooperativity and saturation of calcium binding and cross-bridge formation, which are determined for each motor unit. The parameters, S i and Y , describe the sag and yield properties of slow and fast-motor units (Brown et al., 1999; Brown and Loeb, 2000) using the following equations: 62 _ S i (t) = a S S i (t) T S ; (18) a S = 8 > > < > > : a S1 ;f eff (t)< 0:1 a S2 ;f eff (t) 0:1 (19) _ Y i (t) = 1c Y [1exp( jVcej V Y )]Y (t) T Y : (20) All parameters associated with Eq. 18&20 are same as ones given in Song et al. (2008b) except a S1 and T S , where were set to 20 and 15 ms to replicate similar behavior of sag. Stage 3: the intermediate activation, e A, was then converted into muscle activation, A, using the following rst-order dynamics: _ A = e AA 2 : (21) The value of 2 depends on each motor unit. Determining free parameters: through Stage 1-3, various properties of individual motor units (e.g., contraction time, activation-frequency relationship, etc.) at the optimal length were modeled by adjusting 11 free parameters ( S , C , k 1 , k 2 ,k 3 , k 4 , , N, K, 1 , and 2 ) in the equations described above. We did so in the following manner: (1) Seed parameter sets were constructed such that each parameter set can generate twitch responses with given contraction times and replicate the activation-frequency relationship, similar to what has been modeled previously (Brown et al., 1999). (2) Parameters in the seed set were then adjusted to minimize an error for a given target contraction time between the activation-frequency relationship of the new model and that of the model described by Song et al (Song et al., 2008b) in the frequency range between 0.5f 0:5 and 63 2f 0:5 . Optimization was performed to minimize any deviation from the desired activation-frequency relationship provided by (Brown et al., 1999) while realizing the target contraction time. We used the method used in (Raphael et al., 2010). Described brie y, one of the parameters was randomly selected and perturbed in both positive and negative directions from the original value and then the new value that produced the smallest error was taken for the next iteration. One iteration consists of this process performed for all 11 free parameters and the total of six iterations were performed with progressively smaller amount of perturbation. Length-dependence of activation-frequency relationship: The activation-frequency relationship is known to depend on muscle length (Brown et al., 1999). Therefore, we introduced length- dependence on four of the 11 free parameters, k 3 , k 4 , N and K, using the following relationship with muscle length, L ce , (Kim et al., 2015): x = 0 + 1 L ce ; (22) where x is the predicted value of a given parameter and 0 and 1 are the coecients to be determined. Once the 11 free parameters were determined for the activation-frequency relationship at the optimal length, the four of those parameters (k 3 , k 4 , N and K) were tted to realize the activation-frequency relationship at dierent muscle lengths (0.8L 0 , 0.9L 0 , 1.1L 0 and 1.2L 0 ) given Brown et al. (1999) while other parameters were kept constant. These parameters were chosen based on the method employed by previous studies to simulate length-dependence of force output (Kim et al., 2015; Williams et al., 1998) and able to simulate the length-dependence of the activation- frequency relationship reasonably well (Fig. 15 b). The same method described above was used to t those parameters. Each iteration was performed 10 times at each muscle length and linear regression was used to determine the value of 0 and 1 for each of the four parameters. Module 3: Musculotendon dynamics To simulate the contraction dynamics of a musculotendon unit during an externally isometric contraction (musculotendon unit length = constant), we used the model of a musculotendon unit proposed previously (Brown et al., 1999; Brown and Loeb, 1999; 64 Brown et al., 1996; Song et al., 2008a,b) This model comprises of a mass, a contractile element, two passive parallel elements and a series-elastic element (Song et al., 2008b). The contraction dynamics of this system is described as follows (He et al., 1991): L ce = 1 M m [F 0 F se cos (F ce +F 0 F pe1 )cos 2 ] + _ L ce 2 tan 2 L ce (23) L mt =L ce L ce0 +L se L se0 =constant (24) where M m , , L ce , L se and L mt are muscle mass, pennation angle, muscle length normalized to optimal muscle length, tendon length normalized to optimal tendon length and musculotendon unit length in cm. F 0 determines maximal tetanic force of muscle as described in the next section. These parameters are based on exor carpi radialis muscle (Lieber et al., 1990; Loren and Lieber, 1995) and listed in Table 2. This muscle was chosen instead of intrinsic hand muscles often used in motor unit simulations (e.g., (Fuglevand et al., 1993)) because the muscle architecture is reasonably compatible with the parallelogram assumption needed to simulate musculotendon dynamics (Segal et al., 1991). F ce , F pe1 and F se are force from contractile element, passive element 1, and series elastic element, respectively. F ce is computed as follows: F ce = n X i=1 PT i A i (FL i FV i +F pe2 ): (25) PT i is peak tetanic force of individual motor units. F pe2 is passive, resistive force against shortening. FL i andFV i are force-length and force velocity properties of individual motor unit as described as follows: FL i =exp( L ce 1 ! ): (26) 65 Table 2: Model parameters for muscle based on architectural parameters of exor carpi radialis muscle(Lieber et al., 1990; Loren and Lieber, 1995) Muscle mass, M m (g) 13.5 Optimal muscle length, L ce0 (cm) 5.98 Optimal tendon length, L se0 (cm) 23.0 Pennation angle, (deg) 3.1 FV i = 8 > > < > > : (V max V ce =[V max + (c v0 +c v1 L ce )V ce ]);V ce 0 [b v (a v0 +a v1 L ce +a v2 L 2 ce )V ce ]=(b v +V ce );V ce < 0 : (27) The values of the coecients in the above equations depend on ber types and can be found in Table 1. in Song et al. (2008b). Muscle maximal tetanic force: we determined the maximal tetanic force of muscle (F 0 ) using the following equation given in (Lieber et al., 1990): F 0 = M m cos() L ce0 ; (28) where M m , , and L ce0 are muscle mass, muscle density (1.06 g/cm 3 ), specic tension (31.8 N/cm 2 ) and optimal fascicle length, respectively. Motor unit tetanic force: Maximal tetanic force of individual motor units, PT i , was determined in using the following equation: 66 PT i =F 0 exp(bi) P N n=1 exp(bi) (29) b =log(RP )=N (30) wherei,N andRP are index of motor units, the number of motor units in a pool and the range of peak tetanic force dened in the following section. This generates an exponential distribution of peak tetanic force such that a greater proportion of motor units produce smaller peak tetanic force (Olson and Swett, 1966, 1971). This also results in an exponential distribution of twitch force as documented extensively in previous studies in both cat (Dum and Kennedy, 1980; Olson and Swett, 1971; Wuerker et al., 1965) and humans (Cutsem et al., 1997; Maceeld et al., 1996; Milner-Brown et al., 1973; Monster and Chan, 1977). The range of peak tetanic force, RP , was set to 25-fold as a default based on the ranges of innervation ratio, cross-sectional area of muscle bers, and specic tension reported by Kanda and Hashizume (1992). To test the potential eect of this parameter on force variability, we increased it to 100-fold, commonly used in previous studies (e.g., Enoka et al. (2003); Jones et al. (2002); Moritz et al. (2005); Taylor et al. (2003); Yao et al. (2000); Zhou and Rymer (2004)). Assigning ber types to each motor unit: Although some motor units properties such as peak tetanic force and contraction can be modeled as continuous across motor units as done previously (Fuglevand et al., 1993), other properties can dier in a discrete manner across ber types such as sag and yield as modeled in Eqs. 18&20. For simplicity, we divide motor units into slow-twitch (type S) and fast-twitch (type F) types as in other models (Song et al., 2008a,b). There exists strong empirical evidence that fast-twitch (type F) units generate higher unit force than slow- twitch (type S) units (Bodine et al., 1987; Burke et al., 1973; Chamberlain and Lewis, 1989; Kanda and Hashizume, 1992; Totosy de Zepetnek et al., 1992; Zajac and Faden, 1985). Thus, we assumed that there is no overlap in peak tetanic tension between slow and fast-twitch units for simplicity. We then divided the motor unit pool into two ber types such that 30% of muscle maximal force arises from slow-twitch units and the rest from fast-twitch units, which results in 78 units out of 67 120 being slow-twitch. Contraction time: Contraction time is dened as the time a motor unit twitch response takes to reach its peak amplitude from the baseline force level. This parameter re ects the speed of calcium kinetics and cross-bridge cycling, which determines the speed at which a given motor unit can generate force. Contraction time has been used as an important indicator to dierentiate slow-twitch and fast-twitch units (Burke et al., 1973). The contraction times of fast and slow- twitch units hardly overlap (Burke et al., 1973). Based on previous experimental observations (Andreassen and Arendt-Nielsen, 1987; Bagust, 1974; Bagust et al., 1973; Burke, 2011; Burke et al., 1974; Dum and Kennedy, 1980; Garnett et al., 1979; Milner-Brown et al., 1973; Olson and Swett, 1966, 1971; Wuerker et al., 1965), we modeled the distribution of contraction time to follow a Rayleigh distribution which spans 20 ms to 110 ms (Fig. 14b). We assigned the 78 slow-twitch motor units to slower contraction times with no overlap with fast-twitch motor units. Motor units that generate higher unit force have faster contraction times (Fig. 14 c). This association was assumed based on a universal nding that fast-twitch units generate more force than slow-twitch units (Burke, 2011; McDonagh et al., 1980; Olson and Swett, 1966, 1971; Stephens and Stuart, 1975). Thus, we assumed in our default model no correlation between peak tetanic force and contraction time within and across motor units (Fig. 14 c) as a common trophic mechanism does not seem to exist (Kernell et al., 1983). Twitch-tetanus ratio: Twitch-tetanus ratio is computed as the ratio of twitch amplitude to peak tetanic force. This is an important characteristics that in uences the activation-frequency relation- ship of a motor unit and the amplitude of force uctuations due to unfused tetanic contraction. However, no previous models have explicitly accounted for this and tested its eect on force vari- ability. Here, we assign a twitch-tetanus ratio to each motor unit based on previous experimental observations from cat and rat muscles. First, the average twitch-tetanus ratio of all motor units of 0.23 was determined based on the result from a whole muscle preparation of feline caudofemoralis reported by (Brown and Loeb, 2000) because the measurement obtained from such preparation is least likely distorted by potential eects of a series-elastic element (Heckman et al., 1992). Then, twitch-tetanus ratios were given to 68 Figure 14: Population characteristics of our default motor unit pool. a) Histogram of peak tetanic force of 120 motor units (blue: slow-twitch, red: fast-twitch). The distribution follows an expo- nential distribution where a large portion of motor units produce relatively smaller tetanic force. b) Peak tetanic forces as a function of recruitment thresholds. The relationship assumes the size- principle (Henneman, 1957): smaller units get recruited earlier than larger units. (c) Histogram of contraction time 120 motor units. The distribution follows the Rayleigh distribution which spans 20 ms to 110 ms. d) The relationship between peak tetanic force and contraction time. Slow-twitch units (in blue) have slower contraction time and smaller peak tetanic force. Within each ber type, no correlation between peak tetanic force and contraction time was assumed. e) The relationship between contraction time and twitch-tetanus ratio. Within each ber type, twitch-tetanus ratio is positively correlated with contraction time (correlation coecients, r, are 0.587 and 0.531 for slow-twitch and fast-twitch units, respectively). f) The relationship between contraction time and the frequency at which half the tetanic force is achievedf 0:5 . Consistent with previous experimental data (Botterman et al., 1986; Kernell et al., 1983), those parameters in our default motor unit pool are highly correlated (r = 0.794). 69 individual motor units such that those values range from 0.08 to 0.57 (Bagust et al., 1973; Burke et al., 1974, 1973; Celichows and Grottel, 1993; Devanandan et al., 1965; Dum and Kennedy, 1980; Stephens and Stuart, 1975; Walsh et al., 1978) and the ratio in a given ber-type is positively correlated with contraction time (Bagust, 1974; Bagust et al., 1973). Experimental Design and Data Analysis We ran ten trials at each level of synaptic input (5% and from 10 to 100% with an increment of 10% maximal synaptic input). We added another set of ten trials at 2.5% of maximal synaptic input for the comparison to the experimental data in Fig. 20. Each trial consisted of a 1-sec zero input phase, a 2-sec ramp-up and a 13-sec hold at a given level of synaptic input. The last 10-sec of the hold phase was analyzed to quantify mean, standard deviation (SD), and coecient of variation (CoV) of output force except for the data presented in Fig. 20. For this data set, we used the method compatible with that described in Moritz et al. (2005) to allow for a fair comparison between our result and their result. To this end, the 10-sec hold phase of output force was divided into ten 1-sec segments, the duration of data used in Moritz et al. (2005). The mean of each segment was used for calculation of CoV. Force in each segment was then linearly de-trended and standard deviation was calculated from the de-trended data. CoV was calculated by dividing the standard deviation by the mean force of that segment before de-trending. Ten values of CoV of force for each trial were then averaged and the average CoV of force represents each data point in Fig. 20. All gures present the average of 10 trials in a solid line with SD denoted by a shaded area. To characterize the frequency content of force variability, we computed the power spectrum of output force during the last 10-sec of the hold phase using the (pwelch function in MATLAB), specifying a 0.5 Hz frequency resolution from 0 to 100Hz without any segmentation or overlap. To characterize the SD-input relationship of our new model, we compared it to the theoretical relationship assumed in previous models (Diedrichsen et al., 2010; Harris and Wolpert, 1998; Scott, 2004; Seethapathi and Srinivasan, 2019; Sternad, 2018; Therrien et al., 2018; Todorov and Jordan, 2002; van Beers, 2009; van Beers et al., 2004; Wu et al., 2014). The theoretical SD-input relationship (i.e., signal-dependent noise) assumes that standard deviation (SD) of force increases proportionally 70 with synaptic input levels (i.e., SD/input p ), or SD =cinput p ; (31) as shown by Jones et al. (2002). The slope, c, determines the amplitude of signal-dependent noise and is often tted to experimental observations to replicate observed kinematic variability (Todorov, 2005; Todorov and Jordan, 2002; van Beers et al., 2004). The scaling factor, p, equals to 1 for a proportional relationship which has been assumed universally in most models (Diedrichsen et al., 2010; Harris and Wolpert, 1998; Scott, 2004; Seethapathi and Srinivasan, 2019; Sternad, 2018; Therrien et al., 2018; Todorov and Jordan, 2002; van Beers, 2009; van Beers et al., 2004; Wu et al., 2014). We tested how the value of c used in previous models compares to our results by showing an example value obtained from Jones et al. (2002) and Todorov (2005). From the model by Jones et al. (2002), the value of c was chosen to be 0.02 (i.e., 2% CoV of force). The model by Todorov (2005) converts a model input,u, into muscle force,f through a second order linear lter using the following equation: 1 2 f(t) + ( 1 + 2 ) _ f(t) +f(t) =u(t); (32) where 1 and 2 are 40 msec as originally modeled. u(t) is modeled asu t (1 + c t ) whereu t is the input amplitude at time t, c = 0.5, and t N(0; 1). We varied the value of u t from 0.1 to 1 with an increment of 0.1 and simulated force output of this system with time step of 10 msec for 15 sec. The standard deviation of force was computed from the last 10-sec segment of the time series. Ten trials were run at each value of u t and average standard deviation across the ten trials was computed. Finally, we found the slope of the resulting standard deviation-input relationship by tting a linear equation (polyt function in MATLAB with the degree of 1). The slope corresponds 71 to the value of c and was used in Fig. 20 for comparisons to our model. To test whether the assumption that p equals to 1 holds for our model, we rst calculated the value of c from SD and input at 5% maximal synaptic input obtained from our model (sold black line in Fig. 21). Any deviation of our model from this relationship indicates that standard deviation of force does not increase proportionally with mean force levels (i.e., p6= 1). We chose this method rather than tting c and p as done in (Jones et al., 2002) because it is clear that our data cannot be tted with a linear equation (Fig. 21) Statistical Analysis Statistical analysis was performed in the R environment for statistical computing (The R Founda- tion for Statistical Computing, Vienna, Austria). Details of the statistical analysis performed are given in corresponding results sections. Results The main objective of this study was to re-evaluate the contribution of motor unit properties to force variability (see Introduction, i.e., recruitment and rate coding, and the stochastic nature of motoneuron discharge). New model of a population of motor units improves the simulated behavior of motor units compared to the Fuglevand model. The use of the Fuglevand model often focuses on non-physiological ranges of discharge rates. We show in the left two columns of Fig. 15 characteristics of force outputs from two of the motor units in the original Fuglevand model: one slower unit and one faster unit in terms of their contraction time. Consistent with previous observations (Brown et al., 1999; Kernell et al., 1983; Maceeld et al., 1996; Rack and Westbury, 1969), the output force of a motor unit follows a sigmodial relationship with increasing discharge rates of the units (Fig. 15 b). However, the discharge rates 72 over which the Fuglevand model is used (shaded areas in Fig. 15 b) is highly non-physiological. This range is too broad and shifted to the right for slower motor units, and too narrow for faster motor units. The former largely eliminates the eect of rate coding on force output (e.g., MU 1) and the latter truncates it as some units can only reach a fraction of their peak tetanic force (e.g., ca. 60% for MU 120 shown in 15 b). This choice of discharge rates has important implications as shown below. The second limitation of the original Fuglevand model is its inability to exhibit twitch fusion, particularly for faster motor units (Fig. 15c&d). It is well known that the amplitude of force ripples during unfused tetanic contraction progressively decreases with increasing discharge rates (e.g., Maceeld et al. (1996)). Yet, most motor units in the Fuglevand model do not tend to reach complete fusion (a value close to 100%, see examples in Fig. 15c). In some units, the degree of fusion actually decreases at intermediate discharge rates|which is not seen experimentally. Furthermore, the rate at which the degree of fusion increases with respect to increases in mean force is too slow compared to the known experimental relationship from Maceeld et al. (1996) (Fig. 15d). In contrast, we assumed in the new model that simulated discharge rates correspond to the steepest region of the output-to-frequency relationship of individual motor units (Fig. 15b). As expected, motor units with faster contraction times (4 th column in Fig. 15) require much faster discharge rates to reach peak tetanic force and fusion compared to units with slower contraction times (3 rd column in Fig. 15). Accordingly, the range of discharge rates depends on the contraction speed of individual motor units. In these ranges, the new model is able to show nonlinear, yet monotonically increasing, fusion (Fig. 15c) that closely approximates the fusion vs. mean force behavior described by Maceeld et al. (1996) (Fig. 15d). Moreover, the critical addition of a series-elastic element and the associated contraction dy- namics of motor unit activation and force generation required the model to account for length- dependence of the activation-frequency relationship of motor units (Eqs. 18, 20 and 22) consistent with prior work (Brown et al., 1999; Rack and Westbury, 1969) (Fig. 15b). Due to the length- dependence of motor unit activation, the degree of fusion also demonstrates length dependence (Fig. 15c). Such a dependency might be expected from the kinetics of calcium release, diusion 73 Figure 15: Our new model improves predictions of force production at the individual motor unit level. (Row a) Output of representative motor units, one slower and one faster, from each model to constant synaptic input to their motoneuron at various frequencies. Note the output of the Fuglevand model is force in arbitrary units, whereas that of our model is motor unit activation (0{1) that is then scaled by peak tetanic force to produce force. (Row b) The output-to-frequency relationship of those same motor units. The shaded area represents the range of simulated discharge rates for those motor unit, which for the Fuglevand model does not correspond to the region of the steepest force-frequency relationship. Also, note that our new model includes the length-dependent output-to-frequency relationship described in (Brown et al., 1999; Song et al., 2008b). (Row c) The degree of fusion as a function of discharge rates. In the Fuglevand model, the increase in fusion is not monotonic and some units do not approach complete fusion. These issues are corrected in our new model. (Row d) The degree of fusion attained as output levels increase compared to the experimental observation from Maceeld et al. (1996) shows that the degree of fusion increases too slowly only to rise abruptly at higher outputs in the Fuglevand model, which is corrected in our new model. The dotted identity line is included for reference. 74 and uptake because the region of thick- and thin-lament overlap becomes longer as the muscle shortens, changing the diusion distances to regions on the thin-laments where cross-bridges may form (Brown and Loeb, 1999). To best of our knowledge, there exist no experimental studies that have reported length-dependence of fusion and therefore it remains to be validated. However, the eect of length on fusion is relatively small at intermediate discharge rates and does not seem to aect overall ndings of our simulation results discussed below. Both `onion-skin' and `reverse onion-skin' patterns emerge from our model. Figure 16 (a) shows the relationship between the peak discharge rate and recruitment threshold of all 120 motor units. We nd a signicant positive correlation between them (r = 0.450, p <0.01), which is consistent with some of the previous ndings in humans where the recruitment threshold and peak discharge rate were determined during linearly increasing force up to approximately the participant's maximal voluntary contraction (Gydikov and Kosarov, 1973; Jesunathadas et al., 2012; Moritz et al., 2005; Oya et al., 2009). The same observation applies to the minimal discharge rate in our model (not shown), which is again consistent with some previous observations (Erim et al., 1996; Gydikov and Kosarov, 1973; Jesunathadas et al., 2012; Kukulka and Clamann, 1981; Moritz et al., 2005). It is important to note, however, that in our model there exists (by design, Fig. 13) considerable variability in these relationships. It is possible to sample pairs of motor units that show opposite relationships (i.e., a higher-threshold unit with lower peak discharge rate)|as is the case experimentally. Furthermore, we calculated the discharge rate of all active unit at 10% of synaptic input (U eff = 0:1). When plotted vs. recruitment threshold (Fig. 16 b), we can observe a signicant negative correlation between the recruitment threshold and discharge rate (r =0:247, p <0.01), which is the characteristic pattern of the onion-skin recruitment scheme (e.g., Fig. 2 in De Luca and Hostage (2010)). To further illustrate this point, we randomly sampled 1,000 pairs of motor units from the 7,140 possible pairs in the entire pool (N = 120) and calculated the dierences in their recruitment thresholds and in their peak discharge rates. This analysis shows that approximately 63.2% are consistent with the reverse-onion-skin recruitment scheme (i.e., the higher-threshold unit has a 75 Figure 16: Both `onion-skin' and `reverse onion-skin' patterns emerge from our model. (a) The relationship of peak discharge rate vs. recruitment threshold shows a signicant positive correlation (r = 0.450, p<0.01). This demonstrates that higher-threshold units tend to show higher peak dis- charge rates (reverse-onion scheme features), without us having built that in. (b) The relationship between the discharge rate at 10% maximum input vs. recruitment threshold shows a signicant negative correlation (r = -0.247, p = 0.038). In contrast to (a), higher-threshold units tend to show slower discharge rates at 10% maximum input, demonstrating the onion-skin pattern as the emergent features of rate coding. 76 higher peak discharge rate); and the balance with the onion-skin recruitment scheme. This was signicantly higher than a chance level (p<0.01, binomial test with a probability of 0.5), suggesting that it is more likely to detect the reverse-onion-skin recruitment scheme, but also there is a good chance to nd otherwise. Therefore, our model explains why it is not at all surprising that the most experimental studies report con icting recruitment behaviors (Erim et al., 1996; Gydikov and Kosarov, 1973; Moritz et al., 2005; Zhou and Rymer, 2004). These results suggest that describing motor unit recruitment schemes as either onion-skin or reverse onion-skin (de C. Hamilton et al., 2004; De Luca and Hostage, 2010; Dideriksen et al., 2010, 2012; Feeney et al., 2018; Hu et al., 2014; Jones et al., 2002; Moritz et al., 2005; Taylor et al., 2003; Yao et al., 2000; Zhou and Rymer, 2004) may be a false dichotomy, and may not be necessary. In fact, these modeling choices can lead to non-physiological motor unit behaviors (Fig. 15) and greatly confound our understanding of the sources of force variability. Unfused tetanic contraction is not the cause of motor noise. At physiological discharge rates, a motor unit produces unfused tetanic contraction (Tsianos and Loeb, 2017). This observation has been used as one of the bases of motor noise (Faisal et al., 2008), yet no previous study has directly quantied its contribution to force variability. Fig. 17 shows how the degree of stochasticity, through its interaction with the force generating process of motor units, in uence the overall amplitude of force variability. An increase in discharge variability (i.e., an increase in CoV of ISIs) introduces low-frequency power in a spike train of an individual motor unit while attenuating power associated with the discharge rate and its harmonics (Fig.17a). Because the process of converting a spike train into motor unit force acts as a low-pass lter, the low- frequency component ( 5 Hz) is accentuated while the higher-frequency counterpart is attenuated (Fig. 17b). Furthermore, spatial ltering through summation of motor unit forces across the motor unit population can eectively attenuate the high-frequency force uctuations (i.e., unfused tetanic contraction), yet its eect is limited for the low-frequency component (Fig. 17c). This increase in low-frequency force uctuations causes the overall amplitude of force variability to increase as the stochasticity in motor unit discharge increases (Fig. 17d). These results demonstrate that unfused tetanic contraction by itself has limited eects on the overall amplitude of force variability. Rather, 77 it is the stochasticity in motor unit discharge that determines the extent to which the activation dynamics due to calcium kinetics and cross-bridge turn-over translates into force variability. The Fuglevand model overestimates the contribution of motor unit properties to motor noise. As previously published (de C. Hamilton et al., 2004; Feeney et al., 2018; Hu et al., 2014; Jones et al., 2002; Moritz et al., 2005; Taylor et al., 2003; Yao et al., 2000; Zhou and Rymer, 2004), the Fuglevand model in our hands also exhibits a monotonic increase in SD of force throughout the range of synaptic inputs (Fig. 18b), which disappears once twitch fusion, recruitment scheme, and muscle mechanics are corrected/added. Consistent with our hypotheses, adding physiologically realistic twitch fusion properties greatly reduces SD above 50-60% of maximal synaptic input (red line in Fig. 18b)|and substantially reduces CoV across all levels of synaptic input (Fig. 18c). Furthermore, addition of a series-elastic element amplies both of these reductions up to the mid range of synaptic inputs (Fig. 18b&c). These results suggest that the properties of motor units do not, in and of themselves, suce to produce signal-dependent noise as was proposed (Jones et al., 2002) and continues to be accepted Faisal et al. (2008). Musculotendon mechanics are essential for realistic simulation of the spectral charac- teristics of motor noise. Addition of the incontrovertible element of a tendon and aponeurosis (i.e., series-elastic element) results in the attenuation of high-frequency (> 5 Hz) force uctuations associated with motor unit discharge (Fig. 19). Such attenuation arises from the well-known low-pass ltering eect of the viscoelastic properties of a musculotendon (Baldissera et al., 1998; Bawa et al., 1976; Krylow and Rymer, 1997; Mannard and Stein, 1973), yet it has been ignored in, to our knowledge, most previous models of motor units (de C. Hamilton et al., 2004; Feeney et al., 2018; Hu et al., 2014; Jones et al., 2002; Moritz et al., 2005; Taylor et al., 2003; Williams and Baker, 2009a,b; Yao et al., 2000; Zhou and Rymer, 2004). This also produces smaller CoV for force at synaptic inputs levels below 60% (Fig. 18c). 78 Figure 17: Increased discharge variability causes an increase in force variability through its interaction with the muscle force generating dynamics. a) Power spectrum of a mo- tor unit spike train. Note that increased discharge variability introduces low-frequency (<5 Hz) power. b) Power spectral density of motor unit force. Increased discharge variability increases low-frequency force uctuations while attenuating those associated with unfused tetanic contrac- tion. c) Power spectral density of tendon force. The spatial ltering of motor unit forces selectively attenuates higher-frequency force uctuations associated with unfused tetanic contraction. d) CoV of force with varying degrees of discharge variability. Increases in discharge variability result in increases in the overall amplitude of force variability. 79 Figure 18: The Fuglevand model can overestimate the contribution of motor noise to force variability. Three simulated conditions are presented as follows: Fuglevand model in gray, the new model without a series-elastic element (SSE) and the new model with SSE. a) Mean force as a function of the synaptic input. All three conditions exhibit a non-linear force-input relationship. b) SD of force normalized to the maximal force, plotted as a function of synaptic input levels. The slope and amplitude SD-input relationship of our new models are substantially dierent from those of Fuglevand model. c) CoV for force as a function of synaptic input levels. Our new model predicts substantially lower contribution of motor noise across all synaptic input levels tested here. Addition of a series-elastic element further reduces such contribution (blue line) at certain synaptic input levels. 80 Figure 19: Viscoelastic properties of the contractile element damps high-frequency os- cillations associated with discharge rate of motor units. Power spectra of output force at dierent synaptic input levels (5%, 20%, 40% and 70%) for our new model without a series-elastic element (SSE) in red and with SSE and blue. Note that addition of SSE substantially reduces power at frequencies > 5 Hz (shaded areas). 81 Motor noise cannot fully account for the experimentally observed amplitude of force variability. To further extend our results, we compare the amplitude of motor noise predicted from our new model to an experimental observation of force variability by Moritz and colleagues (2005) and assumptions made in previous theoretical models by (Jones et al., 2002) and (Todorov, 2005). Fig. 20 a&b show that the predicted motor noise by the new model is well below what has been observed experimentally (the black dotted line) across the entire possible range of synaptic input. Furthermore, our prediction highly deviates from the nature of motor noise assumed in previous theoretical models: steep, monotonic increases in SD of force (green and magenta lines in Fig. 20 a) and constant CoV (a green line Fig. 20 b). Note that the data from (Todorov, 2005) would be represented as a horizontal line at CoV of 12.9% in Fig. 20 b, but the data were omitted to maintain the resolution of the gure. These results altogether suggest that motor noise cannot account for the experimentally observed force variability nor justify the specic implementation of motor noise in many recent models of motor control. Motor unit functional organization is not a primary contributor to signal-dependent noise. Changing the values of three critical parameters dening the functional structure of a motor unit pool does not improve the ability of our model to produce realistic signal-dependent noise. In particular, we computed the eect of changing (1) the range of peak tetanic force (PR), (2) the range of recruitment thresholds (U r ), and (3) the number of motor units in a pool (N). These three parameters were proposed to be the primary determinants of signal dependent noise by Jones et al. (2002). On the one hand, Fig. 21 (a) shows that the model produces realistic monotonic increase in force production as synaptic input increases, even when PR, U r , and N are changed dramatically. In contrast, the SD of force does not follow the expected trend along the unity line (p = 1) in Fig. 21 (b) proposed by Jones et al. (2002). Moreover, Fig. 21c shows that the CoV of force also departs from the expected constant level 82 Figure 20: Motor noise cannot fully account for the experimentally observed amplitude of force variability. The amplitude of motor noise predicted by our new model is compared to the amplitude of force variability recorded from the rst dorsal interosseus muscle in 20 partic- ipants reported by Moritz and colleagues (2005). Furthermore, our prediction is then compared to assumptions made in previous theoretical models by (Jones et al., 2002) and (Todorov, 2005). Note that the predicted motor noise is smaller than the experimentally measured force variability (a black dotted line) the entire possible range of synaptic input and our prediction highly deviates from the nature of motor noise assumed in previous theoretical models (green and magenta lines). Figure adapted from Fig 6. in Moritz et al. (2005) 83 Figure 21: Signal-dependent noise is not the by-product of the organization of a motor unit pool. The range of peak tetanic force (PR), the range of recruitment thresholds (U r ), and the number of motor units in a pool (N) were altered to demonstrate the sensitivity of our results. a) Mean force as a function of synaptic input levels. Note that this relationship is non-linear and highly dependent on those model parameters. b) SD of force (% of maximal force) as a function of synaptic input levels. A solid black line represents the theoretical scaling factor of the SD-input relationship (p = 1 in Eq. 31). Also, this result is robust to important model parameters suggested to in uence the relationship (Jones et al., 2002; Moritz et al., 2005). c) CoV for force as a function of synaptic input levels. Only a change in the number of motor units in a pool (orange line) had an appreciable in uence on CoV for force. 84 with increase in synaptic input (Jones et al., 2002). On the contrary, the increased number of motor units in a pool to N = 300 from N = 120 substantially lowered the CoV (and lower SD as well), likely through a more eective spatio-temporal averaging of stochastic force twitches across motor units (Dideriksen et al., 2012). This result is important because the amplitude of motor noise is assumed constant across all muscles regardless of their functional organization (Diedrichsen et al., 2010; Harris and Wolpert, 1998; Scott, 2004; Seethapathi and Srinivasan, 2019; Sternad, 2018; Therrien et al., 2018; Todorov, 2005; Todorov and Jordan, 2002; van Beers, 2009; van Beers et al., 2004; Wu et al., 2014). These results strongly suggest that the functional organization of a motor unit pool is not a primary contributor to the features of signal-dependent noise consistently observed experimentally across dierent muscles (Jones et al., 2002; Slifkin and Newell, 2000). This questions the central role functional organization of motor units plays in theoretical and computational models of muscle function (especially motor noise amplitude) Jones et al. (2002); Todorov (2005). Discussion Physiological Mechanisms of Motor Noise The concept of motor noise is based on two experimental observations: unfused tetanic contraction and stochasticity in motoneuron discharges (Faisal et al., 2008). We demonstrate here that these mechanisms are insucient to account for even the majority of experimentally observed force variability. First, high-frequency force uctuations associated with unfused tetanic contraction have limited eects on the overall amplitude of force variability because they are low-pass ltered through 1) the activation dynamics of muscle bers (i.e., calcium kinetics and cross-bridge turnover), 2) spatial ltering through summation of motor unit forces and 3) the contraction dynamics of an elastic musculotendon unit (Fig.17 & 19). Furthermore, the activation dynamics of muscle bers causes progressive fusion of motor unit twitches as the discharge rate increases, resulting in progressive decreases, rather than increases, in the amplitude of motor noise as force levels increase (Fig. 18). All of these results argue against the signicance of unfused tetanic contraction in the generation 85 of signal-dependent motor noise when motor units discharge periodically and asynchronously during normal recruitment in most muscles (cf. Fig. 3 in (Brown et al., 1999)). This leaves stochasticity in motoneuron discharges as a potentially more important source of motor noise. Stochasticity in motor unit discharges has been assumed to stem from synaptic noise { random uctuations in membrane voltage due to stochastic arrivals of action potentials from many excitatory and inhibitory synaptic sources, all superimposed onto a constant synaptic input (Calvin and Stevens, 1968). Like unfused tetanic contraction, however, stochastic motor unit discharges cannot account for the experimentally observed features of force variability, nor can this mechanism explain the notion of `signal-dependent' noise as a fundamental property of motoneuron activation with which the neural controller needs to contend Harris and Wolpert (1998). Synaptic noise does not follow the relationship with its mean level that has been assumed in prior models (31). It is the variance of synaptic noise (i.e., square root of standard deviation), not its standard deviation, that increases proportionally to the mean input level (Matthews, 1996). Furthermore, the increase in synaptic noise does not necessarily translate into greater discharge variability of motoneurons. In fact, discharge variability due to synaptic noise actually decreases as the mean synaptic input level (and synaptic noise) increases (Matthews, 1996). This prediction is consistent with experimental observations in humans (Moritz et al., 2005; Person and Kudina, 1972; Tanji and Kato, 1973). Synaptic noise might cause high discharge variability due to sporadic discharges of near-threshold units, but such discharges would be likely only at low to mid levels of synaptic input where re- cruitment thresholds of motor units are close to each other and there exist many below-threshold motoneurons. Therefore, this type of sporadic ring becomes irrelevant at higher synaptic input levels in our simulations, where most units are already recruited. Even at low- to mid-levels of synaptic input, hysteresis in motor unit recruitment (lower synaptic current required to de-recruit than recruit a motor unit) may prevent sporadic discharge and promote self-sustained discharge (Binder et al., 2020). Low threshold units, which are most susceptible to sporadic discharge, are the most strongly aected by hysteresis-generating plateau potentials (Lee and Heckman, 1998a,b). In fact, recordings of single motor units in behaving animals and humans (e.g., Fig. 1 in Baweja et al. (2009) and Fig. 1 & 4 in Broman et al. (1985)) rarely if ever show sporadic discharges. 86 Finally, simultaneous changes in the muscle contraction speed would likely compensate for time- dependent changes in motoneuron discharge rates during constant voluntary activation (e.g., ring rate adaptation and fatigue). For example, decreases in the discharge rate of fast-twitch units during a fatiguing contraction are often accompanied by slowing of muscle contraction speed (prolongation of twitch rise- and fall- times) (Bigland-Ritchie et al., 1983). The combination of these changes would likely minimize changes in the amount of motor noise as slowing of muscle contraction speed would increase the degree of fusion at a given discharge rate (Fig. 15). Alternative sources of force variability: uctuating synaptic input due to feedback- driven control Experimentally observed uctuations in force variability are not appropriately characterized as random motor noise, as discussed above. It is critical then, to revisit physiological mechanisms through which force output is actually controlled. Without veridical feedback, motor output (i.e., force or discharge rate of a motor unit) tends to drift (e.g., Ambike et al. (2016); Smith et al. (2018)). It is important to note that the direction of force drift is not random, but rather very stereotypical (Smith et al., 2018). As a consequence, maintaining output at a constant level requires closed-loop control via sensory feedback (e.g. visual, auditory, etc.). Because of the inherent delay in our sensorimotor systems, closed-loop control in- troduces low-frequency (1-2 Hz) uctuations in synaptic drive and resulting muscle force (Baweja et al., 2010, 2009; Keenan et al., 2017; Slifkin and Newell, 2000; Smith et al., 2018). Similarly, pro- prioceptive feedback modulates synaptic inputs in the 5-12 Hz range due to its interactions with musculotendon mechanics (Nagamori et al., 2018). Furthermore, descending inputs may contain oscillations from distinct sources, such as the alpha-band (8-15 Hz) arising from the subcortical structures (e.g., Gross et al. (2002); Grosse and Brown (2003); Soteropoulos and Baker (2006)) and the beta-band (15-30 Hz) and gamma-bands (35-70 Hz) arising from the cortical structures (e.g., Brown et al. (1998); de Vries et al. (2016)). These oscillations from various sources may also add to or modulate the low-frequency (< 5Hz) component of synaptic drive/force (Nagamori et al., 2018; Watanabe and Kohn, 2015). Failure to consider these structured, dynamic uctuations in synaptic 87 input leaves discharge variability to be accounted for erroneously as the consequence of synaptic noise (Matthews, 1996) and modeled as a constant, random process. An additional consequence of an inevitably uctuating level of synaptic input is that these uctuations are often shared across the pool of motor units and cause synchronization among them (Farina and Negro, 2015). Correlated motor unit discharges are expected to have disproportionately large eects on force variability compared to asynchronous motor unit discharges. The eects of this would be largest below 10 Hz (Farina and Negro, 2015; Farina et al., 2014; Negro et al., 2016) due to the low-pass ltering properties of motor units (Fig. 7, cf. (Mannard and Stein, 1973)). Accordingly, studies have found the degree of motor unit synchronization below 5 Hz to be correlated with the amplitude of force variability (Castronovo et al., 2018; Negro et al., 2009; Pereira et al., 2019). Increased force variability due to aging (Castronovo et al., 2018; Pereira et al., 2019) and neurological conditions such as stroke (Lodha and Christou, 2017) and essential tremor (Neely et al., 2015) have been associated with increases in this low-frequency force variability. Furthermore, a recent study showed that the degree of synchronization below 5 Hz increases with contraction intensity (Castronovo et al., 2015), suggesting that increases in force variability at higher forces arise from increased low-frequency uctuations in the common synaptic inputs (rather than increased motor noise) probably due to shifts in high-level control in compensation for drift or fatigue (Smith et al., 2018). Finally, it is important to mention that, although synchronized activity above 5 Hz has limited eects on the overall amplitude of force variability in healthy neuromuscular systems, many neurological disorders have been associated with amplied synchronization/force variability above 5 Hz (e.g., Choudhury et al. (2018); Grosse et al. (2004); Neely et al. (2015); Salenius et al. (2002). Common synaptic inputs can also cause pairs of motor units to discharge nearly simultaneously within approximately 10 ms of each other (short-term synchronization/synchrony, Sears and Stagg (1976)). However, experimental data argue its eect is limited (De Luca et al., 1993), although a simulation study has suggested otherwise (Yao et al., 2000). Furthermore, the strength of short- term synchronization can dier quite drastically across muscles and even in the same muscles across tasks (Loeb et al., 1987). These observations highlight a need for further investigation of the neural mechanisms of short-term synchronization and its consequences on motor control. 88 Theoretical and clinical implications Our ndings argue for a reinterpretation of both motor noise and its impact on motor control and learning. We suggest that most muscle force variability and the resulting kinematic variability re ect properties of control strategies embodied through distributed sensorimotor systems. For example, in contrast to the conclusion made by Harris and Wolpert (1998), the stereotypical, bell- shaped velocity prole observed during reaching can arise in the absence of signal-dependent noise (Tsianos et al., 2014), suggesting that it is a property of biological control systems attempting to achieve accurate movements given biomechanical constraints of non-linear musculoskeletal systems (Gordon et al., 1994a,b). Furthermore, kinematic variability may not necessarily re ect the perfor- mance limitation imposed by the motor system as proposed by optimal control theories (Todorov, 2005; Todorov and Jordan, 2002; van Beers et al., 2004), but rather implies the tendency of the brain to use non-optimal, `habitual' control strategies (de Rugy et al., 2012; Loeb, 2012). This is consistent with experimental observations that various sensorimotor control circuits which in u- ence performance variability may be tuned as needed (as per the precision requirements of a task or dierent portions of a task) or when useful (e.g., purposeful exploration, persistence of excitation and active sensing) (Dhawale et al., 2017; Whiting, 1983). Rather than lumping such variability into a single coecient of variation, spectral analyses of variability should help to understand both the underlying mechanisms and their alterations after disease or injury. The experimentally ob- served tendency of the nervous system to maintain a constant total variance of force is itself in need of explanation. We suggest that force variability is a rich source of information about how the nervous system executes voluntary action; it is mostly not the result of constant, low-level, random noise. 89 Chapter 3: Synaptic noise is not a sucient explanation for force variability. Abstract Many popular theories of sensorimotor control invoke the concept of `noise' to create models that explain experimentally observed kinematic and kinetic variability. However, we and others have shown that the spectral properties of force variability re ect control strategies and properties in- herent to distributed sensorimotor systems. This debate in the literature arises because the phys- iological validity of the mechanisms used to implement noise in those previous theories have not been rigorously tested. In a prior study, we explored motor unit force generating mechanisms and demonstrated that the contribution of `signal-dependent motor noise' (from the stochastic nature of motoneuron action potential discharge and unfused tetanic contraction) is much smaller than previously assumed. Some may suggest yet another simpler explanation: that signal-dependent force variability could arise simply from signal-dependent noise in the motor command, or synaptic noise due to asynchronous, stochastic EPSPs and IPSPs from many synaptic sources. Here we explore that possible explanation, by extending our previous model of the motor unit force gen- erating mechanism to include a physiologically tenable model of the motoneuron spike generating dynamics. We nd that that simpler explanation does not hold. Specically, 1) Synaptic noise in- creases in proportion to the square root of the mean synaptic input level, and not in proportion to the mean synaptic input level assumed in previous theoretical models of signal-dependent noise. 2) Discharge variability of motor units decreases with increasing levels of synaptic inputs (i.e., `noise' decreases with increasing neural drive to muscle). 3) Hysteresis in motoneuron channels due to neuromodulatory inputs attenuates motor unit discharge variability and thereby its contribution to the amplitude of force variability. Importantly, our ndings about neuromodulatory inputs chal- lenge their previously proposed role as a simple gain amplier of noise to account for modulation of force variability with dierent levels of neuromodulatory inputs. These results altogether call into question the simplistic notion of additive noise in the motor systems to account for experimentally observed force variability. Rather, force variability is a rich source of information about the con- 90 sequences and contributions of our hierarchical sensorimotor system to intended and unintended variability in voluntary action in health and disease. Introduction Variability in human movements have informed many theoretical models that attempt to uncover computational theories of human sensorimotor control. Many currently popular models invoke the concept of `signal-dependent motor noise' { standard deviation of muscle force output increases linearly with force levels{ to explain observed kinematic and kinetic variability. However, we and others have shown that the spectral properties of force variability re ect control strategies and properties inherent to distributed sensorimotor systems (e.g., Keenan et al. (2017); Laine et al. (2013, 2014); Nagamori et al. (2018); Shinohara et al. (2005); Slifkin et al. (2000)). This apparent con ict in the current literature arises because the physiological validity of implemented noise in those previous theories and models has not bee rigorously tested. In fact, the concept of noise is based on premature interpretation and lack of validation of computational models of two experimental phenomena: unfused tetanic contractions of muscle and stochasticity in motoneuron discharges. In Chapter 3, we have demonstrated that those two mechanisms cannot account for the experimentally observed amplitude of force variability nor its signal dependence (Nagamori et al., in review), suggesting that force variability and its signal dependence are not a natural by-product of the motor unit force generating mechanism and the organization of the motor unit population, which argue against previous conclusions and interpretations (Jones et al., 2002). However, some might instead argue for another, potentially simpler, explanation that signal- dependent noise in force output requires signal-dependent noise in the motor command to muscle (Jones et al., 2002). This simpler explanation only requires a signicant amount of noise in synap- tic input which follows the statistical properties assumed for signal dependent noise in previous models (Harris and Wolpert, 1998; Jones et al., 2002; Todorov, 2005; Todorov and Jordan, 2002). Furthermore, the degree of the resulting stochasticity in motoneuron discharges must continuously increase for the amplitude of force variability to follow signal-dependent noise (Fig.6 in Nagamori et al. in review). However, while tenable, the physiological mechanisms for such source of noise 91 have not been examined carefully. The most plausible, if not the only, source of signal dependent noise in synaptic input is synaptic noise. Intracellularly recorded motoneurons display apparently random membrane voltage uctu- ations in motoneurons, which is thought to be synaptic bombardment by many asynchronous, stochastic excitatory and inhibitory inputs (i.e., synaptic noise) (Calvin and Stevens, 1968). This observation may stand to reason that signal-dependent noise in force arises from such synaptic noise. However, it is unlikely that the existence of synaptic noise would justify signal dependent input noise. First, the variance of a stochastic process (i.e., Poisson process) is equal to its mean (Tkachenko, 2006). In other words, its standard deviation is proportional to the square root of its mean. Second, some experimental observations of motor unit discharge patterns during volun- tary contraction in humans show that coecient of variation (CoV) of inter-spike intervals (ISIs) decreases with increasing force levels (Matthews, 1996; Moritz et al., 2005). This pattern is likely a consequence of high discharge variability at lower force levels (i.e., lower levels of synaptic in- puts) due to unstable recruitment of near-threshold motor units by synaptic noise (Matthews, 1996). Finally, hysteresis in motoneuron causes de-recruitment current threshold to be lower than recruitment current threshold (Binder et al., 2020)), which may prevent extremely high discharge variability (>50%) of motor units due to their unstable recruitment by synaptic noise, which would otherwise substantially increase the level of `noise' in motor unit discharge and the resulting force variability. To explore these mechanisms in detail, we systematically examine eects of synaptic noise on motor unit discharge patterns and quantify their eects on the resulting force variability. To this end, we extended our previous model (Nagamori et al. in review) to include a realistic model of motoneurons, each speed-matched to muscle bers they innervate. We demonstrate that signal- dependent noise in synaptic input due to synaptic noise cannot, in fact, account for the amplitude of force variability nor for its signal dependence. We do so by demonstrating the following: 1) Stan- dard deviation of noise in synaptic current to individual motoneurons imposed by asynchronous, stochastic synaptic inputs increases in proportion to the square root of mean input levels, rather than mean input levels assumed in the concept of signal-dependent noise (Harris and Wolpert, 1998; Todorov and Jordan, 2002). 2) The degree of stochasticity in motor unit discharges would 92 decrease with increasing input levels in spite of increases in input noise. 3) Hysteresis in motoneu- ron channels due to intrinsic properties of motoneurons prevents sporadic discharges and reduce discharge variability. Our results demonstrate that the experimentally observed force variability is not a by-produce of noise in the muscle force generation mechanism or in the motor command. These results reveal the insuciency of the simplistic assumption that force variability arises from `noise' in our mo- tor systems. Rather, force variability likely re ects control properties of distributed sensorimotor systems. This view suggests that force variability contains a rich source of information about how the nervous system executes voluntary action, and its disruption in healthy aging and neurological conditions. Material and Methods Model of a population of motor units The critical component that we added to our previous model (Nagamori et al. in Review) is an input/output model of the motoneuron. This component is necessary to simulate dynamic interactions between non-stationary, uctuating synaptic input and properties of motoneurons that determine how noise in synaptic input in uences the output action potential discharge patterns, and the resulting force variability. Motor unit model Each motoneuron model, described below, innervates a group of muscle bers, which makes up a motor unit. The model of the motor unit force generating process converts a spike train of motoneuron discharge patterns into motor unit force (Fig. 12). The full descriptions and equations are given in our previous publication (Nagamori et al. in Review), so only brief descriptions are given here. This model consists of two modules. Module 1 turns spike trains into motor unit activation 93 (active force generating state of cross-bridges), A, through a three-Stage process. In Stage 1, individual action potentials (R) trigger calcium kinetics, which is described using ve states, [s], [cs], [c], [f] and [cf] with associated rate constants (k 1 ,k 2 ,k 3 and k 4 ) between those states. Stage 2 converts [cf] into the intermediate activation, e A, through a non-linear lter, which describes cooperativity and saturation of calcium binding and cross-bridge formation with two additional parameters,N andK. We added the sag and yield properties of slow and fast-motor units (Brown et al., 1999; Brown and Loeb, 2000) in this stage. Stage 3 introduces an additional rst-order dynamics of cross-bridge turnover to generate motor unit activation, A. At this point we introduce the known length-dependence of activation-frequency relationship (Brown et al., 1999) by length- dependence on four free parameters, k 3 , k 4 , N and K. Module 2 outputs tendon force, F se , by accounting for the contraction dynamics between a muscle and a series elastic element representing the tendon and aponeurosis (Eq. 3.21). The muscle force, the sum of all motor unit forces, is scaled by force-length and force-velocity relation- ships as well as two passive parallel elastic elements (Eq. 3.21) described previously (Song et al., 2008b). We adjusted model free parameters to simulate various properties of individual motor units (e.g., contraction time, activation-frequency relationship, etc.). This allows us to create a population of motor units that express dierent ber types (i.e., slow and fast) and a spectrum of contraction speed. In this study, we assumed a approximately constant twitch-tetanus ratio of 0.23 based on the result from a whole muscle preparation of feline caudofemoralis reported by (Brown and Loeb, 2000). Motoneuron model The motoneuron model is based on pulse-based approximation of the Hudgkin-Huxely model of a neuron originally proposed by (Destexhe, 1997) and extended to a two-compartment model of motoneuron by (Cisi and Kohn, 2008). The equivalent electrical circuit is presented in Fig.22. The membrane potentials of the dendritic and somatic compartments, V d and V s , respectively, 94 Table 3: Model parameters for the motoneuron model: Rheobase current, I r , (nA) 1 { 32 Membrane specic capacitance, C m , ( F=cm 2 ) 1 Dendrite specic resistance, R d , (k cm 2 ) 15 { 2.5 Soma specic resistance, R c , (k cm 2 ) 0.6 { 0.1 Soma surface area, A c , (m 2 ) 6000 { 15000 Dendrite-soma surface area ratio, p 38 Sodium conductance, g Na , (mV ) 30 Fast potassium conductance, g Kf , (mV ) slow-twitch: 4 fast-twitch: 0.5 Slow potassium conductance, g Ks , (mV ) 167.6 { 41.1 Leakage Nernst potential, K l , (mV ) 0 Sodium equilibrium potential, K Na , (mV ) 120 Potassium equilibrium potential, K K , (mV ) -10 M , M (1=ms) 22, 13 H , H (1=ms) 0.5, 4 N , N (1=ms) 1.5, 0.1 Q , Q (1=ms) 2.52 { 1.08, 0.071 { 0.014 95 Figure 22: Equivalent electrical circuit of the motoneuron model. A motoneuron consists of two-compartments: soma and dendrite. The membrane potentials of each compartment, V s and V d , respectively, are described by the system of coupled dierential equations in Eq. 33. Synaptic inputs, I syn , are injected through the dendritic compartment. Model parameters are described in texts and in Table 3. are described by the following system of coupled dierential equations using the parameters in Table 3: C d dV d (t) dt =I eff (t)g ld (V d (t)E l )g cd (V d (t)V s (t)) (33) C s dV s (t) dt =I ion (t)g ls (V d (t)E l )g cs (V s (t)V d (t)) (34) I ion (t) = g Na m(t) 3 h(t)(V s (t)E Na ) + g Kf n(t) 4 (V s (t)E K ) + g Ks q(t) 2 (V s (t)E K ) (35) C d =C m A d (36) C s =C m A s (37) g ld =A d =R d (38) g ls =A s =R s (39) g c = 2 R i l d r 2 d + R i ls r 2 s : (40) The dynamics of dendritic membrane potential (Eq. 3.1) is driven by the synaptic input current, 96 I eff , and the currents generated through the dendritic leak conductance (Eq. 3.6), g ld , and the coupled conductance (Eq. 3.8), g c . The dynamics of somatic membrane potential (Eq. 3.2) is governed by the currents generated through the somatic leak conductance (Eq. 3.7), g ls , and the coupled conductance (Eq. 3.8), g c , as well as the current from voltage-dependent ionic conduc- tances, I ion , described in Eq. 3.3. We include voltage-dependent sodium conductance, g Na , and fast and slow potassium conductances, g Kf and g Ks , as in a previously described model (Cisi and Kohn, 2008). The dynamics of these voltage-dependent ionic conductances are simulated using the time-dependent state variables,m,h,n, andq with pulse approximation (pulse duration of 0.6 ms) as done previously (Cisi and Kohn, 2008; Destexhe, 1997). Before and after the pulse, the unforced dynamics of the state variables are described by: m(t) =m 0 exp( M (tt 0 )) (41) h(t) = 1 + (h 0 1)exp( H (tt 0 )) (42) n(t) =n 0 exp( N (tt 0 )) (43) q(t) =q 0 exp( Q (tt 0 )); (44) where t 0 is the time at which the previous pulse ended and m 0 , h 0 , n 0 , and q 0 are the values of those state variables at t 0 . During the pulse, the forced dynamics of the state variables are described by: m(t) = 1 + (m 0 1)exp( M (tt 0 )) (45) h(t) =h 0 exp( H (tt 0 )) (46) n(t) = 1 + (n 0 1)exp( N (tt 0 )) (47) q(t) = 1 + (q 0 1)exp( Q (tt 0 )); (48) wheret 0 is in this case the time at which the pulse started and again m 0 ,h 0 ,n 0 , andq 0 are the 97 values of those state variables at t 0 . Model parameters associated with passive electrical properties: The geometry of the motoneuron (i.e., the size of soma and dendrite) and passive membrane properties (i.e., specic capacitance and resistance) determines two passive electrical properties of motoneurons: input resistance and membrane time constant. The range and distribution of those parameters were determined based on previous experimental studies and adjusted to replicate experimentally recorded values of input resistance and membrane time constant (see Results). The specic membrane capacitance,C m , is often assumed constant across most biological mem- branes and equal to 1 F=cm 2 (Moore et al., 1966), which we assumed here for all motoneurons. Much higher values of C m have been reported for motoneurons, ranging from 1.5 to 4 F=cm 2 with a mean value of 2.9 (Moore et al., 1966). However, these estimates seemed to have arisen from an incorrect assumption of uniform specic membrane resistance, R m , across somatic and dendritic compartments (Moore et al., 1966). In fact, we found the C m value of 1 F=cm 2 with asymmetrical R m across compartments yielded input resistance and membrane time constant of motoneurons much closer to their experimentally reported values. The value of R m of a whole motoneuron, estimated from the geometry of the motoneuron and its input resistance to injected current to soma, ranges from 1200 to 8000 =cm 2 (Barrett and Crill, 1974; Moore et al., 1966; Rall, 1959; Ulfhake, 1984). These previous studies assume the con- stant specic membrane resistance for both somatic and dendritic compartments, which seemed to have contributed to higher C m values reported previously (Moore, 1976). Alternatively, it has been suggested the uneven specic membrane resistance between the two compartments where the dendritic specic membrane resistance is much higher than somatic one (Clements and Redman, 1989; Fleshman et al., 1988; Iansek and Redman, 1973; Ito and Oshima, 1965; Rall, 1959; Rose and Vanner, 1988). As discussed above, we found the uneven specic membrane resistance more accu- rately replicated previously reported ranges of input resistance and membrane time constant and therefore assumed as done previously (Cisi and Kohn, 2008). The wide range of the dendrite-soma ratio of specic membrane resistance has been estimated using various experimental measurements (Clements and Redman, 1989; Fleshman et al., 1988; Iansek and Redman, 1973; Ito and Oshima, 98 1965; Luscher and Clamann, 1992; Rall, 1959; Rose and Vanner, 1988). Here, we assumed the specic membrane resistance of soma to range from 0.1 to 0.7 k =cm 2 (Fleshman et al., 1988; Kernell and Zwaagstra, 1989) and the specic membrane resistance of dendrite to be 25 times that of soma (Kernell and Zwaagstra, 1989). To best of our knowledge, there is no experimental data on the distribution of somatic and dendritic membrane resistance of motoneurons and therefore, we used an exponential distribution with their minimal and maximal values shown in Table 3 to replicate the reported distribution of input resistance (Fleshman et al., 1981; Powers and Binder, 1985; Zengel et al., 1985). The experimentally measured surface area of soma,A c ranges from 6000 to 19000m 2 (Barrett and Crill, 1974; Kernell and Zwaagstra, 1989; Luscher and Clamann, 1992; Ulfhake, 1982, 1983, 1984). The soma size measured by the soma diameter is related to the soma surface area (Ulfhake, 1984) and follows a normal distribution across dierent muscle ber types where slower motoneurons (as per conduction velocity and twitch contraction time) tend to be smaller than faster motoneurons (Burke, 2011; Luscher and Clamann, 1992; Ulfhake, 1983, 1984). Accordingly, we created normally distributed somatic surface areas which ranges from 6000 to 19000 m 2 . The size of 1-st order dendrites in terms of their diameters is positively correlated with the size of the soma and it is in turn closely related to the total dendritic surface area (Luscher and Clamann, 1992; Ulfhake and Kellerth, 1981). Thus, we assume the constant ratio of the dendritic surface area to somatic surface of 38, the mean value reported across various cat hind limb motoneurons by Ulfhake (1984) and others (Luscher and Clamann, 1992). Quantication of passive electrical properties of motoneurons: Given the parameters described above and shown in Table 3, we quantied input resistance, R N , and membrane time constant, 0 , of individual motoneurons. Those were measured from voltage changes of the soma compartment in response to 1-nA current pulse of 200-ms duration (Zengel et al., 1985). R N was calculated as the maximal voltage amplitude achieved during the current pulse divided by the amplitude of injected current (i.e., 1-nA). 0 was computed using the method described in detail by Ulfhake (1984). Brie y, we dierentiated the somatic membrane voltage signal with respect to time, log- transformed the resulting signal and then calculated the reciprocal of its slope with respect to time within a 5-15 ms time window after the pulse initiation. 99 Model parameters associated with active electrical properties: Combined with the passive elec- trical properties mentioned above, the voltage threshold and voltage-dependent ionic conductances of a given motoneuron determine the level of synaptic current at which the motoneuron initiates repetitive discharge (I min ), the discharge frequency atI min (MDR), and the slope of the discharge frequency-current relationship. The range and distribution of model parameters associated with those active electrical properties were determined based on previous studies and adjusted to t the desired MDR dened by mechanical properties of muscle bers that individual motoneurons innervate as described in the following. The value of rheobase current (I r ), the amplitude of injected current that initiates repetitive discharge, ranges from 1 to 30-40 nA across various muscles (Fleshman et al., 1981; Frank and Fuortes, 1956; Gustafsson and Pinter, 1984; Heckman and Binder, 1988; Kernell and Monster, 1981; Powers and Binder, 1985; Ulfhake, 1984). Its value is lower for type-identied motoneurons that innervate slow-twitch muscle bers than those that innervate fast-twitch muscle bers (Fleshman et al., 1981; Zengel et al., 1985). Yet, rheobase current across dierent types of motoneurons exist in continuum, rather than discrete groups, and follows a distribution skewed to higher rheobase current (Fleshman et al., 1981; Heckman and Binder, 1988; Powers and Binder, 1985). Accordingly, we assumed rheobase current of individual motoneurons increases with their size from 1 to 32 nA, whose distribution follows a Rayleigh distribution. Rheobase current of a given motoneuron was then multiplied with its respective R N to obtain its voltage threshold, V t h. The slope of the frequency-current relationship of individual motoneurons in response to somatic current injection is approximately constant (1.5Hz=nA) regardless of their size and the contraction speed of the muscle unit they innervate (Kernell, 1979; Schwindt, 1973). Thus, we assumed the constant slope of 1.5 Hz=nA for all motoneurons. In our motoneuron model, slow potassium conductance, g Ks , determines the slope of the frequency-current relationship. We initialized the value of g Ks based on Cisi and Kohn (2008) and then adjusted it such that we achieve the constant slope of the frequency-current relationship of 1.5 Hz=nA for all motoneurons. Other parameters associated with the voltage-dependent ionic conductances were initially set to those from a previous study by Cisi and Kohn (2008). 100 In our simulations, we model all synaptic inputs as dendritic injected currents (Heckman and Enoka, 2012). As such, I min was determined using dendritic current pulses of 4 sec. The values of Q and Q were adjusted to replicate the target minimal discharge rate of a given motoneuron at I min dened by the activation-frequency relationship of the muscle bers that it innervates. We did so by a custom written gradient descent algorithm that adjusts those parameters simultaneously with decreasing step changes for each iteration. Quantication of active electrical properties of motoneurons: To compare our motoneuron model to experimentally reported properties of motoneurons, we determined the amplitude, duration, and half-decay time of after-hyperpolarization (AHP) using the method described in Zengel et al. (1985). To this end, we applied a supra-threshold current pulse (100 nA) of 0.5 ms duration into the somatic compartment. We computed the amplitude of AHP as the minimal somatic membrane potential below the equilibrium potential of 0 mV. The AHP duration was determined as the duration of the somatic membrane potential to return to 0 mV (with 0.1 mV accuracy) from the action potential onset. Finally, the half decay time of AHP was computed as the duration of the somatic membrane potential to return to half the AHP amplitude from the time point of the minimal somatic membrane potential. We determined the slope of the frequency-current relationship for both somatic and dendritic current injections by plotting discharge frequencies as a function of injected current amplitudes ranging fromI min to the level of synaptic current at which the motoneuron reaches its predetermined peak discharge frequency, I max . We used polyt in MATLAB with the order of 1. Model of synaptic inputs to a population of motoneurons Eective synaptic current: The synaptic input to a motor unit population, U (a value between 0 and 1), is transformed into the eective synaptic current, I i eff , to the i-th individual motoneuron using the following equation: 101 I i eff =a i 1exp U b i : (49) The coecients, a i and b i , were determined by tting the above equation for each motoneuron such that I i eff corresponds to I i min when U =U i th and I i max when U = 1. Recruitment thresholds of individual motoneurons,U i th , were determined by dividingI i min by the maximal current required for all motoneurons to reach their peak discharge frequencies (i.e., 114.5 nA in our model). It is important to note that I i eff varies across motoneurons. We chose this method over applying the same synaptic current amplitude to all motoneurons as done in previous studies (Cisi and Kohn, 2008; Elias et al., 2014) for the following reasons. First, the relationship between the synaptic input, U, and discharge rates of individual motor units qual- itatively mimic that observed in human motor units (Fuglevand et al., 2015; Heckman and Binder, 1991b; Kanosue et al., 1979; Monster and Chan, 1977; Moritz et al., 2005): namely rapid accel- eration and saturation of discharge rates of lower-threshold motor units and progressive increases in discharge rates of higher-threshold motor units up to the maximum voluntary contraction (Fig. 23a). Second, the force-input relationship of the motor unit pool follows a sigmodal relationship, consistent with previous experimental data (Brown et al., 1999). Model of stochasticity in motoneuron discharges Discharge patterns of motor units during a constant, isometric force production task display stochas- ticity (e.g., Clamann (1969)). The degree of stochasticity is often quantied as the coecient of variation (CoV) of inter-spike intervals (ISIs), which typically ranges from 8 - 30 % of CoV of ISIs (e.g., Clamann (1969); Matthews (1996); Nordstrom et al. (1992); Nordstrom and Miles (1991)). The sources of such stochastic discharge patterns of motor units can be divided into intrinsic and extrinsic factors at the motoneuron level. The intrinsic factors are related to noise in the spike-generating mechanisms of a motoneuron such as uctuations in the voltage threshold and time-course of AHP. The extrinsic factors concern with uctuations in synaptic inputs, or synaptic 102 Figure 23: Input-output relationships of a motor unit population. a) Discharge rates of ten motor units (ve slow and ve fast-twitch units) as a function of synaptic input levels. b) Force output of a motor unit population as a function of synaptic input levels. 103 noise. Although quantifying the relative contribution of those two sources is critical to under- stand the eects of synaptic activity on force variability, little attention has been given to them previously. Synaptic noise: To model synaptic noise, we rst quantied statistical properties of synaptic noise expected from bombardment of many stochastic excitatory (EPSP) and inhibitory (IPSP) postsynaptic potentials. Here, we assume that 100 excitatory and inhibitory pre-synaptic neurons (an extremely conservative model given that motor units are estimated to receive synapses from 10-20,000 independent sources (Hultborn and Fedirchuk, 2009)) synapse onto a single motoneuron (the lowest threshold motoneuron from the population described above). The activity pattern of those pre-synaptic neurons was modeled as a Poisson process with a given discharge rate in 1- sec time windows. This method has been suggested as an adequate description of the combined synaptic input from many sources insofar as individual synaptic inputs can be described as a renewal process and are independent (Burkitt, 2006). The synaptic current was modeled as excitatory and inhibitory conductances for each synaptic contact as follows: I syn (t) = n X i=1 (g syni (t)(E syni V d (t))); (50) whereg syni is the synaptic conductance of the i-th presynaptic neuron andE syni is the reversal potential of the i-th presynaptic neuron (70 mV for excitatory and -16 mV inhibitory synapses) (Cisi and Kohn, 2008). The time course of g syni was modeled using the following equation (Cisi and Kohn, 2008; Destexhe et al., 1994): g syni (t) =g max r(t); (51) whereg max is the maximum conductance of the synapse (0:1nS) andr t describes the time course of conductance change (Cisi and Kohn, 2008; Destexhe et al., 1994). The same parameters as in 104 (Destexhe et al., 1994) were used (i.e., = 2msec 1, = 1msec 1 and T m ax = 1mM, cf. Eq.4-5 in Destexhe et al. (1994)). We varied the amplitude of synaptic current by altering the discharge rate of both the excitatory and inhibitory presynaptic neurons from 0.5 Hz to 100 Hz to quantify standard deviation of synaptic noise that a motoneuron receives and the resulting motoneuron discharge variability. This balanced state seems to be a default pattern to drive motoneurons (Berg et al., 2007) (however, see (Johnson et al., 2017; Powers et al., 2012)). We applied such synaptic input for 4 s and analyzed the last 3-s segment. We ran 10 trials for each input level and average motoneuron inputs and outputs across the 10 trials are presented for each variable. The relationship between the mean and standard deviation (SD) of synaptic input was characterized using the following equation: SD =amean b +c; (52) wherex andy are the coecients,a,b andc, were found using curve tting in MATLAB. For the rest of simulations presented in this study, we synaptic noise as randomly uctuat- ing synaptic current with a Gaussian distribution superimposed on the eective synaptic current. Statistical properties of such current noise were t to follow those described in Fig. 27 shown in Results. To this end, the standard deviation of synaptic noise was scaled to 80% of the square root of the eective synaptic current amplitude to individual motoneurons. With this noise level in the presence of intrinsic noise described below, the minimum value of CoV of ISIs was found to be8% (7.4-8.8 %), which corresponds to the lowest value observed experimentally in soleus muscle (Matthews, 1996). The desired autocorrelation of synaptic noise (a blue line in Fig. 27d) was achieved by ltering a set of random numbers generated from a Gaussian distribution with a rst-order dierential equation with time constant of 1 ms (a orange line in Fig. 27d). Intrinsic sources of stochasticity in motoneuron channels: It is often assumed that the observed CoV of ISIs in human motor units arise entirely from extrinsic synaptic noise (Calvin and Stevens, 105 1967, 1968; Fuglevand et al., 1993; Matthews, 1996, 1999). Yet, it is likely that noise intrinsic to the spike-generating mechanisms (Eq. 33) also contributes to at least a portion of the stochasticity (Diba, 2004; Hubbard et al., 1967; Mainen and Sejnowski, 1995). To quantify the relative con- tribution of the intrinsic sources to motoneuron stochasticity, we leveraged the method and data presented by (Binder and Powers, 2001). In that study, the authors intracellularly injected the same prole of noisy synaptic input into the same motoneuron on repeated trials and computed the degree of short-term synchronization (see the denition below) between the resulting spike trains across trials. If there is no intrinsic noise, then we should expect perfect synchronization (i.e., all spike timings would be identical) between a pair of spike trains. However, they observed the degree of short-term synchronization was far from perfect ( 40%) and noted the need for noise in the force-generating mechanisms of motoneurons in their subsequent simulations. To assess the extent of intrinsic noise needed in our simulation to replicate those experimental results, we performed a similar simulation following their method. In this set of simulations, we injected a 4-s current step with superimposed noise into the soma compartment of a motoneuron with the lowest recruitment threshold described above. The amplitude of current step was set to I min (i.e., 1:2nA). The superimposed noise was modeled as a white Gaussian noise ltered by a rst-order dierential equation with time constant of 5 ms with the standard deviation 1:1nA. The resultant prole of the noise approximates the autocorrelation and power spectrum of noise reported in Fig. 1 in Binder and Powers (2001). We recorded 10 pairs of trials for two conditions: without intrinsic noise and with intrinsic noise. We then computed the degree of short-term synchronization for each pair. Short-term synchronization (or short-term synchrony) is a time-domain measure of synchronous discharge of a pair of motor units 1 . It concerns the occurrence of near-simultaneous discharges of a pair of motor units (usually within around10 ms with respect to the discharge timing of the reference motor unit) (Sears and Stagg, 1976). Above chance level near simultaneous discharges have been commonly observed within a motor unit pool in humans during voluntary isometric force productions (e.g., Bremner et al. (1991a,b,c); Datta and Stephens (1990); De Luca et al. (1993); McIsaac and Fuglevand (2008); Milner-Brown and Lee (1975); Nordstrom et al. (1992); Schmied et al. (1993)). This type of synchronization is thought to arise from branched axons of common 106 pre-synaptic neurons (Datta and Stephens, 1990; Sears and Stagg, 1976), likely of the central origin (Datta et al., 1991; Fuglevand, 2011; Kirkwood et al., 1982; McIsaac and Fuglevand, 2006). To quantify the degree of short-term synchronization, a cross-correlation histogram was com- puted between a pair of spike trains with a 1-ms binwidth and time lags of100 ms (Fig. 24a). The count of simultaneous discharges in each bin was normalized by subtracting the baseline bin count (mean count in lag times more than40 ms) (Binder and Powers, 2001). Cumulative sum of the normalized histogram (CUSUM) was then generated (Ellaway, 1978) to identify the duration of the peak in the histogram within10-ms time lags (Fig. 24a). Time lags at which the minimal and maximal values of the CUSUM over this range occurred were used to dene the duration of the peak for further analyses. Various indices have been proposed to quantify the degree of short-term synchronization (see the Methods of Nordstrom et al. (1992)). For simplicity, we used two commonly used indices, E and CIS. The index, E, represents the proportion of extra simultaneous discharges within the duration of the peak area dened above with respect to the total number of discharges of reference unit (Datta et al., 1991; Datta and Stephens, 1990), which can be described as the following equation (Binder and Powers, 2001): AB p R ; (53) where A is the sum of added spikes in the peak area with respect to the baseline (i.e., the dierence in the minimal and maximal values of the CUMSUM), is the average count per bin the baseline,B p is the number of bins in the peak, andR is the number of spike counts of the reference unit. The reference unit was chosen arbitrarily from the pair. The second index, CIS, presents the number of extra simultaneous discharges above a chance level (i.e., baseline) per unit time (or second) (Nordstrom et al., 1992). CIS is dened as follows: 1 The dierence between short-term synchronization and coherence (a frequency-domain measure of synchroniza- tion) is nicely illustrated by Schmied et al. (2014). 107 Figure 24: Short-term synchronization to quantify the degree of intrinsic noise in our model of motoneurons. a) Cross-correlation histograms between spike trains of two motoneurons (top) and their cumulative sum of the normalized histograms (bottom). The solid line indicates the average response across 10 trials and shaded lines indicate individual trials. b) Comparison of the values of CIS between our model and the data from (Binder and Powers, 2001). The error bars indicate standard deviation. AB p T ; (54) where T is the duration of a trial in second. As expected, the condition without intrinsic noise produced perfect synchronization (E 1). This suggests that intrinsic noise is necessary to replicate the experimental nding where E 0:4 (Fig. 24b). Thus, we introduced two forms of intrinsic noise into our model of motoneurons as 108 done previously (Binder and Powers, 2001): random uctuations 1) in voltage threshold and 2) in membrane voltage during AHP. Random uctuations in voltage threshold were introduced by adding, at each time step, a random number to the default voltage threshold value drawn from a Gaussian distribution with standard deviation of 0.1 mV. The value of the standard deviation was taken from the value used in a previous study (Binder and Powers, 2001). The second source of intrinsic noise we introduced in the model was in the time course of membrane voltage during AHP. We did so by adding a white Gaussian noise ltered by a rst-order dierential equation with time constant of 5 ms to the AHP-related parameters, Q and Q in Eqs. 41 and 45. These parameters were chosen based on a previous nding by (Diba, 2004) and also our observation that other parameters did not have a signicant impact on discharge variability. The standard deviation of this noise was scaled to 8% of the default values of those parameters. With this set of parameters, we obtained a slightly higher value of E (0:53 0:12) compared to E 0:4 found in the cat motoneuron by Binder and Powers (2001) (Fig. 24b). The value of CIS we found from our simulation was 4:03 0:88, comparable to 4 by Binder and Powers (2001). Note that standard deviation ofCIS values found by Binder and Powers (2001) was slightly larger 1:1 than ours. We found these values reasonable as the injected to the same motoneuron across trials in the experiment may not be perfectly identical and there was some degree of background synaptic noise (Binder and Powers, 2001). Furthermore, the introduction of this intrinsic noise results in 4 % of CoV of ISIs in the absence of synaptic noise (i.e., response to a step injected current), which is similar to 5 to 10 % of CoV of ISIs reported by in a similar experimental condition (Mainen and Sejnowski, 1995; Manuel et al., 2006). Hysteresis in motoneuron channels The activity of voluntary activated motoneurons tends to show hysteresis in their discharge thresh- olds (e.g., Gorassini et al. (2002)). This hysteresis is characterized as lower synaptic current re- quired to de-recruit than recruit a motoneuron (Binder et al., 2020). Such hysteresis promotes self-sustained discharge (Binder et al., 2020) and can prevent sporadic discharge. This function is critical how synaptic noise in eunces discharge variability of motoneurons as it may reduce the contribution of synaptic noise at low- to mid-levels of synaptic input where recruitment thresholds 109 of motor units are close to each other and there exist many below-threshold motoneurons. The primary mechanism for the hysteresis is thought to be plateau potentials due to calcium- mediated persistent inward current (PIC) (Binder et al., 2020; Johnson et al., 2017). Here, we modeled the eect of PIC as a step current superimposed on the eective synaptic current as done in a previous simulation study (Revill and Fuglevand, 2011). We opted for this very simplied implementation of PIC over detailed ones (e.g., Powers and Heckman (2015, 2017)) because its parameters can be more easily manipulated and its results more easily interpreted, and it has been shown to be able to replicate the eect of PIC (Revill and Fuglevand, 2011). The eect of PIC in discharge patterns of motoneurons during sustained contraction simulated in this study is likely largest for low-threshold motoneurons, which most likely innervate slow- twitch muscle bers (Lee and Heckman, 2000). The PIC threshold for those motoneurons is lower or approximately equal to their discharge threshold (Lee and Heckman, 1998a; Li and Bennett, 2003). The eect of PIC can sustain for a long period of time (>10 s) once it is activated (Lee and Heckman, 1998a; Li and Bennett, 2003). The additional synaptic current generated by PIC can be fairly large (up to 30 nA) (Hamm et al., 2010; Lee and Heckman, 1998a, 1999). Deactivation of such current requires relatively strong inhibitory synaptic inputs (Hounsgaard et al., 1988) such as reciprocal inhibition of Ia aerent (average eective synaptic current of -1.66 nA; Heckman and Binder (1991a)). Comparatively, inhibitory action from Renshaw cells, whose average maximum eective synaptic current of -0.42 nA (Lindsay and Binder, 1991), may not be sucient to termi- nate PIC. Therefore, we assume that PIC for low-threshold motoneurons would be active during isometric contractions we simulated here for the entire duration of our simulations (10-sec holding phase). In contrast, the eect of PIC during sustained contraction may be limited to lower-threshold motoneurons despite their comparable PIC current amplitudes occasionally observed (Hamm et al., 2010; Lee and Heckman, 1998a,b). First, some high-threshold motoneurons may not even display PIC (Hamm et al., 2010) or self-sustained discharge longer than 2 sec (Lee and Heckman, 1998b). Furthermore, hysteresis is at best weak, if ever present in higher-threshold motoneurons (Lee and Heckman, 1998a,b). These observations may be due to the faster decay of PIC in those motoneurons 110 (Lee and Heckman, 1998b). Finally, the derecruitment threshold of PIC of those high-threshold motoneurons is slightly higher or equal to their recruitment threshold (Lee and Heckman, 1998a), making them more susceptible to inhibitory synaptic inputs of various strength. Taken altogether, we assume in our simulation that high-threshold motoneurons receive no PIC. Previous experimental observations have not provided any quantitative description we can read- ily apply to our model to relate the presence of PIC (or its amplitude) to individual motoneurons in a muscle. Therefore, we infer such a relationship based on the underlying functions that dierent types of motor units serve. The highly energetically ecient contraction dynamics (Barclay, 1996; Heilmann et al., 1981; Klug et al., 1983; SwartzS and Moss, 1992; Tsianos et al., 2012) and the well-developed oxidative pathway enable slow-twitch units to generate stable force output for a long period of time as in postural control. Self-sustained motoneuron discharges due to PIC are conducive to such a function (Heckman et al., 2005). Additionally, relatively strong hysteresis and reduced sensitivity to synaptic inputs can prevent random dropout of recruited motoneurons due to synaptic noise or inhibitory synaptic inputs. In contrast, fast-twitch muscle bers have adopted faster calcium kinetics and cross-bridge turn-over rates, which allows fast contraction and relaxation (Tsianos and Loeb, 2017), as required in tasks such as jumping and running. For such function, an initial sharp increase in PIC upon recruitment would be useful to accelerate motoneuron discharge rates (Binder et al., 2020; Johnson et al., 2017) and its quick decay would allow maintaining the high sensitivity of motoneurons to excitatory and inhibitory synaptic inputs, which is likely critical for task performance. All of these proposed functions t well with the experimentally observed properties of PIC discussed above. From this perspective, we assumed in our model that only low-threshold motoneurons that innervate slow-twitch bers receive PIC. Experimental data on how quickly PIC reaches its peak amplitude seem equivocal: some suggest immediate activation (Bennett et al., 2001a,b; Li and Bennett, 2003; Li et al., 2004) and others suggest relatively slow activation (>1 s) (Hounsgaard and Kiehn, 1989; Hultborn et al., 2003). Therefore, we made a simplied assumption here that PIC is activated immediately upon recruit- ment of a motoneuron as done previously (Revill and Fuglevand, 2011). This assumption is likely insignicant during constant, isometric contractions we study here. 111 The amplitude of PIC depends on the level of neuromodulatory drive (Lee and Heckman, 2000) and therefore it is dicult to assume its particular value for voluntary actions in humans. Thus, we tested at four dierent PIC amplitudes: 0 (default), 1, 3 and 5 nA. These values were at the lower end of experimentally reported PIC amplitudes (1-30 nA Hamm et al. (2010); Lee and Heckman (1998a, 1999)) and produced comparable values of F measures (see below for the method and Results for comparisons to experimental values). Simulation protocols We ran ten trials at each level of synaptic input (5% and from 10 to 100% with an increment of 10% maximal synaptic input). Each trial consisted of a 1-sec zero input phase, a 2-sec ramp-up and a 13-sec hold at a given level of synaptic input. Data and statistical analysis Force variability: To quantify mean, standard deviation (SD), and coecient of variation (CoV) of output force, we used the method compatible with that described in Moritz et al. (2005) to allow for a fair comparison between our result and their experimental results. To this end, the 10-sec hold phase of output force was divided into ten 1-sec segments, the duration of data used in Moritz et al. (2005). The mean of each segment was used for calculation of CoV. Force in each segment was then linearly de-trended and standard deviation was calculated from the de-trended data. CoV was calculated by dividing the standard deviation by the mean force of that segment before de-trending. Ten values of CoV of force for each trial were then averaged and the average CoV of force represents each data point. All gures present the average of 10 trials in a solid line with SD denoted by a shaded area. Motoneuron discharge variability: The degree of stochasticity in motoneuron discharges was quantied as coecient of variation (CoV) of inter-spike intervals (ISIs). Spike trains (binary signals with ones indicating the occurrence of action potential) of individual motoneurons were obtained from the 10-sec hold phase. ISIs were calculated as a time dierence between two successive 112 Figure 25: F measurement. The top panel shows the force output of the motor unit population in response to a triangular current with peak amplitude of 15% of the maximum synaptic current. The smoothed instantaneous discharge rates of a lower-threshold control unit and higher-threshold test unit are shown in the bottom panel. F was calculated as a dierence in discharge rates of the control unit at recruitment and derecruitment of the test unit as indicated in dotted, vertical lines. spikes. Eects of PIC: Hysteresis in motoneuron discharges induced by PIC was quantied by injecting a slow, triangular current input into the motor unit population model. The triangular current input was consisted of 5-sec rise and fall phases with the total duration of 10 sec, consistent with previous studies (e.g., Lee and Heckman (1998b)). The amplitude of the triangular current was set to 15% of the maximum synaptic current. To quantify the extent of hysteresis in our model, we used F measure, a dierence in discharge rates of the control unit at recruitment and derecruitment of the test unit (Gorassini et al., 2002; Revill and Fuglevand, 2011) (Fig. 25). The control unit refers to a motor unit with lower recruitment threshold in a pair. To this end, we assigned motor units whose recruitment threshold is below 5% of the maximal synaptic input as control units and those whose threshold is between 8-13% as test units (Gorassini et al., 2002). Instantaneous discharge rates of lower threshold control units, computed from two successive discharges, were smoothed with a ve-point moving average centered around that in- stantaneous discharge rate and linearly interpolated to produce a continuous discharge rate prole (Revill and Fuglevand, 2011). This continuous discharge rate prole was used to extract discharge rates of control units at recruitment and derecruitment of test units. The times at which higher threshold test units are recruited and derecruited were dened as the times at which the rst and 113 last spikes were observed in those motoneurons, respectively (Fig. 25). F was then computed as the discharge rate at derecruitment minus that at recruitment. Statistical analysis: Statistical analysis was performed in the R environment for statistical computing (The R Foundation for Statistical Computing, Vienna, Austria). Details of the statistical analysis performed are given in corresponding results sections. Results Setting the parameters for the electrical properties of motoneurons The main objectives of this study was to quantify eects of synaptic noise on motor unit discharge patterns and the resulting force variability. To achieve this objective, it is critical to develop a model of motoneurons that accurately implements the physiological mechanisms of synaptic noise as we understand them, and evaluate its discharge patters in response to non-stationary synaptic inputs. To this end, we rst adjusted parameters of our motoneuron model such that a range of both passive and active electrical properties of individual motoneurons are consistent with experimentally observed values as described below. Input resistance of motoneurons in our model ranges from 0.4 to 4.3 M , consistent with the values observed in a muscle of mixed ber types (e.g., gastrocnemius) (Burke, 1967; Coombs et al., 1959; Fleshman et al., 1981; Frank and Fuortes, 1956; Gustafsson and Pinter, 1984; Heckman and Binder, 1988; Kernell, 1966; Powers and Binder, 1985; Ulfhake, 1984). Its distribution follows a Gaussian distribution slightly skewed to the higher input resistance (Fig. 26a) as shown previously (Fleshman et al., 1981; Heckman and Binder, 1988; Powers and Binder, 1985). Furthermore, the average of membrane time constant of 4.8 ms is compatible with that observed experimentally (Bar- rett and Crill, 1974; Burke, 2011; Ulfhake, 1984). Its distribution follows a Gaussian distribution slightly skewed to the higher input resistance (Fig. 26b) as in Rose and Vanner (1988). Our model shows the distribution of recruitment threshold of motor units determined by their input resistance and rheobase current is skewed toward higher recruitment thresholds where there 114 are many more low-threshold motor units compared to high-threshold ones (Fig. 26c). Such a distribution has been well documented (Duchateau and Hainaut, 1990; Henneman et al., 1965b; Milner-Brown et al., 1973; Stephens and Usherwood, 1977). The time course of AHP plays an important role not only in regulating discharge rates of motoneurons to match the contraction speed of muscle bers that they innervate (Bakels and Kernell, 1993; Kernell et al., 1999) but also in determining eects of synaptic noise on discharge variability (Matthews, 1996; Powers et al., 2000). As such, we compare parameters associated with AHP obtained from our model to their experimental observations (Burke, 2011; Powers et al., 2000; Ulfhake, 1984; Zengel et al., 1985). The distributions of AHP duration, magnitude and half-decay time as well as the range of their values (Fig. 26d-f) are in accord with experimental observations for muscles with mixed ber types (Burke, 2011; Ulfhake, 1984; Zengel et al., 1985). Furthermore, the shape of the distributions reasonably approximates the previously observed ones (Zengel et al., 1985). Our motoneuron model also demonstrates the previously shown relationship of input resistance with other parameters such as AHP duration and the f-I slope. Fig. 26g shows that input resistance is positively correlated with AHP duration (r = 0:53) as shown by Kernell (1966). In contrast, we nd no correlation between input resistance and the frequency-current (f-I) slope (Fig. 26h), which is again consistent with previous observations (Kernell, 1979; Schwindt, 1973). These results conrm that our motoneuron model captures important properties of motoneurons (which have not necessarily been tested rigorously in previous studies) that would likely in uence how non- stationary synaptic input determines their discharge variability. Synaptic noise does not follow the pattern of signal-dependent noise Previous theoretical models have assumed that the standard deviation of synaptic noise increases linearly with mean input levels. However, we now show that the asynchronous, random synaptic in- puts from many excitatory and inhibitory sources, which constitute a major component of synaptic noise, do not follow such a trend. Fig. 27a demonstrates that increases in the amount of synaptic current cause increases in the associated noise, yet its standard deviation does so in a proportion to 115 Figure 26: Electrical properties of motoneurons. a) Distribution of input resistance. b) Distribution of membrane time constant. c) Distribution of recruitment threshold normalized to the maximal eective synaptic current required to recruit all motor units at their peak discharge rates. d) Distribution of AHP duration. e) Distribution of AHP magnitude. f) Distribution of AHP half-decay time. g) Relationship between input resistance and AHP duration. Note that these parameters are positively correlated (r = 0.55), consistent with a previous experimental observation (Kernell, 1966). h) Relationship between input resistance and frequency-current (f- I) slope. Note that there is no association between these parameters, consistent with previous experimental observations (Kernell, 1979; Schwindt, 1973). 116 the square root of its mean levels. This is dierent from the trend in signal dependent noise, where the standard deviation of muscle force output would increase linearly with mean synaptic current. As expected, increases in the amount of synaptic current, generated by increasing discharge rates of presynaptic neurons, lead to higher discharge rates of a given motoneuron (Fig. 27b). How- ever, the increased level of noise does not translate into higher discharge variability (i.e., CoV of ISIs) (Fig. 27c), demonstrating that `noise' in the nal component of our motor systems, which then determine the amplitude of force variability, decreases, rather than increases, with increasing synaptic input levels. These results suggest that a simplied concept of noise is not sucient to describe the theoretical relationship assumed in previous models. Synaptic noise does not generate signal dependent noise As one might expect from the results shown in Fig. 27a&c, standard deviation of force does not show the monotonic increases (Fig. 28) which would have been predicted by signal dependent noise (a green line in Fig. 28 from Jones et al. (2002)). Furthermore, the extent of force variability obtained from our new model deviates substantially from the amount of motor noise used in a previous model to account for experimentally observed kinematic variability (Todorov, 2005). It is important to note that the predicted motor noise by our new model is still lower than the experimentally observed force variability across the entire range of force levels except for the mid range (10-30% of maximal force). These results demonstrate that signal-dependent noise in force output cannot be accounted for simply by signal-dependent noise in synaptic input in contrast to previous simplied assumptions (Harris and Wolpert, 1998; Jones et al., 2002; Todorov, 2005; Todorov and Jordan, 2002). Intrinsic properties of motoneurons facilitate generation of stable force output A detailed analysis of motor unit discharge patterns reveal that sporadic discharge patterns of motor units and the resulting high discharge variability likely resulted in overestimation of the motor noise contribution at the mid range of mean force levels (10-30% of maximal force). The top panel of Fig. 29a shows a raster plot of all active motor units in response to noisy synaptic 117 Figure 27: Statistical properties of synaptic noise. a) The relationship between mean synaptic current and its standard deviation (SD) generated by asynchronous, random synaptic inputs. Note that this relationship is characterized by SD/mean 0:5 . b) Discharge rates of a motoneuron as a function of synaptic currents. c) Discharge variability of a motoneuron as a function of synaptic currents. Note that increasing the amount of synaptic current decreases discharge variability. d) Autocorrelation functions of simulated noise generated by asynchronous, random synaptic inputs and of synthetic noise used in the following simulations. 118 Figure 28: Synaptic noise does not generate signal dependent noise. Standard deviation of force variability is plotted as a function of mean force levels. The predicted motor noise is compared to experimental data by Moritz et al. (2005) and the theoretical relationship assumed in two previous models by Jones et al. (2002) and Todorov (2005). Note that the predicted motor noise by our model is lower than the experimentally observed force variability (a dotted line) at higher force levels (>40%). Also note that our model prediction deviates greatly from the amplitude of motor noise used in previous theoretical models (green and magenta lines). input at 5% of the maximal synaptic input. It is important to note that some units show very sporadic discharges (only a few spikes in a 10-sec period). These sporadic discharges result in very high values of coecient of variation of ISIs (>30% in Fig. 29b). These highly variable discharge patterns arise from random crossing of recruitment threshold of below-threshold units by noisy synaptic input. This eect is expected to be largest at the low to mid range of mean force levels where recruitment thresholds of motor units are close to each other and there exist many below- threshold motoneurons. This is consistent with the above-mentioned result where the contribution of motor noise was overestimated between 10 and 30% of maximal force. These sporadic discharges can be prevented by hysteresis in motoneuron channels that cause synaptic de-recruitment current to be lower than recruitment current, which has been observed experimentally during voluntary contraction in humans (see a recent review by Binder et al. (2020)). The bottom panel of Fig. 29a shows discharge patterns of all active motor units in the presence of such hysteresis induced by 3nA of extra synaptic current injected at the onset of motor unit recruitment. Note that all units discharge very regularly and CoV of ISIs are conned within the range between 8-30% of CoV of ISIs (Fig. 29b), which is consistent with the range often observed 119 experimentally (e.g., Clamann (1969); Nordstrom et al. (1992); Nordstrom and Miles (1991)). Given the above-mentioned eect of hysteresis on discharge patterns of motor units, we predicted that such hysteresis would decrease the amplitude of force variability. To this end, we tested eects of varying degrees of hysteresis by changing PIC amplitude (1, 3 and 5 nA). As expected, increases in PIC enhance the extent of hysteresis as quantied by the value of F measure (Fig. 30a). The stronger hysteresis results in smaller force variability at mean force levels below 50% of maximal force (Fig. 30b), conrming our prediction. It is worth nothing that the PIC amplitude that produced a comparable value of F measure to experimental data by Gorassini et al. (2002) (a green line in (Fig. 30b)) lowers CoV of force to the extent comparable to experimental observations by (Moritz et al., 2005) even at 15% of maximal force, where the eect of synaptic noise was largest. These results suggest that physiologically plausible levels of PIC, which is much lower than the levels typically used in reduced preparations Lee and Heckman (1998a, 1999), can have substantial impact on the ability of our motor system to produce smooth force output. Discussion Physiological mechanisms of noise in synaptic input The predominant, if not the only, source of noise in synaptic input to motoneurons is stochastic patterns of pre-synaptic neurons (Calvin and Stevens, 1968). Such patterns have been observed in neurons in both motor and sensory systems (Churchland et al., 2010; Kuno, 1964; Rudomin et al., 1975). Such stochastic patterns in pre-synaptic neurons generate random uctuation in synaptic current and the resulting membrane voltage of a post-synaptic neurons (i.e., synaptic noise) (Calvin and Stevens, 1967, 1968). However, we demonstrate here that signal-dependent noise at the level of the synapse cannot account for the amplitude of force variability nor for its signal dependence observed experimentally (Fig. 27). First, the standard deviation of synaptic noise increases in proportion to the square root of mean input levels (instead of the linear increase seen in signal- dependent noise). Second, discharge variability of motoneurons due to synaptic noise decreases, rather than increases, with increases in synaptic input levels. Third, hysteresis in motoneuron 120 Figure 29: Hysteresis in motoneuron channels prevents sporadic discharges and reduces discharge variability. a) Raster plots of all active motor units at 5% of the maximal synaptic input. The top and bottom panels represent example motor unit responses to noisy synaptic inputs without hysteresis and with hysteresis, respectively. Note that in the absence of hysteresis, some motor units display sporadic discharges. b) Coecient of variation (CoV) of inter-spike intervals (ISIs) with and without hysteresis. Note that a higher degree of discharge variability in the absence of hysteresis, which is conned within the 8-30% range in the presence of hysteresis. 121 Figure 30: Intrinsic properties of motoneurons facilitate generation of smooth force output. a) The dierence in control unit discharge rates at recruitment and derecruitment of test units (i.e., F measure) as a function of PIC amplitudes. Note that higher PIC amplitudes induce stronger hysteresis in motor unit discharge patterns. b) CoV of force at varying degree of PIC amplitudes compared to experimental data by Moritz et al. (2005). Note that the greater extent of hysteresis reduces force variability at low forces. 122 channels prevents sporadic discharges and reduces discharge variability in motoneurons. These results call into question the commonly invoked simplistic notion of signal-dependent noise in motor commands to account for experimentally observed force variability (Jones et al., 2002). Furthermore, our results from the present and previous studies (Nagamori et al. in Review) together argue against the ability of `noise' to explain variability in human motor behaviors|which is then often assumed in previous theoretical models to explain variability in health and disease (Harris and Wolpert, 1998; Todorov, 2005; Todorov and Jordan, 2002; van Beers et al., 2004). Alternative sources of force variability Synaptic input to a motoneuron may not be a simple summation of asynchronous, stochastic EPSPs and IPSPs from many sources. Many synaptic sources likely defy the assumption that it should be modeled as a stochastic pattern superimposed on stationary mean discharge rates. For example, the mean discharge rate of a single Ia aerent arising from muscle spindle is highly modulated with respect to muscle length and velocity in a very ne time scale (Burke et al., 1978; Loeb and Hoer, 1985; Loeb et al., 1985a,b; Mileusnic and Loeb, 2006; Vallbo et al., 1979). Such modulation is under the control of the fusimotor system that eectively alters the dynamic and static sensitivity of muscle spindle (Mileusnic and Loeb, 2006). Furthermore, the delayed, yet relatively fast, monosynaptic excitatory feedback system that Ia aerents form with motoneurons generates uctuating synaptic inputs time-locked to the underlying musculotendon contraction dynamics (i.e., non-random) and results in force variability at specic frequencies (6-7 Hz) (Nagamori et al., 2018). It is important to note that such temporal dynamics specic to Ia aerents allow appropriate tuning of its feedback gain to decrease the overall amplitude of force variability (Nagamori et al., 2018). These observations demonstrate that eects of Ia aerent inputs cannot be considered simply as an additional source of synaptic noise. Fluctuating synaptic inputs are likely a common feature of distributed sensorimotor systems. The consequence of uctuating synaptic inputs at specic frequencies is that a pool of motor units tend to share such inputs and synchronize at those frequencies. Such synchronized activity of motor units likely have disproportionately large eects on force variability. Especially, the low-frequency 123 component of such synchronization has a predominant eect on the overall amplitude of force variability due to low-pass ltering eects of motor units and musculotendon contraction dynamics (Nagamori et al. in Review). Accordingly, the amplitude of force variability is correlated with the degree of motor unit synchronization below 5 Hz (Castronovo et al., 2018; Negro et al., 2009; Pereira et al., 2019). Moreover, increases in such synchronization patterns seem to account for increased force variability with aging (Castronovo et al., 2018; Pereira et al., 2019), in stroke (Lodha and Christou, 2017) and in essential tremor (Neely et al., 2015). This low-frequency synaptic inputs seem to arise from a closed-loop aspect of force control (Nagamori et al., 2018). Maintaining motor output (i.e., force or discharge rate of a motor unit) at a constant level requires closed-loop control via veridical feedback via visual and auditory channels. Otherwise, motor output tends to drift (Ambike et al., 2016; Smith et al., 2018). Such closed-loop control of muscle force leads to low- frequency (1-2 Hz) force uctuations (Baweja et al., 2010, 2009; Keenan et al., 2017; Slifkin and Newell, 2000; Smith et al., 2018) due to its inherent delay of visuomotor loop and muscle contraction dynamics (Nagamori et al., 2018). As in monosynaptic Ia excitatory feedback discussed above, other proprioceptive feedback sys- tems such as Ib and Renshaw inhibitory feedback likely generate uctuating synaptic inputs (Nag- amori et al., 2018). Eects of each feedback pathway depend on their delay and dynamics of their inputs arising from sensory systems. Furthermore, descending inputs contain oscillations that arise from interactions among various neural structures within the brain (de Vries et al., 2016; Farmer, 1998; Laine and Valero-Cuevas, 2017, 2020) and/or intrinsic properties of neurons ring at a preferred frequency (Llin as, 2009). Such oscillations at dierent frequency bands seem to re ect distinct sources. For example, the -band (8-15 Hz) oscillations originate in the subcortical structures such as brainstem and cerebellum (Gross et al., 2002; Grosse and Brown, 2003; Llin as, 2009; Soteropoulos and Baker, 2006). On the other hand, oscillations in the (-30 Hz) and -bands (35-70 Hz) seem to arise from the cortical structures (Brown et al., 1998; de Vries et al., 2016). Importantly, these high frequency oscillations may re ect the operation of distributed sensorimotor systems that may re ect the underlying control strategies as the degree of synchronization across muscles in those frequency bands depend on coordination requirements of muscles imposed by motor tasks (Laine and Valero-Cuevas, 2017). 124 It is important to note that high-frequency force uctuations due to these oscillatory synaptic inputs constitute a minor portion of force variability in healthy systems due to low-pass ltering eects of motor units and musculotendon dynamics (Nagamori et al. in Review). However, the eects are not limited to their apparent frequency bands. As discussed above, the underlying dynamics of proprioceptive feedback can help attenuate the overall amplitude of force variability by reducing the low-frequency (<5 Hz) component of force variability (Nagamori et al., 2018). Also, the -band oscillations can directly in uence the overall amplitude of force variability by being de-modulated to produce low-frequency (< 5Hz) synaptic inputs (Watanabe and Kohn, 2015). Furthermore, better understanding of the underlying physiological mechanisms for such high-frequency oscillations is critical because many neurological disorders have been associated with amplied synchronization/force variability above 5 Hz (e.g., Choudhury et al. (2018); Grosse et al. (2004); Neely et al. (2015); Salenius et al. (2002). These discussions highlight the importance of considering force variability as emergent properties of distributed and delayed sensorimotor systems, rather than a simple re ection of noise. Implication for eects of neuromodulatory inputs on force variability Our results demonstrate that an increase in the amplitude of PIC due to increased neuromodulatory inputs leads to progressive reductions in the amplitude of force variability at low force levels (Fig. 30). This observation can be explained by two well-known eects of PIC: hysteresis and amplica- tion of synaptic inputs (Binder et al., 2020). Hysteresis in motoneuron channels prevents sporadic discharges and as a result reduces discharge variability (Fig. 29). Amplication of synaptic inputs increases discharge rates of motor units at a given level of synaptic input, causing motor units to produce more fused tetanic contraction (Fig. 4 in Nagamori et al. in Review). However, these observations are in disagreement with a previous observation of eects of neuro- moduatory inputs on force variability in humans (Wei et al., 2014). Wei and colleagues observed an increase in CoV of force (reduction of0.2-0.3% in CoV of index nger adduction force high-pass ltered at 2 Hz) after oral intake of paroxetine (a selective serotonin reuptake inhibitor, which increases the eect of PIC) and its decrease after oral intake of cyproheptadine (a selective sero- 125 tonin antagonist, which decreases the eect of PIC) (Wei et al., 2014). They interpreted this result as consistent with gain control, a fundamental concept in control systems, by neuromodulatory inputs. Based on this concept, eects of neuromodulatory inputs can be described by modulation of the gain, , that is multiplied to the neural command, u such that Force = u. Given the relationship between variance of force and the neural command, 2 Force =Fu, it follows that SD of force increases in proportion to the gain, : Force = p Fu; (55) where F and Force are a scaling constant (or Fano factor) and standard deviation of output force, respectively (Wei et al., 2014). However, describing the above observations using this control systems concept has many fundamental aws. First, it is easy to show that the gain,, does not aect CoV of force, which Wei and colleagues used to quantify increased force variability, as shown in the following derivation: CoV Force = Force Force = p Fu u = p F p u : (56) Eq. 56 further implies that increased CoV of force they observed requires a disproportionate increase in Force with respect to mean force levels|and thus p F with respect to p u. Second, eects of certain physiological mechanisms through which neuromodulatory inputs mod- ify the level of noise through control of the gain of the force-input relationship cannot be captured by a simplied relationship described by Eq. 55. Amplication of synaptic inputs that increases the slope (i.e., gain) of the frequency-input relationship of motoneurons arises from addition of extra synaptic current from voltage-dependent L-type calcium or sodium channels (Heckman et al., 2009), not multiplication of existing synaptic currents. Furthermore, force or eletromyographic 126 output increases with greater neuromodulatory inputs in response to a xed level of synaptic input (Fig. 1 & 2 in Wei et al. (2014)) likely due to lowered recruitment thresholds of motor units via de- polarization of resting membrane potentials (Lindsay and Feldman, 1993) and hyper-polarization of spike thresholds (Heckman et al., 2009), neither of which acts on synaptic inputs and hence cannot increase noise in proportion to synaptic inputs. Eects of neuromodulatory inputs such as serotonin and norepinephrine are not limited to generation of PIC, but also include increases in input resistance of motoneurons (Inoue et al., 1999; Lindsay and Feldman, 1993; Parkis et al., 1995; Powers and Binder, 2001) and decreases in AHP amplitude (Inoue et al., 1999; Powers and Binder, 2001), both of which enhance the gain of the frequency-input relationship of motoneurons (Powers and Binder, 2001). From Ohm's law (V = IR), increased input resistance acts as gain amplication (i.e., greater membrane depolarization for a given synaptic current), consistent with Eq. 55. Although this would result in increased synaptic noise for a given synaptic input level, the signal-to-noise ratio remains constant as shown in Eq. 56, which does not cause the required disproportionate increases in noise relative to signal as stated above. Decreases in AHP amplitude not only increase the gain of the frequency- input relationship (Hounsgaard et al., 1988), but also can directly increase discharge variability of motoneurons (Manuel et al., 2006). However, the eect of neuromodulatory inputs on AHP amplitude may be insignicant in adult humans. Relatively large reductions in AHP amplitude ( 30% of control values) have been observed in neonatal rat motoneurons (Bayliss et al., 1995). Yet, the eects were very limited in juvenile rat motoneurons (1 mV) and require a relatively large dose of serotonin (Inoue et al., 1999). Such an small change in AHP amplitude may increase CoV of ISIs only by1 % (Manuel et al., 2006), which likely has insignicant eects on the amplitude force variability. It is important to note that the eect of neuromodulation depends highly on other factors such as species and motoneuron population (Powers and Binder, 2001), which necessitate further investigation. These considerations altogether suggest that gain modulation of noise is not a plausible expla- nation for changes in CoV of force with systemic variations in serotonin induced by oral intake of serotonin agonist/antagonist as suggested by Wei et al. (2014). Importantly, changes in the gain of the force-input relationship of muscle with neuromodulatory inputs likely aect the behavior of 127 closed-loop control embodied by distributed sensory motor systems, which might help explain such an observation. Theoretical and clinical implications Our results has disproved a common assumption in current theoretical models that variability in force or kinematics arises predominantly, if not exclusively from `noise' in our motor system (i.e., noise in motor unit force generation or generation of motor commands). We instead suggest the motor commands that re ect observed variability are the consequence of integrative actions of the distributed sensorimotor systems. If so, a biological organism can tune the constituent elements of the systems to control the structure of synaptic inputs and predict its consequences, which would allow for arbitrary modulation of output variability (Dhawale et al., 2017). Furthermore, the simplistic concept of noise to explain variability is a conceptual distraction that hinders eorts to understand the actual underlying physiological mechanisms for force variability ,and their alter- ations due to experimental manipulations (as illustrated in an example of neuromodulatory inputs above) and disease. Rather, detailed analysis of the spectral content of synaptic inputs and the resulting force variability is a particularly fruitful approach to reveal information useful to eluci- date the operation of distributed sensorimotor systems, and its disruptions due to healthy aging, neurological disorders, or injuries. 128 Chapter 4: Physiologically-Realistic Sensorimotor Systems Model of Human Force Control Abstract Variability is a common feature of biological sensorimotor behavior. As such, better understanding of the underlying physiological mechanisms can greatly inform theories of human sensorimotor con- trol and its disruptions in disease. However, such an eort has been hampered by over-simplied theories/models to explain observed variability. Those theories do not account for many other aspects of variability observed experimentally and, more importantly and for a variety of reasons, often ignore the very physiology we try to understand. Our previous computational model (Chapter 4) of a motor unit population disproved the popular notion that kinematic variability arises predom- inantly, if not exclusively, from random endogenous noise in the motor system. We also challenged the popular corollary that human sensorimotor control is driven and determined by compensation for the deleterious eects of this endogenous noise. Following up on those results, this Chapter now further demonstrates that a signicant portion of force variability i) likely re ects control strategies implemented through a delayed and distributed sensorimotor systems, and ii) that spectral anal- ysis is useful to uncover the underlying physiological mechanisms of force variability. We present a physiologically realistic model of the sensorimotor control of static force in humans that extends previous models. The critical additions in our model are 1) more realistic noise in the synaptic and force generating mechanisms, and 2) a hierarchical structure of the sensorimotor systems in which canonical segmental proprioceptive feedback pathways modulate the descending command from a higher-level error-driven controller. To test our model's predictions, we quantied the changes in amplitude and spectral content of static force variability while changing the amplitude (visual sensitivity) of the error feedback in healthy male participants (n = 11). Compared to prior models, our new model better explains features of force variability, and uncovers their potential sources. In particular, the experimentally-observed features of the low-frequency component (0.5-3 Hz) of isometric force variability arise naturally from a simple physiologically-plausible error-driven feed- back mechanism attempting to maintain constant force magnitude. Second, adjustments to the 129 modeled fusimotor drive suce to replicate the changes in force variability in the 3-5 Hz range seen when changing the the visual sensitivity of the feedback. Furthermore, we show that increasing the gain of monosynaptic Ia excitation (either at the spinal synaptic or fusimotor levels) decreases the low-frequency power (0.5-3 Hz) while increasing the high-frequency counterpart (3-7 Hz). These cross-frequency eects of Ia excitation can eectively reduce the overall amplitude of force variabil- ity despite no changes in the degree of motor unit discharge variability. Interestingly, combining gain modulation in Ia excitation with the more limited inhibitory eects of Ib interneurons and Renshaw cells can eectively reduce both the low and high-frequency component of force variabil- ity, suggesting an emergent property of spinal circuitry. This novel explanation of the functional modulation of frequency-specic force variability reinforces the notion that force variability is not a simple re ection of endogenous noise in our sensorimotor systems, but a consequence of the in- tegrative actions of, often ignored, hierarchically-distributed sensorimotor interactions. Our study provides a physiologically based approach to uncover a mechanistic understanding of healthy force variability, and its alterations in healthy aging and disease. Introduction Kinematic and kinetic variability is an inherent feature of biological sensorimotor behavior (Newell and Carlton, 1988; Sherwood and Schmidt, 1980; Whiting, 1983). Furthermore, alterations in sensorimotor systems secondary to healthy aging and neurological conditions are often accompanied by amplication of such variability (Chu and Sanger, 2009; Enoka et al., 2003; Gordon et al., 1995; Lafargue et al., 2003; Lodha et al., 2010; Rothwell et al., 1982; Therrien et al., 2016, 2018; Tracy et al., 2005; Vaillancourt et al., 2001a). As such, many previous studies attempted to develop computational theories of sensorimotor control to explain such observations (Dhawale et al., 2017; Diedrichsen et al., 2010; Harris and Wolpert, 1998; Scott, 2004; Seethapathi and Srinivasan, 2019; Sternad, 2018; Todorov and Jordan, 2002; van Beers et al., 2004; Wu et al., 2014) or leverage such observations to uncover governing principles of altered sensorimotor control in disease (Therrien et al., 2016, 2018). However, many theories remain mostly descriptive, and their physiological mechanisms are often elusive. 130 The most predominant view is that observed variability arises from inherent (i.e., endogenous) noise in our sensorimotor systems and/or control strategies attempting to minimize their negative consequence on motor performance (Harris and Wolpert, 1998; Jones et al., 2002; Sherback et al., 2010; Todorov, 2005; Todorov and Jordan, 2002; van Beers et al., 2004). This view is, however, in- consistent with many previous experimental observations that the amplitude of force variability has been associated with the strength of low-frequency (<5 Hz) common synaptic inputs (Castronovo et al., 2018; Negro et al., 2009; Pereira et al., 2019), which may arise from the closed-loop nature of human force control (Nagamori et al., 2018). Accordingly, manipulation of visual feedback is known to aect this low-frequency force variability (Baweja et al., 2010, 2009; Keenan et al., 2017; Slifkin and Newell, 2000; Smith et al., 2018). Moreover, many pathological manifestations of variability aect specic frequencies (Lodha and Christou, 2017; Neely et al., 2015; Tousi and Cummings, 2017), rather than a broader spectrum expected from generic broadband noise. In fact, our pre- vious work has shown that the contribution of noise which is embedded in the synaptic input to motoneurons, or which emerges from the mechanics of motor unit force generation, has much smaller impact on overall force variability than previously assumed (Nagamori et al. in Review). These results strongly emphasize the importance of considering variability arising from control strategies embodied through distributed sensorimotor systems in frequency-specic manners. Spectral analysis of force variability has been used to characterize the eects of various exper- imental manipulations on several aspects of sensorimotor control. However, it is often dicult to provide a unied description of how each source of force variability relates to a frequency content, owing to the lack of standardized frequency ranges. Studies often use an arbitrary range of fre- quencies based either on signicant eects that they nd or previous studies. This not only limits our ability to draw general conclusion from multiple studies, but also risk our interpretations being dependent on statistical artifacts. Furthermore, observed changes in the frequency content of force variability remain descriptive until their causal mechanisms are established. This issue is especially critical because 1) the frequency range most in uenced by a given mechanism may defy the stan- dard intuition that this should be determined by loop delays and 2) there may exist cross-frequency interactions (Nagamori et al., 2018). For example, the eects of manipulating visual feedback sen- sitivity may go beyond visuomotor error corrections (which tend to aect lower frequencies), but 131 may include alterations in sensorimotor integration at the spinal level (which aects low and high frequencies) (Laine et al., 2014; Nagamori et al., 2018). Proper characterization of the mechanisms and disruptions of frequency-specic variability is also clinically informative as many pathological manifestations of increased force variability appear at distinct frequency ranges such as Parkinso- nian tremor and essential tremor (Hallett, 2008). For these reasons, it is critical to characterize the eects of a proposed mechanism on force variability through a physiologically-realistic generative computational model. This is also critical to avoid misconception or mischaracterization of the underlying physiology. To this end, we develop a physiologically realistic model of sensorimotor control systems for hu- man force control that combines and extends our previous models (Nagamori et al. 2018; Nagamori et al. 2020). The critical aspects of our model are 1) addition of more realistic noise in the synap- tic and force generating mechanisms and 2) the hierarchical structure of sensorimotor systems in which canonical proprioceptive feedback pathways regulate the command signal from a higher-level error-driven controller. This model enables, for the rst time, quantication of emergent behaviors that arise from interactions between distributed sensorimotor systems and physiologically-realistic forms of noise. To test our model's predictions, we quantied changes in the amplitude and spectral content of force variability with manipulation of visual feedback sensitivity in healthy male partic- ipants (n = 11). Our results reinforce the notion that force variability is not a simple re ection of noise in our sensorimotor systems, but a consequence of the integrative actions of distributed sensorimotor systems. Our study provides a critical step toward more mechanistic understandings of variability and its alterations in health and disease. Methods We develop a physiologically realistic model of sensorimotor control systems for human force control that combines and extends our previous models (Nagamori et al. 2018, Nagamori et al. in Review). A schematic representation of our model is shown in Fig. 31. In this model, a population of motor units is controlled by a combination of spinal feedback loops and a high-level error-driven feedback controller. Spinal feedback includes monosynaptic Ia excitation, Ib inhibition and Renshaw inhibi- 132 Figure 31: Schematic representation of the model of our sensorimotor system. An visual feedback controller at the brain level sends a descending command signal based on the error perceived between the output force and the desired constant force. Synaptic inputs from spinal segmental feedback pathways (i.e., homologous monosynaptic Ia excitation, Ib inhibition, and Renshaw cell inhibition) are pre-synaptically added to the descending command at the spinal cord to generate the eective synaptic input to the population of 120 motor units. The population of motor units converts this eective synaptic input into output force. tion. Synaptic inputs are integrated at the spinal cord to generate eective synaptic input equally to the population of 120 motor units. Model of a population of motor units The model includes 120 motor units (78 slow-twitch and 42 fast-twitch units, each consisting of a single idealized muscle ber). Full descriptions of the model are given in our previous publication (Nagamori et al., in Review) and Chapters 2 and 3. Therefore, only brief descriptions are given here. The motoneuron model is an extension of a two-compartment (i.e. soma and dendrite) model of motoneuron by (Cisi and Kohn, 2008). This model generates a series of action potentials in response to eective synaptic current,I i eff (see below for details), applied to its dendritic compartment based 133 on the dynamics of dendritic and somatic membrane potentials. Their dynamics are modeled by the system of dierential equations coupled with a coupling conductance between the two compartments (Eq. 3.1-3.8). We used pulse-approximation to simulate the dynamics of voltage-dependent ionic conductances in the somatic compartment (Eq. 3.9-3.12). Model parameters (Table 3.1) were adjusted to replicate various experimentally measured active and passive electrical properties of motoneurons critical for simulating eects of dynamics synaptic inputs on the discharge behavior of motoneurons. These properties include input resistance, membrane time constant, parameters associated with after-hyperpolarization and the frequency-current relationship. Furthermore, those model parameters allowed us to 1) replicate a distribution of recruitment thresholds that closely matched that observed experimentally (see Fig. 4.5) and 2) match the minimum and peak discharge rates of motoneurons to the contraction speed (i.e., contraction time) of the muscle bers that each innervates. Based on the results presented in Chapter 3, we added two forms of intrinsic noise into our model of motoneurons: random uctuations 1) in voltage threshold and 2) in membrane voltage during AHP. We also introduced hysteresis in motoneuron channels due to persistent inward current. We do so by injecting an extra current of 3 nA only into low-threshold motoneurons that innervate slow-twitch bers immediately upon their recruitment. The model of the motor unit force generating process converts a binary spike train of motoneuron action potentials into motor unit force (Fig. 3.1). This model consists of two modules. Module 1 turns spike trains into motor unit activation (active force generating state of cross-bridges) through a three-Stage process. In Stage 1, individual action potentials trigger release of calcium into muscle bers. Calcium kinetics between calcium diusion and reuptake by the sarcoplasmic reticulum determines the value of a state variable, [cf]; a fraction of cross-bridges bound to calcium that can participate in force generation. Stage 2 converts [cf] into the intermediate activation, e A, through a non-linear lter, which describes cooperativity and saturation of calcium binding and cross-bridge formation with two additional parameters,N andK. We added the sag and yield properties of slow and fast-motor units (Brown et al., 1999; Brown and Loeb, 2000) in this stage. Stage 3 introduces an additional rst-order dynamics of cross-bridge turnover to generate motor unit activation,A. At this point we introduce the known length-dependence of activation-frequency relationship (Brown 134 et al., 1999) by length-dependence on four free parameters, k 3 , k 4 , N and K. Module 2 outputs tendon force,F se , by accounting for the contraction dynamics between muscle and a series elastic element representing tendon and aponeurosis (Eq. 3.21). The muscle force, the sum of all motor unit forces, is scaled by force-length and force-velocity relationships as well as two passive parallel elastic elements (Eq. 3.21) described previously (Song et al., 2008b). We adjusted model free parameters to simulate various properties of individual motor units (e.g., contraction time, activation-frequency relationship, etc.). This allows us to create a population of motor units that express dierent ber types (i.e., slow and fast) and a spectrum of contraction speed. In this study, we assumed a approximately constant twitch-tetanus ratio of 0.23 based on the result from a whole muscle preparation of feline caudofemoralis reported by (Brown and Loeb, 2000). Model of hierarchical sensorimotor control systems Our hierarchical sensorimotor control systems consist of nested and delayed feedback loops: a higher-level outer loop representing visuomotor control and then a lower-level inner loop arising from segmental feedback pathways. Visuomotor loop as an error-driven feedback controller: Maintaining force output at a constant level requires continuous closed-loop control via sensory feedback (e.g., visual and/or auditory). Otherwise, force output tends to drift (e.g., Ambike et al. (2016); Baweja et al. (2009); Smith et al. (2018); Tracy (2007); Vaillancourt and Russell (2002)). Participants likely use a form of negative feedback based on error information (Miall et al., 1993). Importantly, the use of such closed-loop control in uences the amplitude and spectral structure of force variability (e.g., Baweja et al. (2009); Tracy (2007)) due to a signicant amount of inherent delay (e.g., 150-200 ms for visuomotor loop (Slifkin et al., 2000)). To quantify such eects, we introduced a high-level, outer feedback loop that describes general a visuomotor error correction mechanism likely being used during force tracking tasks. To model such a mechanism, We used the simplest form of a controllers that adjust its input based on the error between observed and desired force output, an equivalent to a proportional 135 Figure 32: Detailed descriptions of the model of our sensorimotor system. Visuomotor controller is represented as an proportional controller with its gain, K F , acting on the dierence (i.e., error) between the delayed information regarding output tendon force and reference force. Spinal segmental feedback pathways (i.e., monosynaptic Ia excitation, Ib inhibition, and Renshaw inhibition) are added to the descending command signal from the visuomotor controller at the spinal cord with their respective delays and gains. Muscle spindle receives dynamic and static fusimotor drive via -motoneurons. Motoneurons convert synaptic inputs into spikes trains, which are then transformed into motor unit forces with the motor unit population model. The resulting muscle force (the sum of individual of motor unit forces) is then converted into tendon force via a model of musculotendon dynamics. 136 controller. More intricate design of this controller is possible (e.g., Dideriksen et al. (2017)), but we found a simple implementation to be sucient to replicate many aspects of the structure of synaptic inputs and resulting force variability that have been observed experimentally. This outer-loop controller rst estimates the current value of force output based on its delayed feedback at a previous time step (tT C ) where t is the current time step and T C is a delay term using the following equation: F est (t) =F se (tT C ) (1 + est ); (57) where F est and F se are estimated force output and actual tendon force output from the model, respectively. We incorporated uncertainties in this estimation process, which we refer to as esti- mation uncertainty, by adding variability, est . est is a time-series of random numbers generated from a Gaussian distribution whose standard deviation equals to p F se (tT C ). The time-series was ltered by a rst-order dierential equation with time constant of 100 ms. This type of un- certainties may be expected from the following experimental observations on visual force-tracking tasks. When participants perform a force tracking task, they tend to xate their gaze near, but not exactly, the stationary target rather than tracking a cursor that moves vertically with their applied force (Huddleston et al., 2013). Importantly, the visual acuity is shown to decrease as the target distance from a xation point increases (Weymouth et al., 1928). Thus, estimation uncertainties in the cursor position might be expected. The controller adjusts its command signal,C(t), based on the perceived error between the desired force output,F des (t), and the estimated force, F est (t) as described in the following equation: C(t) =K F (F des (t)F est (t)) +C(t 1) (58) 137 The value ofK F determines the gain of this controller (see the Results for the value used for each simulation). The nal output of the controller, i.e., synaptic input to a motoneuron population, is computed as follows: u C =C (1 + u ): (59) We also incorporated error in generation of corrective input, which we refer to as motor uncer- tainty, by adding variability, u . u is a time-series of random numbers generated from a Gaussian distribution whose standard deviation equals to p C(t). The time-series was ltered by a rst- order dierential equation with time constant of 100 ms. Please note that the visual appearance of the error (perceived error amplitude) was manipulated to change the perception of this error, as described below. Monosynaptic Ia excitation: The muscle spindle model originally proposed by (Mileusnic et al., 2006) was used to simulate the ring rate of Ia aerent, FR Ia and the resulting monosynaptic Ia excitatory feedback. This model generates group Ia and II aerent ring rates as functions of muscle ber length (L ce ), velocity (V ce ) and acceleration (A ce ) through three types of intrafusal bers (i.e., bag 1 , bag 2 , and chain bers). Each of intrafusal ber receives ber-type-specic fusimotor drive (dynamic and static) through -motoneuron. Only Ia excitatory feedback was included in this model as done previously (Raphael et al., 2010; Tsianos et al., 2014). Group II aerent activity was omitted because the connectivity and function of interneurons associated with group II aerent are not well understood (Raphael et al., 2010). Disynaptic Ib inhibition: The Golgi tendon organ (GTO) model described in (Elias et al., 2014) was used to simulate the ring rate of Ib aerent, FR Ib and the resulting disynaptic Ib inhibitory feedback. This GTO model converts tendon force into Ib aerent ring rate using a transfer function described in Elias et al. (2014). Although some evidence suggests convergence of Ia aerent and other segmental inputs onto Ib interneurons (Loeb, 1987) and reversal to positive feedback loops (McCrea, 1986), we simplied the model as done previously Elias et al. (2014); Nagamori et al. 138 (2018); Raphael et al. (2010); Tsianos et al. (2014). Renshaw recurrent inhibition: To simulate Renshaw inhibition, we used the model of a Renshaw cell population and its connectivity described in Maltenfort et al. (1998) and the model of Renshaw cell dynamic behavior described in Koehler and Windhorst (1985); Windhorst and Koehler (1983). We down-scaled the Renshaw cell population and their connectivity with motoneurons accordingly due to a smaller number of a population of motor units (N = 120) used in our simulation compared to the previous study (N = 256) by Maltenfort et al. (1998). To this end, we included 30 Renshaw cells according to a 4:1 ratio of the number of motoneurons to that of Renshaw cells. Motoneurons were randomly assigned to a hypothetical 30 by 4 grid (rostrocaudal by mediolat- eral). Renshaw cells were aligned in a single column (a vector of 30 elements) in the rostrocaudal direction. A motoneuron projects onto a Renshaw cell in the same rostrocaudal dimension as well as neighboring ones. Similarly, a Renshaw cell projects to motoneurons in the same rostrocaudal dimension as well as the neighboring ones. The magnitude of synaptic weight, W , decreases with distance in the rostrocaudal direction as described in the following equation: W = 1=(1 + 16(d=d max ) 2 ); (60) where d is the rostrocaudal distance of a target neuron from a projecting neuron and d max denes the maximum distance for synaptic connections. Each neuron reaches up to neurons in 2d max + 1 rostrocaudal dimensions. The synaptic weight of Renshaw cells was constant for all four motoneurons in the same rostrocaudal dimension. The values of d max for motoneuron- Renshaw and Renshaw-motoneuron connections were set to 1 and 7, respectively. With this grid arrangement, each motoneuron synapses onto 2-3 Renshaw cells and each each Renshaw cell receives synaptic inputs from 8-12 dierent motoneurons. Conversely, 8-15 Renshaw cells synapse onto each motoneurons. The elements were randomly assigned such that each motoneuron synapses on 3 Renshaw cells 139 and each Renshaw cell receives synaptic inputs from, on average, 12 dierent motoneurons. R j was then normalized to the number of synaptic connections the j-th Renshaw cell receive from motoneurons. It is most likely the synaptic inputs of motoneurons with dierent recruitment thresholds to Ren- shaw cells are inhomogeneously distributed such that higher-threshold motoneurons have stronger in uence on discharge rates of Renshaw cells (Cullheim and Kellerth, 1978; Hultborn et al., 1988; Windhorst and Koehler, 1983). Accordingly, we adjusted the synaptic weight of each motoneuron- Renshaw connection byK as a function of Rheobase current,I r , of individual motoneurons as done previously (Maltenfort et al., 1998). Such dependence is described in the following equation: K = 1=(1 0:93 (I ri min(I r ))=36): (61) The coecients were chosen such that the ratio of synaptic weight of the highest recruitment threshold motoneuron to that of the lowest recruitment threshold is 5:1, consistent with the model by Maltenfort et al. (1998) and experimental data (Hultborn et al., 1988). The normalized discharge rate (a value between 0 and 1) of individual Renshaw cells, r R (a vector of size M), is computed as follows. We rst computed instantaneous normalized discharge rates of individual Renshaw cells, R, as follows: R =A MR r MN : (62) r M N is a vector of instantaneous discharge rate of individual motor units normalized to their respective maximum discharge rate whose size isN. The matrix,A M R, of sizeM byN determines the connectivity between motoneurons and Renshaw cells. Its elements equal to 1 when the synaptic connection exits, which are then scaled by WK. 140 We then converted R into r R by taking into account the dynamics of Renshaw cell activity (Koehler and Windhorst, 1985; Windhorst and Koehler, 1983) using the following transfer func- tion: r Rj (s) R j (s) = 1 + 1 s (1 + 2 s)(1 + 3 s) exp(s): (63) The time constant parameters, 1 , 2 , and 3 , are 0.14, 0.003 and 0.09, respectively. Their recruitment dependence was removed since we only simulate a single synaptic input level. The value of was set to 1.5 ms, which introduces time delay of 1.5 ms in time domain (Windhorst and Koehler, 1983). The normalized ring rate of individual Renshaw cells, u R (a vector of sizeN and a value between 0 and 1), is computed as follows: FR R =A RM r R : (64) The matrix, A R M, of size N by M determines synaptic connections of Renshaw cells onto motoneurons. Its elements equal to 1 when the synaptic connection exits, which are then scaled by W computed from Eq. 60. Each element of the resulting FR R is then normalized by its total weight. Synaptic integration at the spinal cord: All synaptic sources are integrated at the spinal cord to generate the eective synaptic input, U eff , to the population of motor units. Fig. 33 shows example time-series of each component of our sensorimotor systems model and their respective power spectral density. U eff was computed as a linear summation of all synaptic inputs as follows: 141 Figure 33: Example time-series of each component of our sensorimotor systems model and their respective power spectral density. Panels on the left column show from the top to bottom time-series of Ia aerent ring rate, Ib aerent ring rate, a single Renshaw cell activity, command signal from the error-driven feedback control, eective synaptic input and force output. Panels on the right show their respective power spectrum. 142 U eff (t) =C(t) +K Ia FR Ia (tT Ia )FR Ib u Ib (tT Ib )K R FR R (t): (65) The gain parameter, K, is introduced to modulate the contribution of synaptic inputs from each segmental feedback pathway at the spinal level. Also, the synaptic inputs from monosynaptic Ia excitatory feedback and disynaptic Ib inhibitory feedback are delayed based on their respective conduction velocities (80 m/s for Ia and 60 m/s for Ib with conduction delay of 1 ms (Kandel et al., 2000)) and the distance between the spinal cord and the muscle. The conduction delay along -motoneurons was also introduced by delayingU e ff based on their conduction velocity of 80 m/s (Kernell and Monster, 1981; Zajac and Faden, 1985). Model of synaptic current into individual motoneurons The eective synaptic input, U eff , is transformed into the eective synaptic current, I i eff , to the i-th individual motor unit using the following equation: I i eff =a i 1exp U eff b i : (66) The coecients, a i and b i , were determined such that individual motor units are recruited at their specied fraction of U eff and all motor unit reach their pre-determined peak discharge rates at the maximum eective synaptic input (i.e., U eff = 1) (see descriptions for Eq. 4.17 for more details). Noise: The activity of individual neurons tends to be stochastic, modeled as a Poisson process (Churchland et al., 2010; Kuno, 1964). The standard deviation of synaptic input, u, due to such stochastic discharge patterns of pre-synaptic neurons is related to its mean level as follows (Nagamori et al. in Review): 143 u = p u; (67) where the scaling factor, , determines the amplitude of such noise. Noise added to a signal at each time step was retrieved from a series of time-smoothed random numbers drawn from a Gaussian distribution of a unit standard deviation. Time-smoothing was achieved by introducing a rst-order dynamics with time constant of 5 ms to the series of random numbers. The amplitude of noise, , for each segmental feedback is determined in the following manner. Loeb and Marks (1985) reported the ratio of variance to mean spike frequencies of spindle unit activity to lie approximately between 0.18 and 0.27. Here, we assume the constant ratio to be 0.2 as a default value for simplicity. The variance-mean ratio of 0.2 corresponds to 14.1% and 10% of CoV for discharge rates of 10 Hz and 20 Hz, respectively, which correspond approximately to the average resting discharge variability recorded in human and cat (Burke et al., 1979; Matthews and Stein, 1969; Nordh et al., 1983). Furthermore, the signal-to-noise ratio (an inverse of the variance-mean ratio) is proportional to p n, where n is the number of units involved in the signal transduction (Kuno and Miyahara, 1969; Rudomin et al., 1975). Based on this observation, we derived the value of for each segmental input using the following equation for the relationship between SD of noise and the mean discharge frequency of each propriocetor activity: = p 0:2 n 1=4 : (68) We applied this noise to u Ia ,u Ib andu R . The number of muscle spindles is set to 125 based on the spindle count of exor carpi radialis by Banks (2006). The number of Golgi tendon organs is always smaller in cat muscles (Jami, 1992) and therefore it is set to 80% of the spindle counts (i.e., 100). Since we simulate individual Renshaw cell activities, n = 1 for u R . To account for potential uncertainties in these parameters, we further varied the value of to characterize the eect of noise 144 (See sensitivity analysis in Results). Also, we included noise in I eff to individual motor units to simulate asynchronous EPSPs and IPSPs from other sources not explicitly included in our simulations as well as the stochastic nature of synaptic transmission (Kuno, 1964; Kuno and Miyahara, 1969). The value of for this type of noise was set to 0.8 (i.e., the standard deviation of noise corresponds to 80% of the mean level of synaptic current). Time constant for time-smoothing was set to 1 ms to achieve the desired autocorrelation function of synaptic noise, random uctuations in motoneuron membrane potential due to synaptic bombardment of many asynchronous, stochastic EPSPs and IPSPs (Calvin and Stevens, 1968; Matthews, 1996). Simulation protocols The progression of the modeling is from purely open-loop control to the addition of hierarchical sensorimotor control systems: rst a higher-level visuomotor outer loop and then a lower-level spinal segmental inner loop. The open-loop model without any feedback component (i.e., open-loop, descending motor com- mand without segmental feedback) established the maximal force output and the amplitude and spectral content of force variability in the absence of the sensorimotor control system. We then extended the model to systematically include other elements to characterize eects of each element on the amplitude and spectral content of force variability. Model parameters for individual elements of the sensorimotor systems are shown in Table 4. We ran 20 trials for each condition at 5% of the maximal isometric force output. Each trial consisted of a 1-sec zero input phase, a 2-sec ramp-up and a 17-sec hold at a given level of synaptic input. The last 10-sec segment of the data was analyzed for comparisons to the experimental features described below. 145 Table 4: Model parameters for the sensorimotor systems: Visuomotor Controller Gain 0.00004 { 0.00006 Delay 120 ms Estimation Uncertainties, est 0.07 Motor Uncertainties, u 0.07 Ia Excitation Gain, K Ia 0.00006 { 0.0001 Delay 9 ms Noise, Ia 0.13 Ib Inhibition Gain, K Ib 0.00006 { 0.0001 Delay 12 ms Noise, Ib 0.14 Renshaw Inhibition Gain, K R 0.0125 { 0.05 Delay 1.5 ms Noise, R 0.45 146 Comparisons to experimental data The rst data set against which we compared our results are the experimental data of the amplitude of force variability from the rst dorsal interosseous muscle published by Moritz et al. (2005). We chose this data set because 1) the number of motor units in a muscle has disproportionately large eects on the amplitude of force variability (Fig. 10 in Nagamori et al. in Review) and 2) the estimated number of motor units is available for rst dorsal interosseous (Feinstein et al., 1955), but not for the exor carpi radialis muscle we simulated here. Thus, the utility of these data are mainly to compare the magnitude of force variability for a muscle with well-characterized population of motor units. On the other hand, Moritz et al. (2005) did not provide any data related to the frequency content of force variability|and the frequency content of force variability depends more on the musculotendon dynamics (Fig. 8 in Nagamori et al. in Review). The architecture of rst dorsal interosseous is bipennate, and does not satisfy the simple parallelogram assumption required to simulate musculotendon dynamics (cf. Infantolino and Challis (2010)). In contrast, exor carpi radialis was used in our simulations because its architecture (particularly pennation angle) is reasonably compatible with the parallelogram assumption (Segal et al., 1991). Therefore, in an attempt to completeness, we also compared our results against the exor carpi radialis data from our own experiments to evaluate the magnitude and power spectral density of force variability from a muscle with a parallelogram musculotendon architecture. We asked eleven, young healthy male participants (mean age of 28.43.9) to perform a force tracking task with their wrist exor muscles at 5% of their maximum voluntary contraction (MVC, Fig. 34). Participants produced wrist exion force against a rigid object attached to a force transducer (Model 20E12A4 100N, JR3 Inc., Woodland, CA) while their wrist was restrained by additional supports (Fig. 34c). The force signals were acquired at 1000 Hz using a National Instruments data acquisition system (NI USB-6218, National Instruments, Austin, TX). Only the normal force applied to the transducer's face (i.e., z-direction) was displayed and recorded for the analysis. MVC was determined as the peak force value they achieved in two MVC trials performed prior to the force tracking task. They received verbal encouragement during the MVC trials, but did not receive any other instruction. Only male participants were recruited due to potential dierences 147 in strength, and passive mechanical and tissue properties of their musculotendons (Blackburn et al., 2006; Onamb el e et al., 2007). During a force tracking task, the participants were asked to maintain a constant force level for 15 sec using visual feedback displayed on a computer display placed 1 m in front of them. An example force signal of the last 10-sec segment, which we used for further analysis, is shown in Fig. 34d) The visual feedback of force output normalized to their respective MVC was displayed as a cursor moving vertically on a computer display (i.e., compensatory feedback (Huddleston et al., 2013; Keenan et al., 2017)). The target force level was provided as a horizontal line at 5 %MVC (Fig. 34c&d). Compensatory feedback was chosen over pursuit feedback (a cursor progresses horizontally with time) since it minimizes force variability due to saccadic eye movement tracking a horizontally moving cursor (Huddleston et al., 2013; Keenan et al., 2017). We varied the degree to which visuomotor error corrections are involved by manipulating the sensitivity of visual feedback as done previously (Laine et al., 2014). In the high sensitivity feedback condition, we set the range of force levels displayed on the screen such that a 1 % of deviation corresponds to1.5 degrees of visual angle by adjusting the range of force levels shown on the computer display to3% MVC from the target (Fig. 34c). Visual angle was reduced 20-fold for low sensitivity feedback condition (Fig. 34d). All participants completed 10 trials for each condition. Data analysis Force variability: The amplitude of force variability has been quantied most commonly using standard deviation (SD) or coecient of variation (CoV) of force across the duration of a trial. It is important to note, however, that some studies remove linear trend (de-trend) a force signal before computing SD or CoV (e.g., Moritz et al. (2005); Tracy (2007) while other do not (e.g., Keenan et al. (2017); Slifkin et al. (2000)). De-trending can have substantial eects on these measures (Tracy, 2007) and may complicate comparisons across dierent studies. Thus, we compute SD and CoV of force both with and without de-tending. As we demonstrate, the comparison between these two methods provide important insights into the sources of force variability. To de-trend our simulated force signals, we used the method compatible with that described in 148 Figure 34: Experimental setup. a) The experimental setup. Participants were asked to produce and hold a given magnitude of isometric force against a force transducer using their wrist exor muscles (while keeping their ngers as relaxed as possible). The arm was restrained by supports at the forearm (not shown) and wrist level. b) The last 10-sec of a representative 15-sec force tracking trial targeting 5% MVC. c) High sensitivity visual feedback condition. The full range of the computer display was set to3% MVC from the target, thus even small variability in force were seen as large visual deviations from the target force. d) Low sensitivity visual feedback condition. The full range of the computer display was set to60% MVC from the target. Note that a give cursor deviation from the target force (i.e., tracking error) in (b) is seen 20x larger in the high sensitivity condition (d) compared to the low sensitivity condition (c). 149 Moritz et al. (2005); Tracy (2007). This method was implemented to allow a proper comparison between he amplitude of force variability predicted by our simulations and the experimental data from Moritz et al. (2005). To this end, the 10-sec hold phase of output force was divided into ten 1- sec segments, the duration of data used in Moritz et al. (2005). The mean of each segment was rst calculated. A force signal in each segment was then linearly de-trended and standard deviation (SD) was calculated from the de-trended data. coecient of variation (CoV) was calculated by dividing SD by the mean force of that segment before de-trending. Obtained values were then averaged across 20 trials. To compare the frequency content of force variability between our simulations and our exper- imental data, both data sets were analyzed in the same manner. To this end, we computed the power spectrum density (PSD) of force output (pwelch function in MATLAB) with the frequency resolution of 0.1 Hz. For the low-frequency component of force variability below 3 Hz, we used 5-sec Hann windows with 90% overlap. For the higher-frequency component above 3 Hz, we used 2.5-sec Hann windows with 90% overlap. This method was chosen to keep relative resolution of estimated PSD both at low and high frequencies. Motor unit discharge variability: CoV of inter-spike intervals (ISIs) between successive spikes was used to quantify the degree of discharge variability of motor units. Spike trains of individual motor units were taken from the last 10-sec of the hold phase. Synchronization of motor units: Coherence analysis is used to quantify synchronization patterns of motor units in the frequency domain as done previously (Negro et al., 2016). To this end, all active motor units during the hold phase of simulations were rst identied and divided into two equally sized groups. Spike trains of those units were summed to create two sets of cumulative spike trains. Coherence was computed between the two cumulative spike trains with mscohere function in MATLAB using a 2-sec Hann window with 90% overlap. The frequency resoultion was set to 0.1 Hz. The computed coherence spectrum was then transformed into Fisher's Z-values: Fz =atanh( p c), where c is a coherence value at each frequency. 150 Statistical analysis Statistical analysis was performed in the R environment for statistical computing (The R Foun- dation for Statistical Computing, Vienna, Austria). We used robust statistical methods contained in the WRS2 package (Mair and Wilcox, 2019). For statistical comparisons between a pair of conditions, we used Yuen's test with 0% trimming (yuen for two independent groups and yuend for two dependent groups). For comparisons across more than three conditions, we performed one- way heteroscedastic ANOVA with trimmed means (t1way) with 0% trimming. All pairwise post hoc comparisons were made using lincon function with 20% trimming. This method adjusts the p-values for multiple comparisons such that the family-wise error rate is 0.05 (Mair and Wilcox, 2019). All of these statistical tests allow for un-equal variance, or heteroscedasticity, (Mair and Wilcox, 2019). We report explanatory measure of eect size, , alongside with p-values when avail- able. Values of = 0.10, 0.30, and 0.50 can be interpreted as small, medium and large eect size (Mair and Wilcox, 2019). For our spectral analyses, we performed statistical comparisons at every 0.1 Hz. We chose this method over using pre-specied frequency ranges because power spectral density within a specic range tends to be non-uniform, biasing changes in the dominant frequencies (e.g., Fig. 36) and potentially masking changes in some frequencies. Furthermore, pre-dened frequency ranges used in previous studies tend to be arbitrary and their denitions vary across studies. However, the results for such high-resolution statistical comparisons at all frequencies still need to be taken with caution because no correction was made for multiple comparisons inherent to this analysis (151 comparisons across frequencies up to 15 HZ), which substantially impacts the statistical power. As such, p-values and explanatory measure of eect size, , are provided as a guide to discriminate frequency ranges of interest for future studies. We present the power spectral density of force and z-score coherence with a solid line representing the mean across either participants (experiment) or trials (simulation) and a shaded area representing the standard error. 151 Results We rst present our experimental ndings and compare them to previous studies in the literature. A series of simulation results are then presented to explore possible physiological mechanisms for our experimental ndings, as well those from the previous literature. The progression of the results is from purely open loop force generation, to the addition of nested and delayed feedback loops: rst a higher-level cortical visuomotor outer loop and then a lower-level spinal segmental inner loop. Eects of visual feedback sensitivity on force variability Fig. 35b shows high sensitivity visual feedback is associated with lower power in the very low- frequency (0-0.5 Hz) range (a red shaded area), consistent with a previous experimental ndings (Baweja et al., 2010; Slifkin et al., 2000). This resulted in a large reduction in CoV of force (p< 0:01) as shown in the right panel of Fig. 35a. This is likely due to visual feedback preventing the natural tendency of force output to drift slowly (i.e., at a very low frequency both in the upward and downward direction (Baweja et al., 2010, 2009; Slifkin and Newell, 2000; Smith et al., 2018; Vaillancourt et al., 2001b)) as we see in the case of low sensitivity visual feedback. Consistent with this view, the right panel of Fig. 35a shows that detrending of force signals with 1-sec segments reduced CoV of force, which has also be found in a previous observation (Tracy, 2007). In contrast, high sensitivity visual feedback, where we exaggerate the amplitude of the tracking error, introduces force uctuations in the 0.5-2 Hz range (a blue shaded area), as indicated by the emergence of an additional peak at1 Hz with high sensitivity visual feedback (Fig. 35b). Eects of visual feedback sensitivity on the high-frequency component of force variability is shown in Fig. 35c. We observed a decrease in power at3 Hz as well as its increase in the 4-5 Hz range (a green shaded area) and the 8-9 Hz range. It is worth noting that we observed a large, distinct peak at11 Hz, whose amplitude does not depend on visual feedback sensitivity. 152 Figure 35: High sensitivity visual feedback aects force variability across frequencies. a) Comparisons of the amplitude of force variability (raw and de-trended force signals) between high and low sensitivity visual feedback conditions. Note that CoV of force for raw force signals was signicantly lower in high sensitivity visual feedback compared to low sensitivity visual feedback condition (p< 0:01). No signicant dierence was found for de-trended force signals (p = 0:40). b) Comparisons of power spectral density (PSD) in the low-frequency component below 3 Hz between the two conditions. c) Comparisons of power spectral density (PSD) in the high-frequency (3-15 Hz) component between the two conditions. Note that high sensitivity visual feedback increases power in the 1-2 Hz and 4-5 Hz ranges (green-shaded area) while reducing power in the 0-0.5 Hz range (red shaded area). 153 Estimation and motor uncertainties in visuomotor error correction are necessary to replicate the experimental observed force variability below 2 Hz. Based on our previous study (Nagamori et al., 2018), we predicted that the low-frequency (<2 Hz) component of force variability and its dependence on visual feedback would arise from visuomotor interactions between motor noise and visuomotor loop likely operating during human force tracking tacks. We also predicted that explicit inclusion of these interactions (as we have done in, for example, (Nagamori et al., 2018; Venkadesan et al., 2007)) would be necessary in our models to replicate the experimentally observed overall amplitude of force variability (i.e., CoV of force) (Farina et al., 2014, 2016; Negro et al., 2016). To test these two predictions, we compared three simulated conditions (i.e., open-loop control and two closed-loop conditions: with and without estimation and motor uncertainties in visuomotor loop) against experimental data from a previous study by (Moritz et al., 2005) and from the current study. Fig. 36a shows that CoV of force with and without de-trending of the original force signals. We nd signicant interactions among the three simulated conditions (p < 0:01 for both raw and detrended conditions and = 0:9 and = 1:05, respectively). Contrary to a nding from our previous study (Nagamori et al., 2018), addition of a visuomotor loop without estimation and motor uncertainties by itself does not increase CoV of raw force signicantly (p = 0:92) and does so only slightly for CoV of de-trended force (p< 0:01). Consequently, the resulting CoV of de-trended force is much smaller than the experimental value reported by Moritz et al. (2005). Given these results, we introduced estimation and motor uncertainties in the visuomotor loop, which increases the amplitude of both raw and de-trended force variability signicantly (p< 0:01 for both) to an extent compatible with a previous experimental observation by Moritz et al. (2005). These results suggest that the quality of visuomotor error corrections can play a signicant role in determining the overall amplitude of force variability, consistent with a previous experimental nding (Keenan et al., 2017). Furthermore, addition of an higher level error-driven feedback controller also aects the discharge 154 variability of motor units (Fig. 36b, p < 0:01, = 0:68). More specically, discharge variability of both closed-loop conditions is higher than open-loop condition without any feedback component (p < 0:01 for both comparisons). Furthermore, addition of estimation and execution errors in visuomotor loop further increased discharge variability (p = 0:045). These results suggests that attributing the entire discharge variability to synaptic noise might be problematic when synaptic input is known to uctuate at a slow rate as noted by Matthews (1996). It is also important to note that addition of explicit estimation execution errors in visuomotor loop increases discharge variability only by0.5 % despite disproportionately large eects of added errors in visuomotor loop on the amplitude of force variability (Fig. 36a). This highlights important limitations when directly attributing changes in the amplitude of force variability to changes in discharge variability. Fig. 36c compares the spectral content of force variability in the simulated open-loop condition against the low-sensitivity visual feedback condition in our experiment in which we expected less involvement of visuomotor corrections. It is important to note that the simulated open-loop condi- tion displays a tendency to drift as indicated by a large proportion of power contained below 0.5 Hz, which explains reductions in CoV force force with de-trending (Fig. 36a). The major dierences with our experimental data exist in that our simulated force contains a smaller proportion of power below 0.5 Hz and a higher proportion above 1.5 Hz. These dierences may be due to a greater tendency of the human control system to drift in the absence of visual error corrections as reported extensively in the previous literature (Baweja et al., 2010, 2009; Keenan et al., 2017; Slifkin and Newell, 2000; Smith et al., 2018). Fig. 36d compares the spectral content of force variability in the two simulated closed-loop conditions with and without estimation and motor uncertainties in the visuomotor loop against the experimental high-sensitivity visual feedback condition in which we expect greater involvement of visuomotor corrections. First, we nd addition of the visuomotor loop with no explicit estimation and motor uncertainties (a dark blue line) is not sucient to replicate the experimental power spectrum (a black dotted line). This large error arises from smaller proportional power in the very low-frequency (<1.5 Hz) component and larger proportional power above 1.5 Hz. Our ability to replicate the experimental power spectrum substantially improves by addition of estimation and execution errors in our controller (a orange line in Fig. 36c, as indicated the presence of two 155 characteristic peaks below 2 Hz and lower proportional power in the high-frequency component. For this particular data shown here, model parameters for the visuomotor loop are such that T C = 120ms, est = 0:07, u = 0:07, and K f = 0:00004 in Eq. 57-59. These results corroborate the above mentioned results such that force variability re ects the way in which force output is maintained at a constant force level with visuomotor error correction mechanisms. Monosynaptic Ia excitation aects force variability across the entire frequencies. Here, we extend the above modeling results by adding canonical segmental feedback pathways (monosynaptic Ia excitation, disynaptic Ib inhibition and Renshaw inhibition) to the entire control system. Based on our previous simulation (Nagamori et al., 2018), we predicted that these segmen- tal pathways would be critical to replicate experimental observations of force variability at higher frequencies above 5 Hz. Additionally, we quantied potential eects of these feedback pathways on the lower-frequency (0{5 Hz) component of force variability (Nagamori et al., 2018) that has been been usually attributed only to visuomotor corrections (Miall et al., 1993; Smith et al., 2018; Vail- lancourt et al., 2001a). For the data presented below, we set model parameters for the visuomotor loop such that T C = 120ms, est = 0:07, u = 0:07, and K f = 0:00006 in Eq. 57-59. Monosynaptic Ia excitation: Fig. 37a-b demonstrates that presynaptic gain modulation of monosynaptic Ia excitation at the spinal level (K Ia in Eq. 65) aects force variability across frequencies up to 15 Hz. In the frequency component below 3 Hz shown in Fig. 37a, we nd increased presynaptic gain of Ia excitation decreases power in the 0.5{2 Hz range (a blue shaded area), the same frequency range we nd statistically signicant eects of visual feedback sensitivity in our experiments (Fig. 35b). The eect of gain modulation is limited to the 1.5{2 Hz range for low gain condition (low gain - control, p = 0:066). Its eect extend across the 0.5{2 HZ range with a further increase in the gain as indicated by signicant reductions in average power in this range in high gain condition compared to others (p < 0:01 for both high gain - control and high gain - low gain). In the frequency range above 3 Hz shown in Fig. 37b, notable eects of gain modulation of Ia excitation can be found between 3{7 Hz (green and orange shaded areas). First, we nd a 156 Figure 36: Error-driven feedback with estimation and execution errors is necessary to replicate experimentally observed amplitude and spectral features of force variability. a) CoV of force compared across three simulated conditions: open-loop control, closed-loop control with and without estimation and motor uncertainties. CoV of force was computed with and without de-trending. b) Average CoV of ISIs of motor units in those three simulated conditions. denotes statistically signicant dierence (p < 0:05). c) Power spectrum of force as proportion of total power compared between open-loop condition in simulation and low visual feedback sensitivity condition in experiment. Note that our simulated force output underestimates relative power in the very low-frequency range and overestimates that in higher frequencies. d) Power spectrum of force simulated for two closed-loop conditions compared against high visual feedback sensitivity condition in experiment. Note that addition of estimation and execution uncertainties dramatically improve the predicted power spectrum by our simulations. 157 signicant reduction in the 3{4 Hz range in high gain condition compared to control condition ( p = 0:041). Moreover, addition of Ia excitation signicantly increases power in the 5{7 Hz range (p< 0:01 and p = 0:01 for low gain - control and high gain - control, respectively). We also nd a tendency for the peaks in this frequency range to shift to higher frequencies with increasing gains of Ia excitation (Fig. 37b). The reductions in the low-frequency component of force variability with increasing presynaptic gain of Ia excitation leads to their smaller overall amplitude of force variability (Fig. 37c). We nd signicant interactions for CoV of both raw and de-trended force (p = 0:033 and = 0:38, and p < 0:01 and = 0:62, respectively). Pair-wise comparisons show that there is a signicant reduction of CoV of raw force in high gain condition compared to control (p = 0:048). Moreover, there are signicant reductions of CoV of de-trended force in high gain condition compared to control and low gain conditions (p < 0:01 and p < 0:01, respectively). A reduction in high gain condition compared to low gain condition is marginally signicant (p = 0:052). Note that these changes in the overall amplitude of force variability are not accompanied by changes in CoV ISIs (p = 0:62, = 0:15). These results suggest that higher presynaptic gain of Ia excitation allows generation of smoother force output without aecting apparent `noise' levels in terms of discharge variability of motor units. Fusimotor systems for gain modulation of monosynaptic Ia excitation: The dynamic and static sensitivities of muscle spindle to muscle stretch are controlled by dynamic and static fusimotor drives, respectively (Mileusnic et al., 2006), permitting the gain control of Ia excitation indepen- dently of presynaptic inhibition of Ia aerent terminals on motoneurons (Rudomin et al., 1983; Rudomin and Schmidt, 1999; Seki et al., 2003). Whether or not humans can control-motoneurons independently of the fusimotor -motoneurons (the so-called ' coactivation' is still debated (Jalaleddini et al., 2017; Maceeld and Knellwolf, 2018; Vallbo et al., 1979), although cat seems to be able to (Maceeld and Knellwolf, 2018). Thus, characterization of its eects on force variability is critical to explore such a decoupling of and -motoneurons . To this end, we varied the level of dynamic and/or static fusimotor drives from 0 to 50 Hz. either simultaneously or independently while the other fusimotor drive held constant. 158 Figure 37: Frequency-specic eects of monosynaptic Ia excitation on force variability. Its gain was modulated from low to high (K Ia = 0.00006 and 0.0001, respectively). Control condi- tion refers to a condition in which all proprioceptive feedback pathways are removed (i.e.,K Ia = 0). We ran 20 trials for each condition. a) The low-frequency (0-3 Hz) component of force variability. b) The high-frequency (3-15 Hz) component of force variability. c) CoV of raw and de-trended force. denotes signicant dierences at = 0.05. d) CoV of ISIs. 159 Consistent with the presynaptic gain modulation of Ia excitation shown in Fig. 37, the mod- ulation of muscle spindle sensitivity via the fusimotor system aects both the 0.5{2 Hz and 3{5 Hz ranges of force variability (Fig. 38a,c and d). Below 3 HZ shown as blue shaded areas in Fig. 38a,c and d, increases in fusimotor drives (either both dynamic and static, or dynamic alone) lead to signicant reductions of average power in the 0.5{1 Hz range (p< 0:01, p = 0:014). Their eects on the 3{5 Hz component are more complex. On the one hand, Fig. 38d shows that an increase in dynamic fusimotor drive from 0 to 50 Hz signicantly increases power in the 4{5 Hz range (p = 0:044). It is worth noting that this eect closely matches our experimental observations shown in Fig. 35c where the amplitude of visual error information was manipulated. On the other hand, an increases in static fusimotor drive has an opposite eect on power in the 3{5 Hz range (i.e., a signicant reduction,p = 0:02) while it does not aect power at higher frequencies in general (Fig. 38e). These results help understand the eects of simultaneous increases in both dynamic and static fusimotor drives shown in (Fig. 38b) where there is no clear change in power in the 3{5 Hz range. This is likely due to competing eects of dynamic and static fusimotor drives on the sensitivity of muscle spindle to muscle stretch where increasing dynamic fusimotor drive enhances the dynamic sensitivity of muscle spindles to stretch, which can be counteracted by increasing static fusimotor drive (Mileusnic et al., 2006). Thus, the simultaneous increases in dynamic and static fusimotor drive likely nullify their respective eects on force variability in the 3-5 Hz ranges. These results altogether suggest that the increased visual error in our experiment is accompanied by increased dynamics fusimotor drive. Interestingly, modulation of dynamic and static fusimotor drives have consistent eects on force variability at9 Hz and12 Hz across conditions the tested here (Fig. 38b,d,e) even though these changes are relatively small at higher frequencies compared to lower frequencies below 5 Hz. Most notably, power12 Hz decreases with increasing both dynamic and static fusimotor drives in all conditions. Despite the frequency-specic eects of the fusimotor systems described above, we nd no eects on the overall amplitude of force variability (CoV of raw force: p = 0:15 and = 0:33,p = 0:96 and 160 = 0:01, and p = 0:86 and = 0:04 for simultaneous modulation, dynamic only and static only respectively; CoV of de-trended force: p = 0:13 and = 0:34, p = 0:18 and = 0:3, and p = 0:38 and = 0:2 for simultaneous modulation, dynamic only and static only respectively). Lastly, we nd a signicant increase in CoV of ISIs with an increase in dynamic fusimotor drive (p = 0:020 and = 0:52) while it is not aected by other conditions (p = 0:99 and = 0:0 for both simultaneous modulation and static only). Synchronization among motor units re ects changes in the frequency content of shared synaptic input. Synchronization, as per coherence, between concurrently active motor units in a pool informs the frequency content of their synaptic input (Farina and Negro, 2015; Farina et al., 2014, 2016; Laine et al., 2013, 2014; Negro et al., 2009). Thus, its characterization allows us to dissociate force variability driven purely by synaptic inputs from others secondary to such inputs (such as changes in motor unit discharge rates) or due to random components. To this end, we compared coherence between motor units across our three simulated conditions: a) control and low presynaptic gain conditions for Ia excitation in Fig. 37, b) low and high presynaptic gain conditions for Ia excitation in Fig. 37 and c) low and high dynamic fusimotor drives in Fig. 38. Fig. 39a shows that low presynaptic gain of Ia excitation signicantly decreases coherence in the 2{3 Hz range while increasing it at5 Hz compared to the control condition where there is no Ia excitation. The synchronized activity at5 Hz for the low gain condition is well matched to the peak in the associated power spectrum (Fig. 37b). The signicant reduction of coherence in the 2{3 Hz range is unexpected from the results shown in Fig. 37a, where we did not nd any signicant eects on force variability in this frequency. However, when we test the dierence in average force power in this frequency range, we indeed nd a signicant reduction (p = 0:019, control - low gain), which might have been obscured by the lack of statistical power in the original statistical analysis. A further increase in the presynaptic gain of Ia excitation from low to high decreases coherence in the 0.5{1 Hz range and increases it in the 6{7 Hz range. These results are consistent with changes in force variability in the same frequency ranges(Fig. 37a&b). 161 Figure 38: Frequency-specic eects of spindle sensitivity modulation by the fusimo- tor system. Eects of the fusimotor systems on the low and high frequency components of force variability are shown as follows. a-b) Dynamic and static fusimotor drives are modulated simul- taneously and tested at two levels: 0 Hz and 50 Hz. c-d) Dynamic fusimotor drive is modulated from 0 to 50 Hz while static fusimotor drive is kept constant at 25 Hz. e-f) Static fusimotor drive is modulated from 0 to 50 Hz while dynamic fusimotor drive is kept constant at 25 Hz. 162 Finally, Fig. 39b shows that an increase in dynamic fusimotor drive decreases coherence in the 0.5{2 Hz range and increases it at4 Hz. Once again, the change in the low-frequency component of synchronization patterns describes well the eect observed in force (Fig. 38c). Furthermore, the4 Hz peak in coherence spectrum can explain the emergence of a peak at this frequency in force (Fig. 38d) and suggests the signicant reduction in power at3.5 Hz is a direct consequence of such synaptic input, not due to a change in the frequency content of synaptic input at that frequency. Eects of disynaptic Ib inhibition and Renshaw inhibition on force variability are limited. Inhibitory feedback provided by disynaptic Ib inhibition and Renshaw inhibition have very limited eects on the frequency content of force variability. In the low-frequency component of force variability below 2 Hz, we nd no signicant eects of gain modulation of Ib inhibition or Renshaw inhibition (K Ib andK R in Eq. 65) (Figs. 40a & 41a). On the other hand, we observe a marginally signicant reduction in the 2-3 Hz range with Renshaw inhibition (p = 0:057, high gain - control). The most notable eect of those inhibitions we can observe anywhere in the frequency range of interest is reduction of force uctuations at10 Hz (Figs. 40b & 41b). As shown in Fig. 40c, neither CoV of raw nor de-trended force are aected by gain of Ib inhibition (p = 0:89 and = 0:08, and p = 0:70 and = 0:013, respectively). Similarly, neither CoV of raw nor de-trended force are aected by gain of Renshaw inhibition (p = 0:65 and = 0:14, andp = 0:67 and = 0:16, respectively) (Fig. 41c). Furthermore, CoV of ISIs does not depend on gain of Ib inhibition nor Renshaw inhibition (p = 0:30 and = 0:26 and p = 0:35 and = 0:22, respectively). 163 Figure 39: Synchronization among motor units re ects changes in the frequency content of shared synaptic input. Z-score coherence between two groups of cumulative spike trains is compared between a) control and low gain conditions in Fig. 37, b) low gain and high gain conditions in Fig. 37 and c) low and high dynamic fusimotor drive in Fig. 38. 164 Figure 40: Ib aerent feedback has limited eects on force variability. Its gain was modulated from low to high (K Ib = 0.00006 and 0.0001, respectively). The control condition refers to a condition in which all proprioceptive feedback pathways are removed (i.e., K Ib = 0). a) The low-frequency (0-5 Hz) component of force variability. b) The high-frequency (3-15 Hz) component of force variability. c) CoV of raw and de-trended force. d) CoV of ISIs. 165 Figure 41: Renshaw inhibition reduces 10 Hz force uctuations. Its gain was modulated from low to high (K RC = 0.0125 and 0.05, respectively). Control condition refers to a condition in which all proprioceptive feedback pathways are removed (i.e.,K RC = 0). a) The low-frequency (0-5 Hz) component of force variability. b) The high-frequency (3-15 Hz) component of force variability. c) CoV of raw and de-trended force. d) CoV of ISIs. 166 The combined eects of proprioceptive feedback on force variability are not a simple summation. Here, we tested eects of combined proprioceptive feedback on force variability by comparing two conditions: Ia excitation only and all segmental feedback pathways. The gains of individual feedback pathways are set to the lowest gains presented above. Despite the limited eects of Ib and Renshaw inhibitions on the amplitude an frequency content of force variability, we demonstrate that their addition to Ia excitation signicantly alters the frequency content of force variability (Fig. 42a-b). More specically, addition of those inhibitory feedback pathways signicantly reduces power at1 Hz as well as at4 Hz. As one might expect from the eects on the low-frequency component of force variability, we nd a signicant reduction in CoV of de-trended force (p = 0:040 and = 0:46, Fig. 42c). A reduction in CoV of raw force does not reach statistical signicance (p = 0:11 and = 0:36). Once again, CoV of ISIs does not dier between these two conditions (p = 0:80 and = 0:06, Fig. 42c), suggesting that the observed eects cannot be described simply by changes in the level of `noise' that a motor unit pool displays. Rather, these results suggest that combining multiple proprioceptive feedback pathways may be useful for generating smooth force output and preventing high frequency force uctuations characteristic of Ia excitation. Discussion In this study, we propose a model of physiologically realistic sensorimotor systems of human force control to investigate the sources and their potential interactions that determine the amplitude and frequency content of force variability at much ner details than any previous models. Our model combines our previous a model of a motor unit population (Nagamori et al., in Review) with inner, spinal segmental feedback loops and an outer feedback loop representing visuomotor error correction Nagamori et al. 2018. This enables, for the rst time to our knowledge, quantication of emergent behaviors that arise from interactions between hierarchically distributed sensorimotor systems and physiologically realistic sources of noise. Each component of our model is built based on extensive 167 Figure 42: Combined eects of proprioceptive feedback on force variability are not a simple summation. Combined eects of all proprioceptive feedback pathways are compared to those of Ia excitation only. The gains of individual feedback pathways are set to their respective lowest values used in previous simulations. a) The low-frequency (0-5 Hz) component of force variability. b) The high-frequency (3-15 Hz) component of force variability. c) CoV of raw and de-trended force. d) CoV of ISIs. 168 experimental observations describing the underlying physiological mechanisms. We could call such an approach 'physiomorphic' in contrast to a more popular approach in many theoretical models where the model parameters are often tted to replicate experimental observations. As a result, our model greatly improves our ability to explain experimentally observed features of force variability and uncover their potential sources at much deeper levels than previous models. First, we demonstrate the quality of an error-driven visuomotor feedback mechanism (estimation and motor uncertainties) used to maintain a constant force level determines experimentally observed features of the low-frequency component (0{2 Hz) of force variability and the overall amplitude (Fig. 36). Second, gain modulation of monosynaptic Ia excitation either at the spinal level or through the fusimotor system aects not only the high-frequency component (3-7 Hz) of force variability but also its low-frequency component, thus aecting the overall amplitude (Figs. 37 and 38). Importantly, eects of visual feedback sensitivity on the high-frequency component of force variability we observed experimentally can be well replicated by adjustments of dynamic fusimotor drive (Fig. 38d). Finally, although we nd the eects of Ib and Renshaw inhibitions on force variability are very limited by themselves, their combined eects with Ia excitation further reduce both the low and high-frequency components of force variability (Fig. 42). These results reinforce the notion that force variability is not a simple re ection of noise in our sensorimotor systems, but a consequence of the integrative actions of hierarchically distributed sensorimotor systems. Origin of low-frequency force variability below 3 Hz and its modulation by monosy- naptic Ia excitation The low-frequency range of force variability below 5 Hz is often lumped together as a single fre- quency range and thought to arise from low-frequency common modulation of motor units (vol- untary or otherwise), called `common drive' (de Luca et al., 1982; Farina and Negro, 2015; Farina et al., 2014, 2016). We nd this simplistic characterization problematic, and leaves its source unex- plained. Our detailed spectral analyses reveal much more than such a simple lumped phenomenon, and points to mechanisms to explain its sources. 169 The output force of our model in the simulated open-loop condition shows a minor tendency to drift (Fig. 36a). Yet the extent of such drift is insucient to account for the proportionally larger power fraction below 0.5 Hz we observed in the low sensitivity visual feedback condition in our experiment (Fig. 36c). This discrepancy is likely due to the fact that our feedforward model does not capture the much stronger (and potentially perceptual) tendency of the human control system to drift without veridical feedback such as visual and auditory (Baweja et al., 2010, 2009; Matthews, 1996; Slifkin and Newell, 2000; Smith et al., 2018; Vaillancourt et al., 2001b). We could not improve our open-loop model based on those previous experimental ndings because the physiological mechanisms for force drift are not well understood. However, it is important to note that the amount and speed of isometric force drift and force variability has been shown to increase in patients with Parkinson's disease (Vaillancourt et al., 2001b). Moreover, the increased force variability in stroke survivors is characterized as increased variability below 0.5 Hz (Lodha and Christou, 2017). These observations suggest that variability below 0.5 Hz, which is often treated as an artifact (Jones et al., 2002; Moritz et al., 2005), may contain clinically useful information about human sensorimotor control, which requires further systematic investigations. Furthermore, our results have the important theoretical implication that the assumption of constant synaptic input during production of a constant force, especially in the absence of visual feedback (e.g., Jones et al. (2002)), is likely false. We then show that addition of an error-driven feedback controller representing visuomotor error corrections is critical for replicating force variability below 2 Hz observed in the high-sensitivity feedback condition in our experiment. Although the low-frequency component of force variability is increased by addition of an visuomotor loop, the eect on the overall amplitude of force variability is not, but itself, sucient to account for the overall amplitude of force variability observed experi- mentally (Fig. 36a). In particular, our model strongly suggests that the experimental amplitude of force variability requires implementation of estimation and motor uncertainties in visuomotor loop, which also improves our ability to replicate the experimental power spectrum in high sensitivity visual feedback condition, characterised by the distinct peak at1 Hz (Fig. 36d). These results are consistent with previous experimental observations that visual feedback improves tracking accuracy (Baweja et al., 2009), yet at the expense of increasing force variability1 Hz (Baweja et al., 2010, 170 2009; Keenan et al., 2017; Slifkin and Newell, 2000; Smith et al., 2018; Sosno and Newell, 2005), which results in greater overall amplitude of force variability (Baweja et al., 2009). Furthermore, these results extended those from our previous simulation (Nagamori et al., 2018) such that this low-frequency component of force variability observed experimentally likely re ect the participant's ability to accurately translate visual information into accurate motor command (indicated by esti- mation and motor uncertainties in our model), rather than an attempt to control motor noise with a delayed error-driven feedback. This interpretation contrasts the predominant notion that force variability represents perfor- mance limitation imposed by additive motor noise in an optimal feedback controller as one might predict from previous theoretical models (Harris and Wolpert, 1998; Sherback et al., 2010; Todorov and Jordan, 2002; van Beers et al., 2004). In fact, in our previous model (Nagamori et al., 2018), we had to inject a large amount of articial signal dependent noise into the motor command to replicate the amplitude of force variability comparable to experimentally reported values (i.e., vari- ance to mean ratio of 0.3). On the other hand, the variance to mean ratio of force variability in open-loop condition (i.e., true feedforward motor noise) from this study is only 0.001, suggesting that our previous model overestimated the contribution of motor noise to the overall amplitude of force variability. It is important to note that adding a large amount of signal-dependent noise to account for experimentally observed kinematic variability is the standard practice in popular theoretical models (Fig.9 in Nagamori et al. in Review). Altogether, our results strongly argue against the notion that observed variability is a re ection of noise in our sensorimotor systems that interferes with a control strategy implemented by the central nervous system. Traditionally, this low-frequency component of force variability below 3 Hz and, consequently the a large share of the power of force variability, are considered to re ect the operation of a higher- order control mechanism presumably at the level of the brain (Miall et al., 1993; Slifkin et al., 2000). Previous studies interpreted the 1-Hz modulation of force observed across many studies as evidence of intermittent nature of human feedback control (Miall et al., 1993; Slifkin et al., 2000) since visual processing time is much shorter (120-150 ms) (Carlton, 1992; Slifkin et al., 2000; Thorpe et al., 1996). The model that describes the human feedback control mechanism as an intermittent controller suggests that corrective motor actions at 1 Hz are based on delayed visual information 171 accumulated at6{7 Hz (Slifkin et al., 2000; Sosno and Newell, 2005; Vaillancourt et al., 2001a). However, our results argue against such an interpretation since our visuomotor error-driven feedback modulates force at1 Hz despite its120 ms delay consistent with visual processing time and a continuous update policy. Our results rather suggest that the 1-Hz modulation of force output is due to smoothly modulated synaptic input acting on a musculotendon unit that acts as a low-pass lter (Baldissera et al., 1998; Bawa et al., 1976; Krylow and Rymer, 1997; Mannard and Stein, 1973). This is an important, yet often overlooked, aspect of biological sensorimotor systems. Furthermore, the current view ignores an important aspect of distributed sensorimotor systems in which spinal segmental feedback constantly regulates synaptic inputs to a motor unit population (Raphael et al., 2010; Tsianos et al., 2014). In fact, we observe that this low-frequency component of shared synaptic input (as per coherence) and the resulting force variability can be highly modulated by gain changes in spinal segmental feedback pathways. Most notably, the increased strength of monosynaptic Ia excitation either through gain modulation at the spinal level or through the fusimotor systems greatly reduces force variability (Figs. 37 and 38) and coherence in this frequency range (Fig. 39). It is worth noting that those eects are frequency specic even within in the 0-3 Hz range that has been considered as a single frequency range. For example, modulation of Ia gain at the spinal level reduces synchronized activity and the resulting force uctuations between 0.5 and 3 Hz range (Figs. 37a and 39a-b), while the eects of gain modulation via the fusimotor system is limited to 0.5{1 Hz range (Fig. 38). Furthermore, our results support a previously proposed link between the strength of synchronization between motor units in this low-frequency range and that of Ia feedback where the strength of Ia excitation is negatively correlated with the degree of synchronization in the 0{5 Hz range (De Luca et al., 1993; Garland and Miles, 1997; Laine et al., 2013, 2014; Nagamori et al., 2018; Schmied et al., 2014). Accordingly, the degree of synchronization in the 0{5 Hz range was found to be higher in patients with deaerentation Schmied et al. (2014), suggesting strongly the possible role of Ia excitation in manifestation of the low-frequency force variability. These eects are consistent with a proposed role of Ia excitatory feedback in correcting inevitable small irregularities while attempting to generate smooth movements voluntarily (Burke et al., 1978; Vallbo et al., 1979). These results altogether suggest that classifying human feedback control mechanism as either continuous or intermittent likely undermines the characterization of 172 important mechanisms in both healthy and pathological sensorimotor systems in operation. It is interesting to note that some of our present simulation results are in disagreement with our own previous study (Nagamori et al., 2018) in that inhibitory feedback from Ib interneurons or Renshaw cells alone is not eective for reducing the low-frequency force uctuations. Yet, we now see that those inhibitions can have profound eects once combined with Ia excitation, suggestive of context-sensitive emergent properties of the spinal network. These results suggest that understanding of physiological mechanisms for precision in force control and its disruption due to aging and certain neurological conditions requires considerations of not only a higher-level control mechanism, but also appropriate gain tuning of segmental feedback. Such a comprehensive view is critical since which aging and certain neurological conditions such as stroke and Parkinson's disease are known to aect the integrity of at least segmental feedback (Baudry et al., 2010; Hagbarth et al., 1975; Katz and Rymer, 1989; Thilmann et al., 1991). Physiological mechanisms for high-frequency force uctuations above 3 Hz Our experimental observations are also intriguing in that they also show that eects of error-size manipulation with visual feedback extends beyond 3 Hz and alters force uctuations in the 3{5 Hz range (Fig. 35c). This observation is consistent with previous experimental results (Laine et al., 2013, 2014) although the frequency range in which we observe signicant changes is much more limited to those found by Laine et al. (2013, 2014). Our simulations demonstrate that the increase in the 4{5 Hz power can be explained by increased dynamic fusimotor drive (Fig. 38d). Accordingly, a previous experimental study has demonstrated that stretch re ex amplitude, which re ects the dynamic sensitivity of muscle spindle controlled by dynamic fusimotor drive, is enhanced when participants receive higher sensitivity visual feedback (Nafati et al., 2004), likely due to higher attentional demands imposed by the task (Hospod et al., 2007; Nafati et al., 2004). Interestingly, Laine et al. (2014) showed larger 4{10 Hz force variability with high sensitivity visual feedback, which is accompanied by increased H-re ex amplitude, a measure of the spinal gain of Ia excitation. This observation closely resembles the eect of gain modulation of Ia excitation at the spinal level shown in Fig. 40. Factors that dierentiate selective modulation either at the spinal level or 173 through the fusimotor system are known. Yet, these results at the very least suggest participants likely tune the gain of Ia excitatory feedback according to the task demands. Our results also provide a case in which humans have independent control over dynamic and static fusimotor drives, in agreement with (Vallbo et al., 1979). It is interesting to note that our previous study suggesting increases in static fusimotor drive independent of dynamic fusimotor drive with muscle shortening (Jalaleddini et al., 2017) can now also explain increased force uctuations in the 5-12 Hz range. This corroborates task-dependent, independent turning of dynamic and static fusimotor drive. These results are consistent with the dierential functions of those two components of the fusimotor system. Dynamics fusimotor drive adjusts the dynamics sensitivity of muscle spindle to small uctuations in muscle length (Mileusnic et al., 2006), thereby allowing more precise force control. On the other hand, static fusimotor drive modulates the static bias of muscle spindles, which can prevent muscle spindle going slack with muscle shortening (Mileusnic et al., 2006). The frequency range in which we nd signicant eects of Ia excitation in our study (3-7 Hz) is lower than that often attributed to Ia excitation (between 6{12 Hz, Christakos et al. 2006; Hagbarth and Young 1979; Lippold 1970). For example, Christakos et al. (2006) reported force tremor with the peak frequency of 7.5 1.0 Hz, which can be eliminated by ischemia that is known to preferentially occlude Ia excitation from primary muscle spindles (Lippold, 1970; Sinkjaer et al., 2000). This discrepancy cannot be explained simply by a potential dierence in the conduction delay. We recorded force output from wrist exor muscles, which are more proximal to rst dorsal interosseous muscle Christakos et al. (2006) recorded from. Also, the loop delay of Ia excitation in our model is only 17 ms, similar to the H-re ex latency of exor carpi radialis muscle reported experimentally (Ongerboer de Visser et al., 1984). Despite such a short delay, its eect is seen in the frequency range between 3{7 Hz range (Figs. 37 and 38) likely because the frequency at which Ia excitation modulates synaptic inputs depends more on the dynamics of muscle spindle activity arising from the underlying contraction dynamics which uctuate at a very slow rate (0.5{1 HZ) as shown in Fig. 37a. Instead, this discrepancy may be attributable to architectural dierences between two muscles. For example, wrist exors are relatively compliant due to their large tendon length-to-ber length ratio and high intrinsic tendon compliance (Loren and Lieber, 1995) whereas rst dorsal interosseous muscle is likely much less compliant due to its very short tendon (Infantolino 174 and Challis, 2010). Future study is required to test this possibility, but would likely be useful to narrow down the previously dened large frequency range over which Ia excitation aects force variability|which has confused the ongoing debate on the sources of the 6{12 Hz force variability in humans (McAuley, 2000). Our results show that eects of gain modulation of segmental feedback pathways are very limited above 10 Hz (e.g., Fig. 38b) and our model is not sucient to generate the large peak at11 Hz we observed in our experimental data (Fig. 35c). Although this peak falls within the frequency range traditionally associated with proprioceptive feedback mechanisms (Christakos et al., 2006; Lippold, 1970), it seems to arise from dierent neural mechanisms. We nd that this 10 Hz force variability arises from strong synchronized activity between wrist exors, and it is radically diminished in wrist extensors (see Appendix). Although Ia aerents also project heteronomously to synergistic muscles monosynaptically, their eects are relatively weak compared to their homonymous connections (Pierrot-Deseilligny and Burke, 2005). Furthermore, there is no experimental evidence to suggest that the strength of canonical proprioceptive feedback pathways we included in our model dier between exors and extensors. Accordingly, recent experimental evidence has started to suggest the origin of the 10 Hz force variability in the descending input from the subcortical structures (Gross et al., 2002; Grosse and Brown, 2003; Laine and Valero-Cuevas, 2020; Soteropoulos and Baker, 2006), which may be the source of the11 Hz force uctuations we observed. However, it is important to note that the subcortical origin of the 10-Hz force uctuation does not necessarily rule out the possible role of segmental feedback and spinal interneurons in mediating its expression. For example, inhibitory feedback from Ib and Renshaw inhibition might reduce the in uence of the descending 10 Hz oscillatory synaptic inputs (Fig. 42 and also see Williams and Baker 2009b; Williams et al. 2010). More importantly, the spinal circuitry likely plays an important role in regulating synaptic inputs to multiple muscles (Kargo and Giszter, 2008) and the balance in the strength of synaptic inputs between synergistic muscles is likely under the in uence of the gains of individual elements of the spinal circuitry set by the descending motor systems (Raphael et al., 2010; Tsianos et al., 2014). In keeping with this idea, the degree of synchronization between two synergistic muscles (rst dorsal interosseous and exor digitorum supercialis) at10 Hz as well as at20 Hz was found to be diminished in patients with deaerentation compared to controls (Kilner 175 et al., 2004). Interestingly, synchronization at20 Hz between muscle and electroencephalographic activity over the motor cortex does not seem to be aected by deaerentation (Patino et al., 2008). These experimental observations suggest a fruitful research direction the systematic analysis of synchronization patterns between muscles may reveal coordination strategies of synergistic muscles embodied through the integrative action of hierarchical sensorimotor systems. Clinical implications Amplication of force variability or distortion of its frequency content is an almost universal phe- nomenon whenever neuromuscular control is altered in those conditions, as for example by aging Enoka et al. (2003); Tracy et al. (2005) and neurological diseases (Chu and Sanger, 2009; Lafargue et al., 2003; Lodha et al., 2010; Rothwell et al., 1982; Vaillancourt et al., 2001a). The common prac- tice of simply characterizing such observations as `increased noise' (Chu and Sanger, 2009; Therrien et al., 2016) misses an opportunity to uncover the underlying physiological mechanisms, and their eects on motor impairment and recovery. In contrast to this prevailing view, the above discussion highlights the important, yet often overlooked, need for an integrative approach to understand force variability by considering the hierarchical structure of distributed sensorimotor systems. This is critical because those conditions due to aging and neurological conditions manifest themselves across many levels of such hierarchy. For example, aging may be accompanied by changes at the level of motor units such as a reduction in the number of motor units in a pool (Hunter et al., 2016), which may increase the level of motor noise (Nagamori et al. 2020), but also it is associated with the impaired ability to modulate the gain of Ia excitation in a task dependent manner (Baudry et al., 2010) and altered visuomotor strategies (Keenan et al., 2017). Similarly, many neurological disorders that originate in the supraspinal structures such as stroke and Parkinson's disease aect the gains of spinal segmental feedback (Hagbarth et al., 1975; Katz and Rymer, 1989; Thilmann et al., 1991), which might compound the eects of disruptions in the spinal structures. Importantly, these mechanisms all likely aect the amplitude of force variability in frequency-specic manners, providing an opportunity to dissect them with straightforward but powerful spectral analyses of simple isometric forces. Our results argue that detailed spectral analysis of synaptic input and the resulting force variability likely provide tenable methods to characterize the underlying physiological 176 mechanisms for altered sensorimotor control in healthy aging and neurological conditions. 177 Summary and Future Work Summary In this dissertation, we systematically investigated physiological determinants of force variability during a constant isometric force production in humans. Specically, we attempted to resolve an apparent dichotomy in the current literature: force variability is either an xed, unavoidable conse- quence of noise in motor systems or a re ection of control strategies embodied through distributed sensorimotor systems that is amenable to change. We did so rst by developing a series of novel computational models that address specic aspects of force variability in Chapters 2-4. In Chapter 2, we presented a model of aerented muscle that allowed us to address the previously underestimated importance of closed-loop neuromechanical interactions among motor noise, a error correction mechanism, proprioceptive feedback, and mechanical properties of muscles and tendons. We showed 1) a potential origin of low-frequency force variability associated with co-modulation of motor unit ring rates (i.e.,`common drive'), 2) an in-depth characterization of how proprioceptive feedback pathways suce to generate 5-12 Hz physiological tremor, and 3) evidence that modulation of those feedback pathways (i.e., presynaptic inhibition of Ia and Ib aerents, and spindle sensitivity via fusimotor drive) in uence the full spectrum of force variability. In Chapter 3, we developed a model of a motor unit population that rectied critical limita- tions and assumptions in previous models by adding 1) calcium kinetics that drives fusion of motor unit twitches, 2) coupling between motoneuron discharge rate, cross-bridge dynamics, and mus- cle mechanics, and 3) a series-elastic element to account for the aponeurosis and tendon. These renements greatly improved our ability to estimate motor noise. Using this new model, we demon- strated that the two previously proposed mechanisms for motor noise (i.e., the stochastic nature of motoneuron discharge and unfused tetanic contraction) cannot account for the majority of force variability nor for its signal dependence. In Chapter 4, we introduced a complete model of a motor unit population by adding a model of motoneuron spike generating dynamics to the model described in Chapter 3. Addition of this 178 component was critical to simulate eects of synaptic noise that gives rise to noise in the motor command on discharge patterns of motor units and the resulting force variability. Using this model, we ruled out an alternative explanation of signal-dependent noise in force that it is a natural by- product of signal-dependent noise in the motor command. Finally in Chapter 5, we presented an integrative sensorimotor systems model that combines all elements presented in Chapter 2-4. This model enabled, for the rst time, detailed quantication of emergent behaviors that arise from interactions between distributed sensorimotor systems and realistic noise. We demonstrated that this model was able to replicate and explain experimental observations in both the literature and the current study at much ner and deeper levels than previously possible. Altogether, we concluded that force variability is mostly not the result of constant, low-level, random noise as assumed in today's prevailing theories, but rather it contains a rich source of infor- mation about the consequences and contributions of our hierarchical and distributed sensorimotor system to intended and unintended variability in voluntary action in health and disease. Further- more, we highlighted the integrative approach between computational modeling and experimental observations is critical to establish the causal relationship between a proposed mechanism and an observed phenomenon. To this end, detailed spectral analysis of synaptic input and the result- ing force variability likely provide a tenable method to characterize the underlying physiological mechanisms for altered sensorimotor control in health and disease. Future Work Our work presented in this dissertation concerns only an isometric force production where the length of a `musculotendon unit,' not muscle, is hold constant. The isometric force production was studied here because 1) the amplitude of force variability directly relates to motor performance of certain daily motor tasks such as ne manipulation of a handheld object and 2) it has been the main experimental condition in which the amplitude and spectral structure of force variability are quantied in the previous literature (e.g., Jones et al. (2002); Moritz et al. (2005)). However, many motor tasks such as reaching and walking require non-isometric contractions. As such, many 179 theoretical models have attempted to describe variability in those dynamics movements by invoking the concept of noise in the motor systems. An important, yet untested, assumption in the current literature is that features of force variability (the amplitude and its signal-dependence) during an isometric condition extrapolates to those dynamics movements. Furthermore, as extensively discussed in Chapter 3 and 4, the amplitude of implemented noise in previous theoretical models has been grossly overestimated and requires re-evaluations of previous simulated eects of motor noise. The new model of a motor unit population presented in this dissertation can be used to test these issues by extending it to dynamics conditions because it incorporates known muscle length and velocity-dependence of motor unit activation and muscle force generation mechanisms such as sag, yielding, and force-length and velocity relationships (Brown et al., 1999; Brown and Loeb, 2000). Our models presented in Chapter 2-5 was limited to control of a single muscle. However, an isolated isometric contraction of a single joint action in most cases involves coordinated activity of multiple muscles. For example, the wrist exion action we studied in Chapter 5 involves exor carpi radialis and exor carpi ulnaris muscles. One of the few exceptions for this is the well-studied index abduction in which rst dorsal interosseous is assumed to be the only muscle involved. Future work is necessary to address this issue because it is important for an accurate estimation of the motor noise contribution as it depends highly on the number of motor units involved (Fig. 21). An increase in the number of muscles involved eectively increases the number of motor units that are active at a given level of force, thus making the temporal and spatial summation of motor unit twitches more eective to reduce the contribution of motor noise. Moreover, it is not only essential for the completeness for modeling the biological system we try to understand but also important for advancing our understanding of distributed sensorimotor control and its consequence in health and disease as discussed below. First, the distributed control of multiple muscles directly concerns with the problem of muscle redundancy. Muscle redundancy for isometric force production arises because the musculoskeletal system often has more muscles than necessary to produce a given net torque level at given single degree of freedom. The above mentioned wrist exion is the very example of muscle redundancy where we have two muscles to perform a single joint action. One of the solutions to this apparent 180 problem is application of the optimality principle to the biological control (Cohn et al., 2018; Loeb, 2012) such that the central nervous system optimize task performance by minimizing a cost function that often includes the sum of the squared muscles activations (Diedrichsen et al., 2010; Harris and Wolpert, 1998; Todorov and Jordan, 2002). Thus, optimal (feedback) control theory may predict the work load imposed on individual muscles would be scaled in proportion to their relative mechanical contributions to a given task. Fagg et al. (2002) shows an empirical support for this theory by demonstrating that muscle activation patterns of ve wrist muscles are well aligned to their respective pulling directions. However, such an observation merely shows the consistency between a prediction and an observation and does not necessarily provide the causal relationship. In fact, a recent experimental observation argued against such an interpretation and rather suggested muscle activation patterns are not optimally tuned to minimize muscles activations, but rather rely on habitual control strategies based on already learned patterns (de Rugy et al., 2012). Loeb (2012) proposed a good-enough control as an alternative view to the optimal control, which suggests observed sensorimotor behaviors re ect stored motor programs found to be useful by trial-and- error learning, rather than optimal solutions computed online (Loeb, 2012). Importantly, emergent properties of the regulatory function of spinal circuits seem to facilitate such learning processes by providing many good enough solutions (Raphael et al., 2010; Tsianos et al., 2014). From this perspective, activation patterns of synergistic muscles may be highly dependent on synaptic inputs from spinal circuits re ecting task-dependent sensory feedback arising from the periphery. Further work that extends our current single muscle system into a multiple-muscle system may directly test this prediction. Moreover, as noted in the Discussion in Chapter 5, the relatively large peak at10 Hz in wrist exion force we observed cannot be explained by our model of a single muscle and seems to require considerations for shared synaptic input between the two wrist exors. In fact, we nd the activity of those muscles are strongly correlated at this frequency and the strength of synchronization is much weaker between wrist extensors (See Appendix). The most plausible origin of such input is that from the subcortical structures (Gross et al., 2002; Grosse and Brown, 2003; Laine and Valero-Cuevas, 2020; Soteropoulos and Baker, 2006). It is interesting to note that this exor bias is a predominant feature of abnormal synergistic activation patterns post-stroke (Dewald and Beer, 181 2001; Miller and Dewald, 2012; Twitchell, 1951) and a recent study shows muscles expressing such abnormal synergy is highly coupled at10 Hz (Lan et al., 2017). This abnormal synergy has been suggested to arise from over-activation of the brainstem structure in which the reticulospinal tract originates (Baker, 2011; Ellis et al., 2012). Accordingly, the activation of the same brainstem structure through acoustic startle generates bilateral synchronization between muscles at10 Hz and the strength of such synchronization is stronger for proximal muscles (Grosse and Brown, 2003), consistent with the proposed in uence of the brainstem descending motor pathways (Lawrence and Kuypers, 1968; Lemon, 2008). These results altogether may suggest the 10 Hz synchronized activity we observed between wrist exors has the same brainstem origin. However, this simplistic interpretation does not account for other experimental observations suggesting the involvement of other subcortical structures (Laine and Valero-Cuevas, 2020; Soteropoulos and Baker, 2006) and the heightened 10 Hz synchronization in patients with Parkinson's disease (Laine and Valero-Cuevas, 2020). Furthermore, recent experimental observations showed that the strength of this 10 Hz shared synaptic input across muscles is not xed as expected from such a simplistic interpretation, but can be modulated by coordination requirements of the task (de Vries et al., 2016; Laine and Valero-Cuevas, 2017). An eort to identify the neural mechanisms for the 10 Hz shared synaptic input and its ampli- cation in stroke and Parkinson's disease might be further facilitated by not focusing on a single neural structure but rather considering the hierarchical nature of sensorimotor systems as we have done throughout this dissertation. It is important to note that the subcortical origin of the 10-Hz force uctuation does not necessarily rule out the possible role of the spinal circuitry in mediat- ing its expression. For example, inhibitory feedback from Ib and Renshaw cells might reduce the in uence of the descending 10 Hz oscillatory synaptic inputs (Fig. 42 and also see Williams and Baker 2009b; Williams et al. 2010). Moreover, the above-mentioned exor bias might be mediated by the underlying spinal connectivity responsible for exor/withdrawal re ex (Pierrot-Deseilligny and Burke, 2005). Finally and most importantly, the spinal circuitry likely plays an important role in regulating synaptic inputs to multiple muscles (Kargo and Giszter, 2008), which is likely modulated by incoming sensory signals both monosynaptically and through spinal interneurons and in uenced by the gain setting of individual elements in the circuitry set by the descending motor 182 systems (Raphael et al., 2010; Tsianos et al., 2014). In keeping with this idea, signicant10 Hz synchronization between two index exors muscles (rst dorsal interosseous and exor digitorum supercialis) found in healthy individuals was diminished in patients with deaerentation (Kilner et al., 2004). Notably, this10 Hz synchronization was weak or absent in other pairs such as rst dorsal interosseous and extensor digitorum communis. A similar, but much widespread, observa- tion was found as synchronization at20 Hz (Kilner et al., 2004). Interestingly, synchronization at 20 Hz between a muscle and electroencephalographic activity over the motor cortex does not seem to be aected by deaerentation (Patino et al., 2008), suggestive of dierent neural mechanisms. This pertinent role of the spinal cord in muscle coordination raises a possibility that expression of the 10 Hz shared input and its disruption may be at least partially the result of the underlying spinal connectivity and how the descending motor systems sets its gains to achieve a desired motor action. This possibility might be tested by extending our models to a multi-muscle system and adding elements for the proposed mechanisms in this dissertation. 183 Appendix Flexor-bias of the 10 Hz shared drive to synergistic muscles We asked twelve healthy male individuals (mean age of 27.24.1) to perform a constant isometric force production at 20% of their MVC for 60 sec. They performed this task with wrist exors and wrist extensors for a single trial each. Visual feedback was displayed on a computer screen placed 1 m in front of them to aid production of a constant isometric force at the target level. MVC was determined as the peak force value they achieved in two MVC trials performed prior to the force tracking task. They received verbal encouragement during the MVC trials, but did not receive any other instruction. Force was measured using a 6 degree-of-freedom force-torque sensor (Model 20E12A4 100N, JR3 Inc., Woodland, CA). The force signals was acquired at 1000 Hz using a National Instruments data acquisition system (NI USB-6218, National Instruments, Austin, TX). The only force in the z-direction was used for visual feedback. We recorded electromyogrphaic (EMG) activity of wrist muscles: exor carpi radialis (FCR), exor carpi ulnaris (FCU), extensor capri radialis (ECR) and extensor carpi ulnaris (ECU). Surface EMG electrodes with bandwidth of 20-460 Hz (Biometrics, Newport, UK) were placed on their respective muscle bellies. EMG signals were acquired at 1000 Hz using a Biometrics LTD DataLINK system (Biometrics, Newport, UK). Both systems were triggered by a custom written MATLAB program (The Math Works, Natick, MA). Recorded EMG signals were high-pass ltered at 150 Hz using 4-th order butterworth lter and then rectied. EMG signals from the last 5-sec data segment were used for further analysis. The degree of shared neural drive between muscles was quantied using coherence analysis (Laine and Valero-Cuevas, 2017). Coherence spectrum of a pair of the high-pass ltered and rectied EMG signals was computed using the mscohere function in MATLAB using 2-s rectangular windows without an overlap. The frequency resolution was chosen to be 0.5 Hz. Raw coherence values (C) were transformed into standard z-scores using the formula,Z = [atanh( p C= p 1=2L)]bias, where 184 L is the number of disjoint segments and bias is the average Z-transformed coherence between 100 and 500 Hz (Laine and Valero-Cuevas, 2017). Comparisons of the frequency content of the shared neural drive between wrist exors and exten- sors show that wrist exors receive much stronger shared neural drive between 8-15 Hz as indicated by higher coherence values (Fig. 43a). This observation was further conrmed by comparisons of average coherence in this frequency range within each individual (Fig. 43b). Every participant demonstrated stronger synchronization in wrist exors compared to extensors and the dierence was highly signicant (p < 0:01). These results suggest a dierence in the shared neural drive received by pairs of muscles depending on their muscle actions. 185 Figure 43: Flexor-bias of the 10 Hz shared drive to synergistic muscles. a) Comparisons of coherence spectra between wrist exors and extensors. Note that the highly synchronized activity in the 8-15 Hz between wrist exors, which is almost absent between extensors. b) Comparisons of coherence in the 8-15 Hz range between the two muscle pairs. 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Abstract (if available)

Abstract This dissertation systematically investigates physiological determinants of force variability during constant isometric force production in humans. I use an integrative approach between computational modeling of human sensorimotor systems and experimental observations. Despite the long history of research on this topic, there are still continuing debates about the fundamental sources of force variability and their relative importance. Here, we attempt to resolve an apparent dichotomy in the current literature: force variability is either a fixed and unavoidable consequence of inherent noise in sensorimotor systems vs. a malleable consequence of control strategies embodied through hierarchically distributed sensorimotor systems. To this end, we present a series of physiologically realistic models of sensorimotor systems that allows us to perform 1) critical evaluations of previous assumptions and the underlying physiology and 2) direct quantification of the relative importance of individual sources of force variability and their interactions to explain experimentally observed features of force variability. We do so by comparing our model predictions against published and our own new experimental observations. We show that force variability is mostly not the result of constant, low-level, random noise as assumed in today's prevailing theories, but rather it contains a rich source of information about the consequences and contributions of our hierarchical and distributed sensorimotor system to intended and unintended variability in voluntary action in health and disease. Furthermore, we highlighted the integrative approach between computational modeling and experimental observations is critical to establish the causal relationship between a proposed mechanism and an observed phenomenon. To this end, detailed spectral analysis of synaptic input and the resulting force variability likely provide a tenable method to characterize the underlying physiological mechanisms for altered sensorimotor control in health and disease.

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Asset Metadata

Creator Nagamori, Akira (author)

Core Title Experimental and model-based analyses of physiological determinants of force variability

School School of Dentistry

Degree Doctor of Philosophy

Degree Program Biokinesiology

Publication Date 07/21/2020

Defense Date 04/22/2020

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag computational modeling,force variability,motor noise,motor unit physiology,muscle mechanics,muscle model,OAI-PMH Harvest,sensorimotor control,spinal circuit,synchronization

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Valero-Cuevas, Francisco (committee chair), Finley, James (committee member), Loeb, Gerald (committee member)

Creator Email a.nagamori0517@gmail.com,nagamori@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-336395

Unique identifier UC11665408

Identifier etd-NagamoriAk-8717.pdf (filename),usctheses-c89-336395 (legacy record id)

Legacy Identifier etd-NagamoriAk-8717.pdf

Dmrecord 336395

Document Type Dissertation

Rights Nagamori, Akira

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA