University of Southern California Dissertations and Theses

Spinal-like regulator for control of multiple degree-of-freedom limbs

(USC Thesis Other)

Spinal-like regulator for control of multiple degree-of-freedom limbs

PDF

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content
SPINAL-LIKE REGULATOR FOR CONTROL OF MULTIPLE
DEGREE-OF-FREEDOM LIMBS

by

Giby Raphael

A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(BIOMEDICAL ENGINEERING)

August 2009

Copyright 2009 Giby Raphael

ii

DEDICATION
To my dearest parents – PRB & GB.

iii

ACKNOWLEDGEMENTS
I gratefully thank my primary thesis advisor and committee chair – Professor
Gerald E. Loeb. Your step by step guidance made this challenging and initially
unsolvable research problem tractable. Most of all, you have taught me how to
patiently and intelligently approach a seemingly impossible problem and I am sure
that the lessons I learned as your student will be the cornerstone for all my future
endeavors in life. You also taught me by example the essential skills of a successful
project manager and the importance of having personal and well tested convictions in
all areas of life. Thanks for believing in me.
I thank the members of my thesis committee – Professors: Gerald E. Loeb,
Francisco J. Valero-Cuevas, James Gordon, Stefan Schaal and Rahman Davoodi. I
appreciate your valuable guidance and suggestions. Thanks also for your
encouragement and motivation. They added a lot of value to this process.
Special gratitude and thanks to all my colleagues who contributed to this
thesis: George Angelos Tsianos for your work on servocontrol models, Mansish Kurse
and Jonathan Weisz for helping me to mathematically optimize the model, Dr. Dan
Song for your work on Virtual Muscle model and part-time students – He Zheng,
Jonathan Yoke, and Matthew Haley for contributions during your internships at our
lab.
I also thank all my lab members – Dr.Hilton Kaplan, Dr.Milana Mileusnic, Dr.
Wei Tan, Dr. Dan Song, Dr.Veronica Santos, Rahul Kaliki, Markus Hauschild,
Djordje Popovic, Nicholas Wettels, and Jeremy Fischel. Thanks for your timely
iv

suggestions and above all, thanks for all the fun we had in the last many years. I also
want to mention the members of the BION team with whom I closely interacted
during the initial years of my program – Ray Peck, Jesse Singh, Sandarsh Mohan
Kumar, Paul Buckley and Ed Ramos.
This research was made possible by the funding from Alfred E. Mann Institute
for Biomedical Engineering and by the NSF Engineering Research Center for
Biomimetic MicroElectronic Systems at the University of Southern California.
And last, by most certainly not the least, I thank my ever-patient,
unconditionally supportive family. I could always count on your support and prayers
all the way across the oceans in India! Thanks for your love and continuous
encouragement throughout this process. Without you this would not have been
possible.

-Giby Raphael

v

TABLE OF CONTENTS
DEDICATION ........................................................................................................ ii
ACKNOWLEDGEMENTS ................................................................................... iii
LIST OF TABLES ................................................................................................ vii
LIST OF FIGURES ............................................................................................. viii
ABSTRACT ....................................................................................................... xviii
CHAPTER 1: INTRODUCTION ............................................................................ 1
i. ORGANIZATION OF THE THESIS .............................................................. 3
ii. BACKGROUND AND SIGNIFICANCE ....................................................... 4
iii. PREVIOUS EFFORTS TO MODEL SENSORIMOTOR CONTROL ......... 14
CHAPTER 2: METHODS ..................................................................................... 30
i. COMPONENT MODELS .............................................................................. 31
a. Muscle model .......................................................................................... 31
b. Computation of muscle force direction ................................................... 35
c. Proprioceptive sensory models................................................................ 37
d. Visualization tool .................................................................................... 41
ii. INTEGRATED BIOMECHANICAL MODEL ............................................. 42
a. Anatomy and mechanics of the human wrist joint .................................. 43
b. Modeling wrist mechanics ...................................................................... 46
iii. MODEL OF THE SPINAL CORD ................................................................ 49
a. Computational model of the interneuron ................................................ 49
b. Modeling the spinal pathways ................................................................. 52
iv. COMMAND MODEL .................................................................................... 68
v. OPTIMIZATION OF RUNTIME .................................................................. 72
CHAPTER 3: INTUTIVE OPTIMIZATION OF SPINAL CORD ....................... 74
i. STABILIZING RESPONSE TO FORCE PERTURBATION ....................... 76
ii. RAPID VOLUNTARY MOVEMENT TO POSITION TARGET ................ 82
iii. OUTPUT OF ISOMETRIC FORCE TO A TARGET LEVEL ..................... 88
a. Pulse force trajectory ............................................................................... 90
b. Step force trajectory ................................................................................ 94
iv. ADAPTATION TO VISCOUS CURL FORCE FIELDS .............................. 97
v. CONCLUSION ............................................................................................ 100
CHAPTER 4: MATHEMATICAL OPTIMIZATION OF SPINAL CORD ....... 102
i. STABILIZING RESPONSE TO FORCE PERTURBATION ..................... 105
vi

ii. RAPID VOLUNTARY MOVEMENT TO POSITION TARGET .............. 109
iii. OUTPUT OF ISOMETRIC FORCE TO A TARGET LEVEL ................... 114
iv. ADAPTATION TO VISCOUS CURL FORCE FIELDS ............................ 119
v. CONCLUSION ............................................................................................ 124
CHAPTER 5: SYSTEMATIC ANALYSIS OF CONTROL INPUTS ............... 128
i. PROPRIOSPINAL PATHWAY .................................................................. 131
ii. MONOSYNAPTIC Ia PATHWAY ............................................................. 136
iii. RECIPROCAL Ia INHIBITION PATHWAY ............................................. 140
iv. Ib INHIBITION PATHWAY ....................................................................... 144
v. RENSHAW PATHWAY ............................................................................. 148
vi. CONCLUSION ............................................................................................ 152
CHAPTER 6: COMPARISON TO SERVOCONTROL SCHEME ................... 154
i. STABILIZING RESPONSE TO FORCE PERTURBATION ..................... 155
ii. RAPID VOLUNTARY MOVEMENT TO A POSITION TARGET .......... 157
iii. OUTPUT OF ISOMETRIC FORCE TO A TARGET LEVEL ................... 160
a. Force step trajectory .............................................................................. 160
b. Force pulse trajectory ............................................................................ 162
iv. ADAPTATION TO VISCOUS CURL FORCE FIELDS ............................ 163
v. CONCLUSION ............................................................................................ 165
CHAPTER 7: CONCLUSION AND FUTURE WORK ..................................... 168
i. CONCLUSION ............................................................................................ 168
ii. FUTURE WORK ......................................................................................... 170
a. Improvements in the current model ...................................................... 170
b. Model of shoulder-elbow motion .......................................................... 173
c. Clinical implications ............................................................................. 176
BIBLIOGRAPHY ................................................................................................ 179

vii

LIST OF TABLES
Table 2.1 : The action and grouping of the muscles controlling the human
wrist joint. .................................................................................................................. 45
Table 2.2 : Parameters of the biomechanical model .................................................. 47
Table 3.1: Comparing results (kinematic parameters) from the simulation with
similar experiment. .................................................................................................... 87

viii

LIST OF FIGURES
Figure 1.1: Sherrington’s stretch reflex circuit ............................................................ 5
Figure 1.2: A simple analogy of the role of the spinal cord to marionette
puppets (Loeb, 2001) ................................................................................................. 11
Figure 1.3: Hierarchical relationship of motor control (Loeb, 1990, 1999) .............. 13
Figure 1.4: VITE model (Cisek et al., 1998) ............................................................. 14
Figure 1.5: FLETE model (Van Heijst et al., 1998) .................................................. 16
Figure 1.6: Block diagram of FLETE model with realistic biomechanical plant ..... 18
Figure 1.7: Modeled circuitry with cells representing motoneurons and
interneurons. Numbers in italics are the approximate number of terminals
received by the cell. (Bashor, 1998) ......................................................................... 19
Figure 1.8: The modeled architecture (Maier et al., 2005) ........................................ 21
Figure 1.9: The structure of the model by Lan (1997). ............................................. 23
Figure 1.10: System block diagram (Lan et al., 2005) .............................................. 24
Figure 1.11: The optimal gains between feedback sensors and 10 muscles for
mid-stance of cat hind-limb (He, et al., 1991). Afferent sources, identified
along horizontal axis, connect at intersections with all muscle groups
identified along vertical axis. The positive gains are shown above and negative
gains below mirror-line. Note the small arrows that denote the tendency for
homonymous feedback to be positive from spindle (Ia) afferents and negative
from Renshaw cells (RC). Also, note the significant gain distribution along the
heteronymous connections. ....................................................................................... 26

ix

Figure 1.12: Schematic design of the hierarchical model used to test postural
maintenance by a pair of antagonist muscles. The symbols are: S*-expected
sensory feedback, E-error feedback, C – command signal, U-motoneuronal
activity,S-sensory feedback ....................................................................................... 28
Figure 2.1: Block diagram of mathematical functions comprising the Virtual
Muscle model (Cheng, Brown & Loeb, 2000) .......................................................... 33
Figure 2.2: Input-Output variables of the Virtual Muscle (VM) model. ................... 34
Figure 2.3: Structure of the muscle spindle model (Mileusnic et al. 2006) .............. 39
Figure 2.4: Structure of the intrafusal fiber model (Mileusnic et al. 2006) ............... 39
Figure 2.5: Input-output relation of an ensemble of Golgi tendon organs
represented as a piece-wise linear static relation. ...................................................... 40
Figure 2.6: Block diagram of the integrated simulation environment. ...................... 42
Figure 2.7: Angles of rotation about the wrist. (A) Extension, (B) Flexion, (C)
Radial deviation, (D) Ulnar deviation. ...................................................................... 45
Figure 2.8: Biomechanical model used in the simulation ......................................... 47
Figure 2.9: Plot of muscle length variation during extension ................................... 48
Figure 2.10: Neuron model (approximating the behavior of a cluster of linear
neurons) ..................................................................................................................... 51
Figure 2.11: Illustration showing the integration of inputs at the motoneuron.
Note that there is no direct descending input to the motoneuron. ............................. 54
Figure 2.12: The Propriospinal interneuron and its homonymous and
heteronymous connections (for a single muscle). ..................................................... 55
Figure 2.13: The Monosynaptic Ia excitation pathway and its homonymous
and heteronymous connections (for a single muscle). ............................................... 57
x

Figure 2.14: The Reciprocal Ia Inhibitory interneuron and its homonymous
and heteronymous connections (for a single muscle). ............................................... 59
Figure 2.15: The Ib Inhibitory interneuron and its homonymous and
heteronymous connections (for a single muscle) ...................................................... 61
Figure 2.16: The Renshaw interneuron and its homonymous and
heteronymous connections (for a single muscle) ...................................................... 63
Figure 2.17: Partial view of the Spinal-Cord model showing the connections
between two ‘Partial-Synergist’ muscles. The pathways shown are:
Monosynaptic Ia-pathway, (PN) Propriospinal pathway, (Ia) Reciprocal Ia-
inhibitory pathway, (Ib) Ib-Inhibitory pathway, and (R) Renshaw pathway.
The ‘SET’ gains adjusted the background activity in the spinal cord and the
‘GO’ gains initiated and maintained the transition to a new state. ............................ 67
Figure 2.18: Partial view of the Spinal-Cord model showing the connections
between ‘True-Antagonist’ muscles. The pathways shown are: Monosynaptic
Ia-pathway, (PN) Propriospinal pathway, (Ia) Reciprocal Ia-inhibitory
pathway, (Ib) Ib-Inhibitory pathway, and (R) Renshaw pathway. The ‘SET’
gains adjusted the background activity in the spinal cord and the ‘GO’ gains
initiated and maintained the transition to a new state. ............................................... 68
Figure 2.19: The command model. The ‘SET’ gains adjusted the background
activity in the spinal cord and the ‘GO’ gains initiated and maintained the
transition to a new state. ............................................................................................ 70
Figure 3.1: Rotation of the resting hand about the wrist joint (extension-
flexion axis) in response to a perturbation (100N, 10ms) at one second .................. 78
Figure 3.2: (A) Output from the motoneuron exciting the extensor muscles,
(B) Output from the motoneuron exciting the flexor muscles, (C) Muscle force
modulation of extensors, (D) Muscle force modulation of flexors. .......................... 79
Figure 3.3: (A) Primary output of muscle spindles attached to the extensor
muscles, (B) Primary output of muscle spindles attached to the flexor muscles,
(C) Output from GTO attached to the extensor muscles, (D) Output from GTO
attached to the flexor muscles. .................................................................................. 80

xi

Figure 3.4: (A) Command input to the spinal cord. The SET input was applied
for 1 sec and then GO inputs were applied to the propriospinal interneurons to
produce movement to the 30
0
, (B) External force perturbation (100N, 10ms)
applied at 2.3 sec of simulation time (C) Extension of the wrist to position
target of 35
0
and resistance to perturbation. .............................................................. 83
Figure 3.5: (A) Output from the motoneuron exciting the extensor muscles,
(B) Output from the motoneuron exciting the flexor muscles, (C) Muscle force
modulation of extensors, (D) Muscle force modulation of flexors. .......................... 85
Figure 3.6: (A) Primary output of muscle spindles attached to the extensor
muscles, (B) Primary output of muscle spindles attached to the flexor muscles,
(C) Output from GTO attached to the extensor muscles, (D) Output from GTO
attached to the flexor muscles. .................................................................................. 85
Figure 3.7: Experimental setup (Ghez and Gordon, 1987) ....................................... 90
Figure 3.8: (A) Experiment: the target force (60N) and the force impulse
generated by the subject. (B) Simluation: the net force calculated from the
force generated by the muscles. ................................................................................. 92
Figure 3.9: (A) Agonist EMG generated in the experiment for force impulse
task (B) Corresponding Antagonist EMG (C) Motoneuron output in simulation
to extensor muscles (D) Corresponding motoneuron output to flexor muscles
(E) Extensor muscle force modulation in simulation (F) Corresponding Flexor
muscle force modulation ........................................................................................... 93
Figure 3.10: (A) Target force (60N) and force produced by the subject in the
experiment (B) The net force generated in the simulation (60N) (C) EMG
generated by agonist muscles in experiment (D) Corresponding EMG
generated by antagonist muscles (E) Motoneuron output to agonist muscles in
simulation (F) Corresponding motoneuron output to antagonist muscles. ................ 96
Figure 3.11: (A) Schematic representation of the 2 degree-of-freedom robotic
manipulandum used in the experiment (Scheidt et al., 2001) (B) graphical
representation of the perpendicular force field presented to the subjects
(Scheidt et al., 2001). ................................................................................................. 98

xii

Figure 3.12: (A) Command input to the spinal cord. Initial SET command to
set the background activity and step functions as GO commands to initiate
movement (B) Rotation of the hand in the intended axis (extension) (C)
Deflection of the hand due to perturbation (radial/ulnar axis). Note that after
adaptation (manual tuning of gains) the deflection reduced from 40
0
to less
than 7
0
(D) After-effects in the radial/ulnar axis after the removal of
perturbing force. ........................................................................................................ 99
Figure 4.1: Rotation of the hand about the wrist joint (extension-flexion axis)
in response to a external force perturbation (100N, 10ms) at one second
simulation time. After optimization, the response improved significantly and
produced zero oshoot…. .......................................................................................... 107
Figure 4.2: Rotation of the hand about the wrist joint in response to external
force perturbation (100N, 10ms) at one second simulation time about a
random axis. (A) Rotation about extension-flexion axis (B) rotation about
radial/ulnar deviation axis ....................................................................................... 108
Figure 4.3: Learning curve that shows change in cost with the step size used in
annealing curve. One-iteration constitutes one pass through all the gains. In all
simulations the cost converged to a low value in a single iteration and did not
change significantly in subsequent iterations. This indicates there are many
stable local minima…………………………………………………………… ...... 108
Figure 4.4: (A) Extension to 450 and response to perturbation. The desired
trajectory is a ramp with an input rise time of 100ms. (B) Rotation in the
Radial/Ulnar deviation axis. The desired trajectory is to have zero movement
in that axis. (C) Output of the agonist and antagonist motoneurons exciting the
Extensor and Flexor muscles, during the ramp and during perturbation ................ 111
Figure 4.5: Learning curves from multiple starting points in state space
(random values for the control inputs), for 10 extension tasks. .............................. 111
Figure 4.6: Comparison of a subset of gains for two equally good solutions ......... 112
Figure 4.7: Pair-wise separations of all 10 solutions in state space, ordered
according to their distance from each other after optimization. Cost difference
between the pairs before and after optimization is also shown. .............................. 113
Figure 4.8: Pair-wise separations of all 10 solutions in state space, ordered
according to their distance from each other before optimization. ........................... 113
xiii

Figure 4.9: (A) Pulse force to the target level without overshoot. (B) Tracking
step force trajectory ................................................................................................. 115
Figure 4.10: (A) Output from the muscle spindles attached the agonist and
antagonist muscles (B) Output from Golgi tendon organs ...................................... 116
Figure 4.11: (A) Output of motoneurons exciting agonist and antagonist
muscles for 80ms rise time pulse force trajectory. Note the conspicuous
antagonist burst,(B) motoneuron output for 200ms rise time pulse force
trajectory. Note that the antagonist burst is missing for larger rise time. (C)
motoneuron output for 200ms rise time step force trajectory, (D) motoneuron
output for 500ms rise time step force trajectory. ..................................................... 116
Figure 4.12: (A) Rotation about the intended direction (40
0
extension) after
adaptation. (B) Negligible deflection in the radial/ulnar axis in the presence of
perturbing force. (C) Rotation about the intended direction (from another
random starting gain values). (D) Adaptation with negligible deflection in
radial/ulnar axis (E) After-effects in the intended direction after the removal of
perturbation. (F) After-effects in the radial-ulnar axis after removal of
perturbation. ............................................................................................................. 120
Figure 4.13: (A) Output from muscle spindles attached to all the four muscles,
(B) Output from Golgi tendon organs attached to all the four muscles, (C)
Motoneuron output exciting all the four muscles after adaptation, (D) Muscle
force modulation (extensors and flexors) after adaptation. (E) Motoneuron
output during after-effects (F) Muscle force modulation during after-effects. ....... 121
Figure 4.14: Learning curve for all the tasks simulated. Note that for every
task the maximum cost reduction is obtained in the first iteration. One-
iteration is defined as one pass through all the gains in the spinal cord .................. 126
Figure 5.1: Propriospinal pathway .......................................................................... 131
Figure 5.2: Deviation from mean values for Propriospinal pathway gains in
Response to perturbation task (Mean: 0.0013, Std Deviation: 0.0728) ................... 131
Figure 5.3: Deviation from mean values for Propriospinal pathway gains in
Rapid voluntary movement to position target task (Mean: -0.0361, Std
Deviation: 0.0903) ................................................................................................... 132
xiv

Figure 5.4: Deviation from mean values for Propriospinal pathway gains in
Voluntary output of isometric force to a target level task with Step force
profile (Mean: -0.011, Std Deviation: 0.1684) ........................................................ 132
Figure 5.5: Deviation from mean values for Propriospinal pathway gains in
Voluntary output of isometric force to a target level task with Pulse force
profile (Mean: 0.0013, Std Deviation: 0.140) ......................................................... 133
Figure 5.6: Deviation from mean values for Propriospinal pathway gains in
Adaptation to viscous curl force fields task (Mean: 0.0486, Std Deviation:
0.2442) ..................................................................................................................... 133
Figure 5.7: Monosynaptic-Ia Pathway .................................................................... 136
Figure 5.8: Deviation from mean values for Monosynaptic Ia pathway gains in
response to perturbation task (Mean: 0.0062, Std Deviation: 0.011) ...................... 136
Figure 5.9: Deviation from mean values for Monosynaptic Ia pathway gains in
Rapid voluntary movement to position target task (Mean: 0.039, Std
Deviation: 0.081) ..................................................................................................... 137
Figure 5.10: Deviation from mean values for Monosynaptic Ia pathway gains
in Output of isometric force to a target level task with step force profile
(Mean: -0.077, Std Deviation: 0.072) ...................................................................... 137
Figure 5.11: Deviation from mean values for Monosynaptic Ia pathway gains
in Output of isometric force to a target level task with pulse force profile
(Mean: -0.010, Std Deviation: 0.132) ...................................................................... 138
Figure 5.12: Deviation from mean values for Monosynaptic Ia pathway gains
in Adaptation to viscous curl force fields task (Mean: -0.007, Std Deviation:
0.193) ....................................................................................................................... 138
Figure 5.13: Reciprocal-Ia Inhibition Pathway ....................................................... 140
Figure 5.14: Deviation from mean values for Reciprocal-Ia Inhibition pathway
gains in response to perturbation task (Mean:-0.087, Std Deviation: 0.0271) ........ 140
xv

Figure 5.15: Deviation from mean values for Reciprocal-Ia Inhibition pathway
gains in Rapid movement to a position target task (Mean: 0.016, Std
Deviation: 0.0859) ................................................................................................... 141
Figure 5.16: Deviation from mean values for Reciprocal-Ia Inhibition pathway
gains in Output of isometric force to a target task with Step force profile
(Mean: -0.015, Std Deviation: 0.153) ...................................................................... 141
Figure 5.17: Deviation from mean values for Reciprocal-Ia Inhibition pathway
gains in Output of isometric force to a target task with Pulse force profile
(Mean: -0.065, Std Deviation: 0.118) ...................................................................... 142
Figure 5.18: Deviation from mean values for Reciprocal-Ia Inhibition pathway
gains in adaptation to curl force fields task (Mean: -0.029, Std Deviation:
0.245) ....................................................................................................................... 142
Figure 5.19: Ib-Inhibition pathway .......................................................................... 144
Figure 5.20: Deviation from mean values for Ib-Inhibition pathway gains in
Response to perturbation task (Mean: -0.0043, Std Deviation: 0.093) ................... 144
Figure 5.21: Deviation from mean values for Ib-Inhibition pathway gains in
Rapid movement to position target task (Mean: 0.0067, Std Deviation: 0.085) ..... 145
Figure 5.22: Deviation from mean values for Ib-Inhibition pathway gains
Isometric force to target level task with Step force trajectory (Mean: 0.062, Std
Deviation: 0.125) ..................................................................................................... 145
Figure 5.23: Deviation from mean values for Ib-Inhibition pathway gains
Isometric force to target level task with Pulse force trajectory (Mean: -0.054,
Std Deviation: 0.0109) ............................................................................................. 146
Figure 5.24: Deviation from mean values for Ib-Inhibition pathway gains in
Adaptation to viscous curl force field task (Mean: -0.019, Std Deviation:
0.279) ....................................................................................................................... 146
Figure 5.25: Renshaw pathway ............................................................................... 148
Figure 5.26: Deviation from mean values for Renshaw pathway gains in
response to perturbation task (Mean: 0.01918, Std Deviation: 0.068) .................... 149
xvi

Figure 5.27: Deviation from mean values for Renshaw pathway gains in Rapid
movement to position target task (Mean: -0.002, Std Deviation:
0.089)…………….. ................................................................................................. 149
Figure 5.28: Deviation from mean values for Renshaw pathway gains in
Output of isometric force to a target level task using Step force trajectory
(Mean:-0.052, Std Deviation: 0.168) ....................................................................... 150
Figure 5.29: Deviation from mean values for Renshaw pathway gains in
Output of isometric force to a target level task using Pulse force trajectory
(Mean: -0.066, Std Deviation: 0.118) ...................................................................... 150
Figure 5.30: Deviation from mean values for Renshaw pathway gains in
Adaptation to viscous curl force fields task (Mean: -0.25, Std Deviation:
0.313) ....................................................................................................................... 151
Figure 6.1: Wiring diagram of the servo-controller ................................................ 154
Figure 6.2: (A) Extensor Carpi Ulnaris (ECU) muscle force modulation, (B)
Flexor Carpi Radialis (FCR) muscle force modulation, Note the large co-
contraction in the muscles. (C) The best perturbation response along the x-
axis, (D) Corresponding response along the z-axis.(E) Typical perturbation
response along x-axis, (F) Corresponding response along z-axis. .......................... 156
Figure 6.3: (A) Extensor Carpi Ulnaris (ECU) muscle force modulation, (B)
Flexor Carpi Radialis (FCR) muscle force modulation (C) The best
performance: extension (D) Corresponding Z-axis deviation (E) Typical
performance: extension (F) Corresponding Z-axis deviation .................................. 159
Figure 6.4: (A) Output of the motoneuron to extensor muscle, (B) Extensor
muscle force modulation (C) Best solution (D) Typical solution for brief rise
time force trajectory…… ........................................................................................ 161
Figure 6.5: (A) Output of the motoneuron to extensor muscle, (B) Extensor
muscle force modulation (C) Best solution (D) Typical solution for brief rise
time force trajectory ................................................................................................ 163
Figure 6.6: (A) Extensor Carpi Ulnaris (ECU) muscle force modulation, (B)
Flexor Carpi Radialis (FCR) muscle force modulation (C) The best
performance: extension (D) Corresponding Z-axis deviation (E) Typical
performance: extension (F) Corresponding Z-axis deviatio .................................... 164
xvii

Figure 6.7: Comparison of the performance between the Spinal cord model
and the simple servo-control scheme for all the tasks. The variability in
experimental data from literature (Ghez et al., 1987; Wierzzbicka et al., 1991;
Liles et al., 1985) is also shown. Two sets of experimental data are shown for
the Force Pulse task 1) with least amount of co-contraction, 2) with significant
co-contraction (dotted). ........................................................................................... 166
Figure 7.1: Representative Renshaw pathway in the spinal circuitry for the arm
model (Tsianos et al., 2009) ................................................................................ 175
Figure 7.2: Parameter distribution in the spinal circuitry for the arm model
(Tsianos et al., 2009) ............................................................................................... 175
Figure 7.3: Block diagram of the BION controller ................................................. 177

xviii

ABSTRACT
The performance of motor tasks requires the coordinated control and
continuous adjustment of myriad individual muscles. The basic commands for the
successful performance of a sensorimotor task originate in “higher” centers such as the
motor cortex, but the actual muscle activation and resulting torques and motion are
considerably shaped by the integrative function of the spinal interneurons. The relative
contributions of brain and spinal cord are less clear for reaching movements than for
automatic tasks such as locomotion. We have modeled a two-axis, four-muscle wrist
joint with realistic musculoskeletal mechanics and proprioceptors and a network of
spinal circuitry based on the classical types of interneurons (propriospinal,
monosynaptic Ia- excitatory, reciprocal Ia-inhibitory, Renshaw inhibitory and Ib-
inhibitory pathways) and their supraspinal control (via biasing activity, presynaptic
inhibition and fusimotor gain). The modeled system has a very large number of
control inputs, not unlike the real spinal cord that the brain must learn to control to
produce desired behaviors. We then programmed this model to emulate actual
performance in four very different but well-described behaviors: 1) stabilizing
responses to force perturbations; 2) rapid movement to position target; 3) isometric
force to a target level; 4) adaptation to viscous curl force fields. We found that
relatively simple and generally intuitive step-changes in a small subset of descending
controls could reproduce each of these behaviors. We then optimized the control
inputs using a gradient descent algorithm. Even though the algorithm started with
random values for the control inputs, the model converged rapidly to produce
xix

physiologically realistic outputs. Our general hypothesis is that the real task of the
brain is to configure the spinal circuitry to minimize interventions by the brain during
the task.
1

CHAPTER 1: INTRODUCTION
A movement system, by definition, needs a controller and an actuator at a
minimum. In order to achieve a desired response in the face of external perturbations,
it also needs a feedback system via sensors. Compared to engineered systems,
biological movement systems face some unique challenges that any theory of
sensorimotor control must address (Loeb et al., 1999): 1) Biological muscles produce
large, instantaneous changes in output force when kinematic conditions change, but
respond sluggishly when neural activation changes. 2) Biological feedback circuits
have few “private-line” paths whereby command or feedback signals can be routed
selectively to individual actuators. Instead, the signals from large numbers of noisy
sensors of diverse physical variables converge with the signals from many command
centers before they are routed to motoneurons. 3) Animals usually find it more
valuable to perform sub-optimally, but adequately in the widest possible range of
circumstances rather than to perform optimally, but only for nominal conditions.
In the past years, there have been several attempts to model the sensorimotor
system for voluntary movement (examples explained below). These theories were
often built on various assumptions and unproven hypotheses, owing to the difficulty in
validating the basic principles through experiments. Realistic physiological systems
are versatile and may be capable of serving several different functions. There is
extensive interaction between the different components of the system and often the
controller and the plant characteristics themselves change to adapt to new
2

requirements. All these challenges make it extremely difficult to postulate a unified
theory of sensorimotor control, but each attempt is hopefully a step forward towards a
better understanding of the complicated system behind voluntary movement.
In this thesis our primary goal was to investigate the role of the spinal cord in
voluntary movement. The initial commands for sensorimotor tasks originate in the
higher centers such as the motor cortex, but it is well known through numerous
experimental observations that these commands are considerably modified by the
spinal cord, before they reach the muscles that actuate the movement. The division of
labor between the brain and the spinal cord is unclear; the major focus in recent years
has been to understand the motor cortex in isolation from the rest of the neural and
mechanical subsystems with which it interacts. This isolationist approach was
necessary to gain an initial understanding of the kinds of information that might be
processed in motor cortex, but we submit that it isn’t going to tell us what role the
motor cortex actually plays in sensorimotor control.
Researchers working on motor cortex have already started to include models of
the musculoskeletal system (Scott, 2004), but most continue to ignore the subsystem
that actually connects the motor cortex to the limb – the spinal cord. We have
developed an explicit model of the spinal cord circuitry from the various types of
interneurons that have been described in the literature. We then programmed this
model to emulate actual performance of a wide range of tasks as described in human
subjects and nonhuman primates. We confirmed that the spinal cord circuitry in itself
possesses the capability of performing a wide range of tasks with relatively simple
3

inputs from the cortex. This property of the spinal cord will substantially change and
may generally simplify the learning of motor tasks by the brain.

i. ORGANIZATION OF THE THESIS
Chapter 1 provides the context for the research work described in this thesis. It
also draws into focus the significance of this thesis. The chapter concludes with a
detailed review of the prior work done on modeling sensorimotor control of voluntary
movement.
Chapter 2 discusses the work involved in developing the spinal cord model and
the musculoskeletal plant. A brief explanation of all the components used to build the
biomechanical model is provided. The pathways in the spinal cord that we modeled
are explained in detail, along with the corresponding justifications and citations in the
literature.
Chapter 3 describes the results from the intuitive optimization of the spinal
cord for four different tasks that we selected to replicate in our model.
Chapter 4 describes the results from the mathematical optimization of the
spinal cord model using gradient descent algorithm for the same four different tasks
that we modeled in chapter 3. The common features across all the tasks, the learning
curve etc are compared in the conclusion.
4

Chapter 5 describes the systematic analysis of the control inputs (gains) in the
spinal cord. The analysis is also followed by our assumptions and inferences from the
strategy used by the control algorithm to optimize the gains.
Chapter 6 compares the spinal cord model with a much simpler classical servo-
control scheme. The results from both models are compared in detail.
Chapter 7 describes the conclusion and the suggested future work.

ii. BACKGROUND AND SIGNIFICANCE
The main idea behind this thesis was first suggested in 1906 by none other than
Sir Charles Sherrington! In 1906 Sir Charles Sherrington published “The Integrative
Action of the Nervous System”, which was a collection of ten lectures delivered two
years before at Yale University in the United States. In this monograph Sherrington
summarized two decades of painstaking experimental observations and his
interpretation of them. It settled the then-current debate between the “Reticular
Theory” versus “Neuron Doctrine” ideas about the fundamental nature of the nervous
system in mammals in favor of the latter, and it changed forever the way in which
subsequent generations have viewed the organization of the central nervous system
(Burke, 2007).

In L
reflex pathw
1.1). It ha
descending
With the lim
that can pr
voluntary m
In its literal
how close h
In L
which he no
calls ‘the p
paths, from
converges f
Fig
Lecture II, a
ways, he pro
ad three seq
‘propriospin
mited experim
roduce repe
movements re
l sense his c
he came to th
Lecture IV, S
otes is a ‘ma
principle of
m afferent rec
from multip

gure 1.1: She
s an indicat
ovided a prop
quential ne
nal neuron’
mental resou
etitive move
epresent cha
conclusion w
he real truth,
Sherrington
ain problem
the common
ceptors, and
ple afferents
errington’s st
tion of his i
posed circui
eurons: the
and the ‘fi
urces he had
ement patter
ains of reflex
was proven f
single hand
begins his c
m in nervous
n path’. He
d public path
s. The mot
tretch reflex
ideas about
it diagram fo
afferent ‘r
inal commo
d then, his v
rns and he
x responses l
false, but th
dedly, a centu
consideratio
co-ordinatio
clearly dis
hs, i.e. neur
toneurons th
x circuit
the central
or the stretch
receptive ne
on pathway’
iew did not
went on t
inked togeth
his thesis dra
ury ago.
on of how re
on’ and intro
tinguishes b
rons that rec
hat activate
organization
h reflex (Fig
eurons, a l
motor neur
include circ
o propose
her by the br
aws attention
eflexes inter
oduces what
between priv
ceive input
muscles w
5
n of
gure
ong
ron.
cuits
that
rain.
n to
ract,
t he
vate
that
were
6

necessarily represented as the ‘final common path’ but in addition he also suggested
other common pathways that receive convergent inputs, which he called ‘internuncial
neurons’ (which are now commonly called interneurons). He argued that since both
‘allied reflexes’ and ‘antagonistic reflexes’ acted through the same final common path
the motoneuron, the central mechanisms must involve interactions between
interneuronal common paths that received convergent afferent inputs and also
descending inputs. In fact, his assumption is very close to our hypothesis in this thesis.
We intentionally left out any direct descending connection to the motoneuron in our
spinal cord model, but instead the outputs from the proprioceptive sensors and the
descending inputs mixed in the interneurons to produce all the results discussed in this
thesis.
The importance of direct cortico-motoneuronal (CM) pathway has been much
highlighted in the history of motor control research (Lemon et al., 2004), but the
indirect pathway to the motoneurons via the spinal interneurons has mostly been
neglected, in spite of numerous experimental observations that such pathways
contribute the majority of inputs to the motoneurons (Galfan et al., 1964; Alstermark
1985; Kitazawa, 1993). Direct corticomotoneuronal projections are largely absent in
opossums, rodents, cats and lower primates (Rathelot and Strick, 2009) that are
capable of fairly sophisticated, voluntary limb movements for prey-catching and/or
food-handling. Even for the hand and finger muscles of rhesus monkeys, for which
there are strong, direct corticomotoneuronal projections, these direct pathways to the
muscles may be influenced by presynaptic inhibition and by the integration of direct
7

afferent and interneuronal synaptic activity in the motoneurons themselves,
Sherrington’s final common path for motor control.
Many studies have shown that in cats the descending excitation from the
cerebral cortex is transmitted to forelimb motoneurons via two different routes:
interneuronal systems in the same segments as the motoneurons, called as the
segmental interneurons, and propriospinal interneurons that are located outside the
segment of the motoneurons, in C3-C4 segments for the upper extremity (Alstermark
et al., 1985; Illert M. et al., 1977; 1978; 1984; Kitazawa et al., 1993). By analyzing
behavioral effects of partial lesions in cortico- and rubro-spinal pathways, they have
shown that the C3-C4 propriospinal inteneurons are primarily involved in mediating
the descending command for ‘target reaching’ and that the segmental interneurons are
involved in mediating the descending commands for ‘food taking’, which involved
reach and grasp of food with digits.
Most of the pyramidal tract output related to reaching tasks appears to synapse
on a variety of spinal interneurons (Isa et al., 2007). Those spinal interneurons receive
convergent input from many modalities and sources of primary somatosensory
afferents (as well as extrapyramidal descending activity) and they exert divergent
excitatory and inhibitory influences on many motoneuron pools (McCrea, 1986; Fetz
et al., 1989). The role of indirect CM pathway was systematically investigated again
recently (Is et al., 2007) in macaque monkeys. Lesions were made on the corticospinal
tract at the border between C4 and C5 segments, thus interrupting the direct CM
connections, while preserving a major portion of the descending axons of the C3-C4
8

propriospinal interneurons. The monkey was able to reach and grasp the food piece
between the tips of the index finger and thumb (‘precision grip’) on the first day
following the lesion. However, the independency of the fingers was affected, which
later improved after a few days. These results suggest that, if the CM connection is
interrupted, indirect pathways can transmit the commands for controlling dexterous
finger movements. It should be noted that our argument is not against the significance
of direct CM pathway, but rather, is to emphasize the prominence of the indirect CM
pathway, as suggested by experimental evidence.
Most studies of arm and hand movements in primates have focused largely on
the role of the motor cortex. They have correlated the activity of populations of
neurons in motor cortex with measureable features of the concomitant motor behavior.
Various research teams have selected different features, including end-point
kinematics (the position and/or velocity of the hand in space), end-point kinetics (the
forces applied by the hand to a manipulandum) or muscle activity (measured as
rectified and filtered EMG). The implication has been that reasonably strong
correlations and predictions constitute support for the hypothesis that motor cortex
computes and encodes the selected feature as an essential step in transforming the
intent to perform a particular behavior into the reality of its execution. One popular
extension of this concept is that the brain contains an internal model of the
musculoskeletal plant which allows it to compute detailed, feedforward commands to
that plant (e.g. Kawato, 1999).
9

Over many years (e.g. Fetz, 1992), there has been a growing appreciation that
there are some fundamental problems with this interpretation of correlations and
predictions from such experimental data (Loeb, 2009 unpublished):
1) All of these measureable features of motor behavior (and others that can be
imagined but have not been employed) tend to be highly correlated with each
other as an inevitable result of the physical design of the musculoskeletal
system of the limb itself. Any particular correlation that may be observed is
not necessarily a reflection of either its causation by or computation within the
motor cortex.
2) The particular transformation that best predicts the selected output features
from the input neural activity tends to change if an independent aspect of the
task changes (e.g. rotation of tuning vectors for kinematics of the hand if the
posture of the proximal arm is changed; Scott & Kalaska, 1995). This apparent
instability of the motor cortical transform is inevitable if the same neurons
actually receive or compute information related to those “independent”
aspects.
3) The transforms are always computed on a highly selected subset of the neural
activity actually present in motor cortex. The design of the electrodes
introduces an unknown sampling bias, half or more of the intermingled units
are summarily rejected because their activity appears to be uncorrelated with
an abbreviated set of “search” movements, and many others are rejected
because a high degree of cross-correlation among cortical units tends to result
10

in “overfitting” and poor generalization to new datasets (e.g. Westwick et al.,
2006).
4) The transforms are generally computed from tightly constrained tasks for
which the animal is highly over-trained. This experimental standardization has
been necessary to allow pooling of neural activity recorded over time and
across animals, but it may not reflect the true functionality of motor cortex
under more natural conditions.
5) The motor cortex receives substantial somatosensory input from the thalamus
and from somatosensory cortex, but the nature and strength of this feedback is
itself likely to be modulated by motor cortex and many other descending
projections (Ajemian et al., 2008).
We submit that most of these problems arise from trying to understand the
motor cortex in isolation from the rest of the neural and mechanical subsystems with
which it interacts. This isolationist approach was necessary to gain an initial
understanding of the kinds of information that might be processed in motor cortex, but
it isn’t going to tell us what role the motor cortex actually plays in sensorimotor
control. This thesis draws attention to the two other subsystems that are often
neglected by researchers: 1) the musculoskeletal system, and 2) the subsystem that
actually connects the motor cortex to the limb – the spinal cord.
11

From a robotics engineering standpoint, the graceful control of multiarticulated
limbs equipped with slow, non-linear muscles is still a baffling concept. Even though
the biological solution is complex and incompletely known, the vertebrate spinal cord
provides an existence proof that such control is, indeed, possible. One of the simple,
yet highly relevant, analogies of the importance and the general role played by the
spinal cord was suggest by Loeb (2001), in comparing it to marionette puppets (Figure
1.2).

Figure 1.2: A simple analogy of the role of the spinal cord to marionette puppets (Loeb, 2001)
12

The obvious analogy is that the marionette puppet represents the
musculoskeletal system and the operator represents the brain, leaving the deceptively
simple handheld pieces on which the strings (muscles) are attached to represent the
spinal cord. The puppeteer has to thoroughly integrate the intrinsic mechanical
properties of the puppet with the range of movements and tasks it must perform. The
mechanics of the puppet account for the trajectory of motion in response to particular
pattern of tugs from the strings. The handheld control operated by simple movements
of puppeteer’s hand, causes simultaneous tugs and relaxations on many strings.
Reactive forces from the puppet and any external obstacles propagate backward to the
control, where they result in additional, complexly distributed movements of the
strings that occur more rapidly than the voluntary reaction time of the operator. The
operator can change the nature of these reactions by adjusting the orientation and
stiffness of his/her grip on the control and by modifying the linkage itself by
repositioning moveable parts of the control during the performance.
The marionette analogy corresponds well with the hierarchal control strategy
of motor control suggested by Loeb (1990, 1999), that provides the big picture view
based on which this thesis was formulated (Figure 1.3). Following the conventions of
control engineering, the brain is represented as an adaptive controller that learns what
descending commands to send to the spinal cord. The spinal cord essentially functions
as a programmable regulator, a multi-input, multi-output system of distributed
interconnections with adjustable gains. The regulator mixes the multiple inputs from
the sensors into the multiple outputs to the actuators of the musculoskeletal system –
13

the plant. The nonlinear intrinsic properties of the muscles themselves contribute
substantially to the stabilization of various behaviors, particularly those involving
perturbations (Preflexes as defined in Brown and Loeb, 2000).

Figure 1.3: Hierarchical relationship of motor control (Loeb, 1990, 1999)

In this thesis we have explored the role and the capabilities of the spinal cord
in detail. Many of the results that emerged were unexpected and the emergent
properties of the spinal cord model showed surprisingly robust and physiologically
realistic characteristics. Moreover, the model was relatively easy to control in
14

comparison to other engineering models with similar structure and multi-dimensional
parameter space. The results discussed are evidence supporting our hypothesis
regarding the role of the spinal cord in voluntary movement. Our proposal is that the
real task of the brain is to learn to use this programmable regulator to achieve its goals,
rather than to seize direct control of the individual muscles themselves.

iii. PREVIOUS EFFORTS TO MODEL SENSORIMOTOR
CONTROL
Cisek, Grossberg and Bullock (1998) suggested the Vector Integration to
Endpoint (VITE) model as a cortico-spinal model of reaching and proprioception
under multiple task constraints. It was a computational model that incorporated model
neurons corresponding to already identified cortical cell types (Figure 1.4). A brief
description of the model is provided below.

Figure 1.4: VITE model (Cisek et al., 1998)
15

DVV (Desired Velocity Vector) in area 4 (primary motor cortex) serves as a
volition-sensitive velocity command which activates lower centers including gamma-
dynamic motor neurons. A voluntarily scalable GO signal gates DV (Difference
Vector) input from posterior parietal area 5 to the DVV. The DVV command is
integrated by a tonic cell population in area 4 whose activity serves as an Outflow
Position Vector (OPV) to lower centers including alpha and static gamma motor
neurons. OFPV (Outflow Force Position Vector) cells in area 4 enable graded force
recruitment to compensate for static and inertial loads using inputs from cerebellum
and integrated spindle feedback. PPV (Perceived Position Vector) is computed in area
5, where it is derived by subtracting spindle based feedback from a copy of OPV. The
final arm movement difference vector (DV) is computed in parietal area 5 from a
comparison of TPV (Target Position vector) and PPV. Inertial Force Vector (IFV)
phasic cells reduce velocity errors and SFV (Static Force Vector) compensates for
static loads such as gravity.
The model showed results such as: 1) maintaining accurate proprioception
while controlling reaches to spatial targets, 2) exertion of force against obstacles, 3)
maintaining posture despite perturbations, 4) compliance with an imposed movement,
4) and compensation for static and inertial loads. However, the model was a lumped
representation and completely neglected, among other things, the intrinsic properties
of the lower levels of the sensorimotor system including the dynamic mechanical
properties of the muscle (Brown and Loeb, 2000) and the actual connectivity of the
16

interneuronal circuitry of the spinal cord, etc. This made it difficult to appreciate the
relative contributions of individual elements and connections in their myriad details.
Van Heijst et al. (1998) developed a model that contained populations of spinal
interneurons and motoneurons responsible for reciprocal control of a simple antagonist
pair of muscles operating a hinge joint (Figure 1.5). They showed that the system
could self-organize its synaptic weights to produce stable control of sinusoidal
movement trajectories. This work was built on previous work from this group on the
FLETE (Factorization of Length and Tension) model of such reciprocal pathways
(Bullock and Grossberg, 1992; Bullock and Contreras-Vidal, 1993).

Figure 1.5: FLETE model (Van Heijst et al., 1998)

17

There were two signals to the motoneuron pools, A
1
to the agonist pool (Ag)
and A
2
to the antagonist pool (An). The combination (A
1
, A
2
) defined the motor
intention, i.e. the desired joint angle. In addition to these two signals there was a co-
contraction signal P, which fed to both motoneuron pools. The joint stiffness is
specified by setting the values as A
1
+P and A
2
+P for the agonist and antagonist pools
respectively. Two pools of Ia-interneurons were added to prevent saturation of the
motoneuron pools as each of them inhibited the motoneuron pool in the opposite
channel. They also inhibited each other to avoid saturation in themselves. The
recurrent inhibition from the Renshaw neuron pool also helped to avoid saturation in
the motoneuron pool, especially when size-principle was imposed on the
motoneurons. The Renshaw neuron pool received excitation from its own motoneuron
pool and sent inhibitory signals back to the same motoneuron pool, to the ipsilateral Ia
interneuron pool and to the contralateral Renshaw pool.
Although consistent with some of the known spinal cord circuitry, the FLETE
model was a lumped model and did not include most of the circuits identified in the
spinal cord. The modification of Van Heijst et al. (1998), added features such as: 1)
self-organization of weights by a Hebbian learning rule, 2) motoneuron pools that
obeyed the size-principle of recruitment, 3) and inhibitory interneurons analogous to
Renshaw cells. The modifications still left out well-known interneurons and
connections in the spinal cord and, most importantly, did not address the extension of
this model to control multi-muscle, multiple degrees of freedom joints.
18

We tried to modify the FLETE model to control the movements of a realistic
biomechanical plant that used accurate mathematical models of muscles and sensors.
The control task was the posture and movement of a model of the wrist joint, with two
degrees of freedom and operated by four muscles (Figure 1.6), similar to the
biomechanical plant we later used with the spinal cord model described in this thesis.
We were unable to train the FLETE model using unsupervised learning to control the
muscles and the model did not interface well with realistic models of proprioceptive
sensors.

Figure 1.6: Block diagram of FLETE model with realistic biomechanical plant
19

Figure 1.7: Modeled circuitry with cells representing motoneurons and interneurons. Numbers
in italics are the approximate number of terminals received by the cell. (Bashor, 1998)

Bashor (1998) created a large-scale model of some spinal reflex circuitry to
study the dynamic interactions among neuronal populations during simple behaviors.
The properties and populations of the neurons and terminals were derived from the
literature, mainly on cat spinal cord. The resulting model contained roughly 2300
20

neurons and about 600000 connections. The interneurons were defined as groups of
cells and each cell in the source population had the possibility of contacting any cell in
the target population. The program had 12 such cell populations driven by fiber
populations. The populations and their interconnections are shown in Figure 1.7. The
MN populations projected to a pair of imaginary antagonistic muscles. The other ten
cell populations in the model represented the known and functionally characterized
interneuronal populations.
The model was primarily used to study how synaptic integration could account
for simple reflexes, such as stretch and Golgi tendon organ reflexes. It was reported to
produce behaviors such as agonist excitation and reciprocal inhibition, but it was never
integrated to a biomechanical model that could simulate any physiologically realistic
movements.
Maier et al. (2005) developed a model of the corticospinal and rubrospinal
circuitry that they trained to duplicate the temporal patterns of recruitment in an
antagonist pair of muscles operating a single DOF wrist joint. The modeled
architecture (Figure 1.8) was comprised of 4 basic unit modules: cortical, rubral and
segmental units and muscle afferents. The step-tracking target position input was
relayed to the cortical module, whose output units [P] project their activity to the
rubral and segmental population. Both cortical and rubral units connected to the
segmental level, which included alpha and gamma motor units projecting to the
muscle. Muscle afferents projected back to the segmental and supraspinal populations.
In addition, a torque feedback was given to the muscle afferents.
21

Figure 1.8: The modeled architecture (Maier et al., 2005)

They did not model the musculoskeletal mechanics but they included a
spindle-like sensor of muscle stretch whose gain could be modulated by a single type
of fusimotor activity. They used back-propagation to adjust gains so that the recursive
loops between spinal cord and brain would produce the alternating patterns of
antagonist motor unit firing observed during alternating flexion-extension wrist
movements, as well as reflex responses to muscle-length perturbations.
22

Artificial neural networks have been applied to the control of point-to-point
arm movements (Lan et al., 1994; Todorov 2000; Fetz et al., 1990; Abbas 1997). The
feed-forward ANN controller was used to learn and store optimal patterns (based on
minimum effort criteria) of muscle stimulation for a range of movements. However, it
was trained for only single joint arm movement. Furthermore, the learned network
architecture did not resemble the detailed and definite structure of the spinal cord, but
instead it functioned more like an open-loop pattern generator.
Lan (1997) proposed a controller for multi-joint arm movements that employed
a hierarchical control structure and included a simple spinal network with reciprocal
inhibition and feedback from spindle-like sensors (Figure 1.9). The model had a
central controller (brain) that represented the motor goal in the Cartesian space and
computed joint equilibrium trajectories and excitation signals so as to minimize effort.
At the lower level, a neural network mimicking the spinal cord (Lan et al., 1994)
translated the excitation signals and equilibrium point (EP) trajectories into control
commands to three pairs of antagonist muscles for a simplified two-joint arm model.
23

Figure 1.9: The structure of the model by Lan (1997).

The model showed results such as: 1) arm movements whose dynamic and
kinematic features were similar to those of voluntary arm movements, 2) near-straight
path with a smooth bell-shaped velocity curve for fast movements, 3) ‘N’ shaped
equilibrium trajectories , 4) triphasic excitation signals, 4) dynamically modulated
joint stiffness, 5) and, adaptation to external load changes. Essentially the model
extended the hierarchical control structure and the minimal effort criterion function to
multi-joint movements on the basis of the equilibrium point hypothesis (Feldman et
al.,1986,; Latash et al., 1991;1991;1992).
24

The same author later proposed the alpha-gamma model (Lan et al., 2005), in
order to investigate the plausible roles of spinal proprioceptive feedback in movement
control (Figure 1.10). The model included minimalistic spinal circuitry (stretch reflex,
reciprocal inhibition and recurrent inhibition) and was tested on a simple joint
controlled by a pair of antagonist muscles. They suggested the regulation of the
equilibrium position of a joint as a contribution of the spinal reflexes, on a similar
paradigm as the equilibrium point hypothesis.

Figure 1.10: System block diagram (Lan et al., 2005)

The model had two pairs of inputs, one to the alpha neuron and the other as
static gamma inputs, corresponding to the descending commands. The descending
excitation signals were transformed into muscle control inputs at the alpha motor
25

neuron pool to produce a movement. Peripheral information, such as the length and
velocity of muscle contraction, was fed back to the model to regulate the activities of
motor neuron pools. The other circuits included muscle autogenic reflex, reciprocal
inhibition to the opposing muscle, as well as recurrent inhibition from the Renshaw
cells. The external load was modeled as a constant torque applied to the joint that
moved in the horizontal plane.
Both these models were not tested on a realistic biomechanical plant with
accurate models of muscles and sensors. Also, it was not clear how the same
principles could be extended to control multiaxial joints such as the wrist, where the
muscles effectively switch pairings as synergists and antagonists depending on the
direction of movement. The lambda equilibrium-point hypothesis of Feldman (1966)
constitutes a general model of spinal cord coordination of posture and movement in
simple antagonist muscle systems (Feldman et al., 1998), but it considers only a very
limited subset of known spinal circuitry. It remains controversial whether it can be
extended to tasks involving control of force (Ostry and Feldman, 2003).
The first application of engineering tools for the design of optimal neural
controllers was by He, Levine and Loeb (1991). They simulated the feedback
regulation of standing posture in response to small perturbations on a dynamical
model of the cat hind limb. The model had a three-joint limb, moving only in the
sagittal plane and was driven by ten musculo-tendon actuators equipped with force
and length sensors similar to biological proprioceptors. The musculoskeletal model
incorporated realistically nonlinear elements, but the authors reasoned that its behavior
26

was locally linear for small perturbations, enabling them to compute optimal gain
settings for all sensors onto all actuators using engineering methods for the design of
linear quadratic regulators (LQR). They reported that a relatively natural-appearing
response to perturbations during standing could only be achieved by including in the
performance criterion the feedback information from all the available modalities of
sensors. Their model also produced connectivity and gain matrices (Figure 1.11) that
had striking similarities to the known pathways in the spinal cord responsible for
various reflexes.

Figure 1.11: The optimal gains between feedback sensors and 10 muscles for mid-stance of cat
hind-limb (He, et al., 1991). Afferent sources, identified along horizontal axis, connect at
intersections with all muscle groups identified along vertical axis. The positive gains are
shown above and negative gains below mirror-line. Note the small arrows that denote the
tendency for homonymous feedback to be positive from spindle (Ia) afferents and negative
from Renshaw cells (RC). Also, note the significant gain distribution along the heteronymous
connections.
27

However, the model had many limitations: 1) LQR theory is valid only for
locally linear conditions, so the gains must be recomputed for different phases of
movements such as stance and swing; 2) Computation of a LQR depends on the
complete, weighted specification of the cost matrix to be minimized; 3) There must be
a sensor or an estimator for each state variable in the dynamical system but, in reality,
state variables such as joint angles and velocities are probably estimated from
unknown combinations of other sensors such as muscle spindles (Scott and Loeb,
1994); 4) The LQR gains represent only the incremental output to the muscles in
response to the perturbations, not the total muscle activation that includes the nominal
output of the unperturbed system in response to central commands and ongoing
sensory feedback; 5) The LQR gains cannot be resolved into the contributions from
discrete populations of spinal interneurons as described in the neurophysiological
literature. The spinal cord model described in this thesis incorporates key principles
from the LQR model of the spinal cord, such as indirect control of motoneurons,
hierarchical structure between controller and regulator (Loeb et al., 1990), and
homonymous and heteronymous network between the sensors and actuators, while
seeking to overcome these limitations.
Loeb, Brown and Cheng (1999) laid out a more general approach to
apportioning motor control among the hierarchical elements of neocortex, spinal cord
and musculoskeletal plant. Their model had the fewest number of elements that were
capable of expressing the general emergent properties of the hierarchy, which
consisted of a task planner and adaptive controller (brain) at the highest level, a
28

programmable regulator (spinal-cord) in the middle and the musculoskeletal plant at
the lowermost level (Figure 1.12). In the highly simplified examples chosen to
illustrate the emergent properties of such hierarchical systems, the plant consisted of a
single DOF hinge joint operated by a pair of antagonistic muscles equipped with force,
length and velocity sensors. The spinal cord model consisted of two or three
interneurons, the output of which depended on the product of a corresponding
command signal from the brain times the sum of the sensor signals. The interneurons
operated linearly with no threshold or rectification.

Figure 1.12: Schematic design of the hierarchical model used to test postural maintenance by a
pair of antagonist muscles. The symbols are: S*-expected sensory feedback, E-error feedback,
C – command signal, U-motoneuronal activity, S-sensory feedback.
29

Even though the modeled system was highly simplified, they showed the
importance of the lower levels in the hierarchy in three modeled tasks: acquiring a
target in the face of random torque-pulse perturbations, optimizing fusimotor gain for
the same perturbations, and minimizing postural error versus energy consumption
during low versus high frequency perturbations. They reported that the emergent
properties of the lower levels maintained stability in the face of feedback delays,
resolved redundancy in over-complete systems, and helped to estimate loads and
respond to perturbations. It was apparent that the controllability of a particular
musculoskeletal plant in a particular task depended on the availability of a set of
interneurons with reasonable connectivity. Also, changing the conditions for which a
given task was optimized resulted in a similar nominal activation of the motoneurons
in the absence of perturbation, but a completely different distribution of activity
among the available interneurons driving those motoneurons. Thus the integration of
the descending inputs and the reflex signals depended on the availability of the
corresponding interneurons and the complex connectivity between them. It was also
shown that spinal-like regulators composed of poorly chosen interneurons may
produce only inadequate or unstable control programs, regardless of the gains (Loeb et
al., 1989). These early systems models provided the foundation for the expanded
model of spinal circuitry investigated in this thesis.

30

CHAPTER 2: METHODS
Voluntary movement and its control in humans and other animals involve
complicated interactions between the nervous system (controller) and the
musculoskeletal system (plant). In order to understand the integrated system and to
investigate the mechanisms involved, movement must often be studied in its entirety.
Experimental studies are often difficult and are largely limited in their ability to
expose the workings of internal components which are not directly observable. Thus,
researchers have to rely on accurate and complete computer models of the
neuromusculoskeletal system (NMS) that can extend and complement experimental
studies. Even though models inevitably simplify reality, a viable compromise is often
reached between experimentally realistic model specifications and computational
complexity.
This thesis was made possible because of the preparatory experimental and
modeling studies done by Dr.Gerald E. Loeb and his students/colleagues over the past
30 years. They built detailed and easily deployable mathematical models of the
components of the musculoskeletal system and developed an integrated modeling tool
to facilitate the study of the control of movement. The simulation environment was
built using Matlab/SimulinkTM, which provides a convenient software platform for
building component models, integrating them into a system model and for performing
simulation and analysis. The modeling task involved two major parts: 1) building a
realistic biomechanical plant, and 2) building the integrated spinal cord model. All
31

components that we used in our model and the integrated system are discussed briefly
in the following sections.

i. COMPONENT MODELS
a. Muscle model
Virtual Muscle
TM
(Cheng, Brown and Loeb, 2000) is a Matlab-based tool that
provides a simple graphical user interface for creating Simulink blocks of
mathematical models of muscles. The original model was based on extensive
experimental data set collected from feline muscles (Brown et al. 1996, 1999, 2000)
and was later scaled to human fiber-type models according to published data. The
model provides a more accurate description of muscle force production than any
previous models, accounting for the interactive effects of length, velocity and
activation over the physiological ranges of each. It is based upon the premise of
modeling specific fiber-types in the muscle and then summing the effects of different
populations of fiber-types to create a whole muscle.
The structure of the model is shown in Figure (2.1). It consists of an active
contractile element with activation, length and velocity dependencies in parallel with a
viscoelastic element for passive force. The total fascicular force is applied in series to
the inertial mass of the muscle and a series elastic element for tendon and aponeurosis.
A brief description of the constituent physiological components is given below.
32

F
PE1
– Passive elasticity, component 1. Stretch of muscle fascicles results in
elastic recoil due to stretching of connecting myofilaments within sarcomeres and
endomysial connective tissue between muscle fibers.
F
PE2
- Passive elasticity, component 2. Active shortening of a muscle below a
certain length results in elastic resistance due to compression of the thick filaments
against the Z-plates of the sarcomeres.
FL – Force-Length. The tetanic, isometric force-length relationship due to
changes in filament overlap.
FV – Force-Velocity. The tetanic force-velocity relationship, which is thought
to be due to population changes in the angles of cross-bridges attachment and the
corresponding dependence of force upon cross-bridge angle.
Af – Activation-frequency. The isometric activation-frequency relationship,
which is presumably due to the effects of stimulus frequency on mean intracellular
calcium concentrations as well as the effect of calcium concentrations on cross-bridge
attachment probabilities.
L
eff
– ‘Effective’ length. This element introduces a time lag between the actual
fascicle length and the ‘effective’ fascicle length, which is one of the inputs into the
function describing the Af relationship. This parameter was later removed (Song et al.
2008) to solve the infinite oscillation problem introduced by the delayed length
dependency of activation-frequency relationship.
33

F
eff
–‘Effective-frequency’. This element introduces a time lag between the
actual stimulus frequency and the ‘effective’ frequency, which is another input into the
function describing the Af relationship and accounts for the rise and fall times.
S –Sag. This element accounts for declining force output during sub-maximal
stimulus trains in fast-twitch muscle fibers.
Y- Yielding. This element accounts for nonmonotonic effects of stretching on
force output during sub-maximal stimulus trains in slow-twitch muscle fibers.

Figure 2.1: Block diagram of mathematical functions comprising the Virtual Muscle model
(Cheng, Brown & Loeb, 2000)

The inputs to the model are 1) Neural Activation: The recruitment element of
the muscle model converts this activation input into an effective firing frequency of
the motor units of the muscle. This value is normalized to be between 0 and 1. 2)
34

Musculotendon path length (m): This value is calculated from the skeletal dynamics in
the integrated simulation environment. 3) Frequency (pps): This input is only required
for FES recruitment scheme and is the stimulus frequency applied to the motor pool.
The outputs from the model are 1) Force (N): The force output from the model
is measured at the series elastic element. 2) Other optional outputs include, Force
(normalized), Activation (as in input), Fascicle Length (normalized) and Fascicle
Velocity (normalized).

Figure 2.2: Input-Output variables of the Virtual Muscle (VM) model.

The model was later modified (Song et al. 2008) to incorporate the following
changes. 1) A new muscle recruitment algorithm called ‘Natural Continuous’ was
introduced as a continuous approximation of the discrete recruitment of slow and fast
motor units used in the previous model. The new algorithm used a single motor unit
for each fiber-type that was frequency modulated and weighted to approximate the
size-ordered physiological recruitment used in the previous recruitment scheme. The
force output during smooth recruitment and derecruitment was thus predicted without
Virtual Muscle
TM
Neural
Activation
Musculotendon
Path Length
Frequency
Force (N)
Force
(normalized)
Activation
(input)
Force-Length
(normalized)
Force-Velocity
(normalized)
35

having to specify a large number of independently recruited units. This eliminated the
oscillations that emerged in closed-loop simulations due to the discontinuity in the
output force modulation. It also reduced the number of states for each integration step
in the model, thus improving the computational efficiency. 2) Another recruitment
algorithm called ‘Intramuscular FES’ was also implemented which modeled the
muscle force production as a result of intramuscular Functional Electrical Stimulation
(FES). It included a single motor unit for each fiber-type with its frequency of firing
specified by an input variable representing stimulus frequency. Force output was
weighted equally among the motor units to approximate the random fiber-type
recruitment reported for intramuscular electrical stimulation. 3) The Virtual Muscle
model was reformulated into state space representation and implemented in C
language using CMEX S-function in Simulink. This reduced the computational load of
the model significantly. 4) An additional output, muscle stiffness was added to the
model by partial differentiation of the muscle force with respect to the fascicle length.
The second-generation Virtual Muscle model, with the new additions, has improved
stability and computational efficiency and is more suitable for the analysis of neural
control of movement.
b. Computation of muscle force direction
Muscles generate force along their effective lines of pull. The joint torques that
actually move the limb must be computed by multiplying this force by the moment
arm. The computation of moment arm in biological systems is complex because the
36

muscles often cross joints to one or more axes of rotation and their paths are often
constrained by boney protuberances and the bulging of adjacent muscles. During
movement, this gets even more difficult as the length and the moment-arm change
continuously. However, in our simplified biomechanical model (described in the next
section) the muscle forces acted along straight line paths between their attachment
points and thus the computation of the moment arms was relatively easy.
However, in order solve this problem for future, more realistic models, we
developed a computational method to represent the paths of the muscles in
musculoskeletal models. The algorithm was based on the modified version of a
previously developed method called the ‘Obstacle-Set algorithm’ (Garner et al. 2000,
2001). The method was based on the premise that the resultant muscle force acts along
the locus of the transverse cross-sectional axis of the muscle called the ‘centroid’. The
centroid (muscle path) is assumed to be a frictionless elastic band, which moves freely
over anatomical constraints such as bones and neighboring muscles. The anatomical
constraints, referred to as ‘obstacles’, were represented in the model by regular-
shaped, rigid bodies such as spheres and cylinders. The obstacles, together with the
centroid defined an obstacle set. The assumption was that the path of any muscle can
be modeled using one or more of the obstacle sets.
A brief description of the algorithm is as follows. Four points were defined
along the centroid, two ‘fixed’ points that coincided with the attachments points of the
muscles on the bones and two ‘via’ points on the anatomical constraint or obstacle.
The coordinates of the ‘via’ points on the obstacle were computed with reference to
37

the fixed points on the bone. The ‘via’ points were set to be active based on a
wrapping algorithm that verified whether the muscle actually passed through them.
The algorithm continuously computed the coordinates of all active ‘via’ points during
movement. The active points were then used to find the muscle path and the new
muscle length was calculated as the shortest distance between them. The direct
computation of moment arm was thus avoided and the torque at the joint was
computed from the direction, point of application and magnitude of the muscle force.
c. Proprioceptive sensory models
Proprioceptive sensors such as muscle spindles and Golgi tendon organs are
the primary feedback mechanisms for the control of movement. Muscle spindles are
found in most vertebrate skeletal muscles lying in parallel to the extrafusal fibers.
They act as stretch receptors and their main function is to signal changes in the length
of the muscle within which they reside. The CNS thus uses them to sense information
critical for sensorimotor regulation and servocontrol, such as the position and velocity
of the limbs during movement. The Golgi tendon organs (GTO) are placed in series
between muscle fibers and tendon and aponeurosis. They function as tension-sensitive
mechanoreceptors and supply the CNS with information regarding active muscle
tension and the state of contraction of the muscle. We used previously developed
models of muscle spindles and GTOs in our simulations. Brief descriptions of the
models are provided below.
38

Muscle spindles (Mileusnic et al. 2006): The model consisted of two intrafusal
fiber models, bag 1 and combined bag 2 plus chain, reflecting their common fusimotor
drive (Figure 2.3). The three intputs to each intrafusal fiber models were the fascicle
length, the velocity and the relevant fusimotor input. The bag 1 fiber received dynamic
fusimotor control and was primarily responsible for velocity sensitivity of the spindle.
The bag 2 fiber and chain fibers were innervated by the static fusimotor control and
contributed mainly to length sensitivity. The model computed two outputs, namely,
primary (Ia) and secondary afferent (II).
Each intrafusal fiber was modeled with the same structure shown in (Figure
2.4) and was divided into a polar region and a central sensory region. The sensory
region was modeled as a pure elastic element (K
SE
), whose strain was linearly related
to afferent firing rate. The polar region was modeled as a spring (K
PE
) with a parallel
contractile element. The contractile element consisted of the active force generator and
the damping element. For each intrafusal fiber model, the equations for tension within
polar and sensory regions were combined into a nonlinear, first order differential
equation representing the net mechanical state. The primary afferent output was
obtained by summing the outputs of bag 1 and bag 2 plus chain intrafusal fiber
models, while secondary afferent output was obtained only from the bag 2 plus chain
intrafusal fiber models.
39

Figure 2.3: Structure of the muscle spindle model (Mileusnic et al. 2006)

Figure 2.4: Structure of the intrafusal fiber model (Mileusnic et al. 2006)

We improved the robustness and the computational efficiency of the original
spindle model by smoothing the output functions and by implementing it in a state-
space formulation using CMEX S-functions in Matlab. The intrafusal activation levels
of the gamma dynamic and gamma static inputs were scaled to a value between 0 and
1, to keep it consistent with the range of other input values used in the integrated
simulation environment. We used only the primary output (Ia) of the spindles in our
simulations which was sufficient to provide both position and velocity feedback to the
controller. The spindle secondary afferents provide essentially pure muscle length
40

information, but relatively little is known of their interneuronal circuitry in the spinal
cord.

Figure 2.5: Input-output relation of an ensemble of Golgi tendon organs represented as a
piece-wise linear static relation.

Golgi tendon organs (Crago et al. 1982; Mileusnic and Loeb 2006; 2009): The
steady-state input-output relation for the tendon organ was found to be linear when the
motor-unit rates were confined to the physiological range. From the total response of a
group of 19 tendon organs specified by Crago et al., we derived a piece-wise linear
static relation between the total muscle force and the afferent firing of the GTO
ensemble (Figure 2.5). The higher slope at the lower levels of muscle forces is due to
the initial recruitment of tendon organs, which makes the predominant contribution to
the total response at low forces.
41

d. Visualization tool
The display tools provided by Matlab show the results from the simulations as
waveforms or as data logs. Even though they were adequate for testing and debugging,
for more realistic demonstrations of the results and the task environment we used the
Musculo-Skeletal Modeling Software (MSMS, Davoodi et al. 2003). The MSMS
provides a comprehensive software platform for representing and visualizing
biomechanical models. The simulations can be run in a computer or using stereoscopic
displays in virtual reality that enables the subject to interact with computer-simulated
environments.
The integrated simulation environment is shown in Figure 2.6. The output
from the controller excited the muscles. The force from the muscle model was
converted to torque by multiplying it with the moment arm. The muscle length was
computed from the skeletal dynamics and was used as another input to the muscle
model. The sensory models used the different outputs from the muscle model and
provided proprioceptive feedback to the controller during closed-loop simulations.
The skeletal dynamics block converted the resulting torque to rotation at the joint. The
output motion was then visualized using MSMS.

42

Figure 2.6: Block diagram of the integrated simulation environment.

ii. INTEGRATED BIOMECHANICAL MODEL
Most of the prior work on biologically controllers for movement used simple
biomechanical models, with reduced degrees of freedom and limited physiological
verisimilitude. In our simulations we wanted to use a reasonably challenging and
physiologically realistic biomechanical plant in order to test the spinal cord model. We
thus chose the 2 degrees-of-freedom human hand movement about the wrist joint. The
human hand represents a mechanism of great complexity and intricate design. Its
functional capabilities and utility arises largely from its specific structural
Virtual
Muscle
Moment
Arm
computation
Skeletal
Model
&
Sensors
MSMS
Database
(Muscle parameters, Obstacle
parameters)
Force
Torque Motion
Muscle Length Orientation
Controller
Activation
Feedback
Fascicle velocity/length
43

characteristics. In this thesis we deal only with the rotation of the human hand about
the wrist joint. Whole volumes have been written on hand anatomy, and a detailed
description is beyond the scope of this thesis. However, the basic construction of the
bones and the neuromuscular apparatus for governing motions and forces at the wrist
joint (from Taylor et al. 1955) is briefly described below.
a. Anatomy and mechanics of the human wrist joint
The bones of the hand are grouped as the carpus and the digits. The carpus
comprise of eight bones which make up the wrist and the root of the hand. The carpal
bones are arranged in two rows, those in the more proximal row articulating with
radius and ulna bones of the forearm. Between the two is the intercarpal articulation.
The bony conformation and ligamentous attachments are such as to prevent both
lateral and dorsal volar translations but to allow participation in the major wrist
motions.
Most of the muscles of hand and wrist lie in the forearm and, narrowing into
tendons, traverse the wrist to reach insertions in the bony or ligamentous components
of the hand. Generally, the flexors arise from the medial epicondyle of the humerus, or
from adjacent and volar aspects of the radius and ulna, and then course down the
inside of the forearm. They are, therefore, in part supinators of the forearm. The
extensors of wrist and digits originate from the lateral epicondyle and parts of the ulna,
pass down the dorsal side of the forearm, and thus assist in pronation.
44

Three principal nerves serve the muscles of the hand. 1) Radial nerve - attaches
to the extensors of wrist, thumb and fingers. 2) Median nerve – Flexors of the wrist
and fingers, abductors, opponens, and flexors of thumb. 3) Ulnar nerve – all other
intrinsic muscles of the hand. Each of these major nerve trunks diverges into countless
smaller branches ending in the papillae of the palmar pads and dorsal skin.
The wrist joint, composed of the radiocarpal and intercarpal articulations, has
an elliptical rotation field with the major axis in the dorsal volar excursion and the
minor in the ulnar radial. No significant torsion occurs. The muscles traversing the
wrist include those inserting into the carpus and metacarpus and those mediating
flexion and extension of the phalanges. The latter contribute to the wrist action,
particularly under loads. In such cases, the finger muscles develop reaction against the
object held (or within the hand itself if the fist is clenched) and add their forces to
wrist action. The action and grouping of these muscles are given in Table 2.1. The
rotation of the bones during the wrist movement is too complex as they occur at
several articulating surfaces and the virtual axes of rotation lie distal to the contact
surfaces owing to gliding motions in the joints. However, they can be approximated
into four simple movements (Figure 2.7) by assuming the rotation is about a fixed
center: 1) Extension (dorsiflexion), 2) Flexion (volar flexion), 3) Radial deviation
(radial flexion), 4) Ulnar deviation (ulnar flexion).

45

Action Muscles
Extension Extensor digitorum communis, Extensor carpi ulnaris, Extensor carpi radialis
longus, Ext c radialis brevis, Ext indicis proprius, Ext pollicis longus
Flexion Flexor digitorum sublimis, Flexor digitorum profundus, Flexor carpi ulnaris,
Flex c radialis, Flex pollicis longus, Abductor pollicis longus, Palmaris longus.
Radial
Deviation
Extensor carpi radialis longus, Abductor pollicis longus, Extensor indicis
brevis, Extensor pollicis longus, Ext carpi radialis brevis, Flexor carpi radialis
Ulnar
Deviation
Flexor carpi ulnaris, Extensor carpi ulnaris

Table 2.1 : The action and grouping of the muscles controlling the human wrist joint.

Figure 2.7: Angles of rotation about the wrist. (A) Extension, (B) Flexion, (C) Radial
deviation, (D) Ulnar deviation.

46

b. Modeling wrist mechanics
We modeled the wrist mechanics within a reasonable approximation to the real
joint as explained in the previous section. A lumped biomechanical model was
adequate for our simulations and we neglected the other intrinsic details of the human
hand. The hand was approximated as a cone with realistic mass and moment of inertia
derived from an average human hand. The cone was connected to a stationary model
of the forearm using a universal joint, capable of rotation about two axes. The muscles
that operate the wrist joint were approximated as four identical Virtual Muscle models
with comparable morphometric and physiological parameters. At one end, the muscles
originated from the same spot on the stationary forearm but they were attached
symmetrically, 90
0
apart from each other on the base of the cone at the other end. The
initial position of the model was such that the apex of the cone pointed downwards to
the ground with the stationary forearm aligned with the vertical y-axis (as shown in
Figure 2.8). The muscles actuated the cone about the x and z axis analogous to
extension-flexion and radial-ulnar deviation movements of the hand, respectively.
The rotations of the cone were limited to 75
o
using a viscoelastic stop. The change in
length of the muscle and hence the change in moment arm during the range of
movements was tuned to be within a reasonable approximation to experimental
observations (Loren et al. 1995; Gonzalez et al. 1997; Lemay et al.1996; Clinical
Mechanics of Hand, 1985; Herrmann et al. 1996; Gillard et al. 1999), but obviously
the profile was symmetric (Figure 2.9) with respect to the neutral position due to the
symmetrical structure of the model.
47

Figure 2.8: Biomechanical model used in the simulation

Table 2.2 : Parameters of the biomechanical model

Model Parameters Values
Mass of cone (grams) 400
Height of cone (cm) 12
Radius of cone (cm) 3.5
Length of forearm (cm) 25
Mass of forearm (grams) 500
48

Figure 2.9: Plot of muscle length variation during extension

Unlike the simple elbow joint that is commonly used in many studies, the wrist
joint requires solving a few reasonably challenging and unique problems by the
control system. The extra degree of freedom obviously adds to the complexity but
even more distinctively the muscles controlling the wrist movement switches their
functional relationships based on the type of task performed. For example during wrist
extension the extensor muscles function as agonists and the flexor muscles function as
antagonists, but during radial/ulnar deviation, the extensor muscles (as well as the
flexor muscles) oppose each other. The spinal cord circuitry is intimately related to
these functional relationships between muscles, so these dynamic relationships
required a new classification and corresponding synapses/connections in our spinal
cord model. In our terminology we decided to call the adjacent muscles as ‘Partial-
synergists’ as they switch from agonist to antagonist based on the type of task and we
called the diagonal muscles that were farthest apart from each other and that always
49

opposed each other (for example, Extensor Carpi Radialis and Flexor Carpi Ulnaris),
‘True-antagonist’. We modeled both synergistic and antagonistic circuits between the
partial-synergists and let the control algorithm adjust the gains, thereby establishing
functional synergist and antagonist relationships as required for each task.

iii. MODEL OF THE SPINAL CORD
The first task before building the model of the spinal cord was to review the
extensive literature published over the past 100 years in order to understand the
various types of interneurons, their connectivity and their functions. It was quite
humbling to realize the time and effort spent by numerous researchers from Charles
Sherrington and Paul Hoffman to many others in the past century, some of whom
spent their entire lifetime trying to identify and explain the connections in the neuronal
maze. We modeled all the classical interneurons identified in the spinal cord and
handcrafted the modeled connections between them as explained below.
a. Computational model of the interneuron
A single neuron is commonly modeled as a discrete time McCulloch-Pitts
neuron (McCulloch and Pitts, 1943) or as a continuous time Leaky Integrator Neuron
(Chrstodoulou et al., 2002; Knight et al., 1972; Burkitt et al., 2003) with a state
equation and an output equation. The multi-compartmental integrate-and-fire neuron
(Fleshman et al., 1988; Rall et al., 1992; Cisi et al., 2008) goes one step further and
incorporates details such as ionic channels and conduction delays between them.
50

When building models with large number of neurons such as ours, however, a trade-
off has to be made between biological realism and computational complexity.
We assumed the relation between the input (depolarization) and the output
(firing rate) of a single neuron to be linear. A pool of these neurons with a range of
different resting biases has an input-output function similar to an s-curve, a medium
slope initial rise where new neurons start firing, a steep middle where all the neurons
are active and contributing, followed by a saturation plateau where all the neurons
have been recruited and an increasing percentage of them have reached their
physiologically maximal firing rate (Figure 2.10). Thus, the interneuron that has the
same characteristics of a pool of neurons was modeled as an s-curve in our simulation.
In order to reduce the computational complexity of the model we used the same
function to model all the interneurons. We also ignored the state function for the
individual neurons, assuming that on the time scale of operation in our tasks the
temporal dynamics of a single neuron were insignificant. This simplification reduced
the computational load even further and made the simulations converge in realistic
time. The computational complexity of the model is one of the most critical factors in
our simulations because often hundreds of iterations are required to be performed
before the optimization algorithm converges to a desired solution. The s-curve was
represented by a simple sigmoidal transfer function shown below, with the input
varying from -1 to +1 and the output clipped between 0 and +1. The function has two
parameters a & b which set the slope and range of the curve respectively.

51

1
1

Figure 2.10: Neuron model (approximating the behavior of a cluster of linear neurons)

Even though the model was highly simplified compared to the other neuron
models, it agrees well with cellular physiology, the nonlinear properties of which lead
to the types of “reflex-gating” described when biological systems switch between
behavioral states. The gains at the input had multiplicative properties even though
mathematically they were added to the input signals, due to the nonlinear nature of the
s-curve. Three types of synapses were defined (excitatory, inhibitory, and selective
synapse that switched its type based on the task). Presynaptic inhibition that
modulated the efficacy of the synapses was also accounted for by adding another input
to these neurons that can be controlled via descending supraspinal commands. The
52

presynaptic inhibition inputs have so far been kept constant for all the interneurons to
reduce the number of control parameters in the model. The control inputs to
interneurons were further differentiated as: 1) bias input that controlled the
background activity in the interneuron and 2) supraspinal descending input that
initiated movement. We called these inputs as ‘SET’ inputs and ‘GO’ inputs
respectively (explained in a later section).
b. Modeling the spinal pathways
While a great deal is known about the spinal interneuronal circuitry, the
information is certainly incomplete and it is not clear how it can be extended to
multimuscle, multi-degree-of-freedom linkages, such as the wrist joint. Much of it
must be extrapolated from literature based largely on the hindlimb of the cat rather
than the arm of the primate. We extracted the structure and properties of the known
pathways in the spinal circuitry from the literature and modeled five classical types of
interneurons and their homonymous and heteronymous connections:
1. Propriospinal pathway
2. Monosynaptic Ia excitatory pathway
3. Reciprocal Ia inhibitory pathway
4. Renshaw inhibitory pathway
5. Ib inhibitory pathway
Based on evidence from the literature (see chapter 1), we strongly believe that
there are relatively few direct descending inputs to the alpha motoneurons in the motor
units of the muscles controlling the wrist movement. Hence in our model, all the
53

supraspinal descending inputs projected only to the interneurons; the integrative
outputs of the interneurons converged at the alpha motoneuron as shown in Figure
2.11. This is a key distinction compared to the other models but it is more consistent
with the actual neuro-anatomy. This also appears to be the first model of spinal cord
circuitry to deal with the shifting patterns of synergistic and antagonistic muscle
activity that necessarily accompany multimuscle, multi-degree-of-freedom systems.
Intuitively, all these novel features add to the complexity of the model and increases
the difficulty of the control problem. However, we found that the intrinsic properties
of the model actually helped it to converge faster and produce physiologically realistic
results. We also accounted for the often neglected heteronymous connections between
the interneurons as found in experimental observations to have significant
contributions.

54

Figure 2.11: Illustration showing the integration of inputs at the motoneuron. Note that there is
no direct descending input to the motoneuron.

1. Propriospinal Pathway (Figure 2.12):
The neurons in the spinal cord that function as the link between the higher
centers and the motoneurons can be classified as ‘segmental’ interneurons, that are
located at each segmental level and ‘propriospinal’ neurons (Alstermark et al.,1992;
Lundberg, 1992), that are connected over multiple segments, rostral to the
motoneurons. There is mounting evidence that, in macaque monkeys (Sasaki et
al.,2004) and in humans (Pierrot-Deseillingy, 2002), a substantial part of the cortical
command for movement is transmitted through the propriospinal pathway. The
‘updating hypothesis’ (Illert et al.,1978) suggests that a motor command initiated in
higher centers is always reshaped en route to the motoneurons by integration in the
55

propriospinal system. Also, there have been claims that the command for target-
reaching is mediated by propriospinal neurons, whereas that for object-taking is
transmitted by segmental interneurons (Alstermark et al.,1981b). In accordance with
the above principles, we activated the descending inputs of the propriospinal
interneurons using step inputs to achieve movement of the hand in our simulations.

Figure 2.12: The Propriospinal interneuron and its homonymous and heteronymous
connections (for a single muscle).

56

Organization and pattern of connection: There is evidence of excitation from
muscle spindles 1a afferents to the proprispinal interneurons (Malmgren et al.,1988a).
This may provide for additional servo-assistance to motoneurons at the propriospinal
level. There may also be excitatory contribution from Ib afferents and cutaneous
afferents. (Gracies et al.,1991; Burke et al.,1992a). We extended the heteronymous
propriospinal axonal projections such that the ‘true-antagonist’ always inhibited, but
the ‘partial-antagonist’ excited the motoneuron.

2. Monosynaptic Ia Excitation Pathway (Figure 2.13):
The monosynaptic Ia excitation pathway, responsible for the ‘stretch reflex’
(Sherrington, 1910; Matthews, 1972), is unique in its simplicity and is one of the
earliest known connections in the spinal cord. But the extent to which the spinal
stretch reflex is involved in normal movement is still not completely clarified (Eccles
et al., 1959). In our simulations the homonymous Ia afferent feedback played a major
role in shaping the output from the alpha motoneurons, especially during perturbations
and fast movements. This agrees well with the facts that the Ia afferents are the largest
and most rapidly conducting peripheral nerve fibers (conduction velocity- cats:
120ms-1, humans: 70ms-1) and often have the earliest effects on the motoneurons due
to shorter feedback paths.
57

Figure 2.13: The Monosynaptic Ia excitation pathway and its homonymous and heteronymous
connections (for a single muscle).

Organization and pattern of connection: The evidence of homonymous
monosynaptic Ia excitation of the motoneurons was also observed for the wrist
muscles (Malmgren et al., 1988a). The heteronymous Ia connections, believed to have
evolved to assist locomotion, often identified the muscles in a ‘myotatic unit’ (Lloyd,
1946; Meunier et al., 1993) that formed a synergistic unit. In our model the projections
58

from the heteronymous ‘partial-antagonists’ formed an excitatory connection at the
motoneurons when it functioned as a synergist in the movement.

3. Reciprocal Ia Inhibition Pathway (Figure 2.14):
Sherrington in 1897 demonstrated that the contraction of a muscle is
accompanied by relaxation of its antagonists and coined the term ‘reciprocal
inhibition’. Later intracellular recordings established evidence of this disynaptic
pathway (Eccles et al., 1956) and demonstrated that the activity in this pathway can
inhibit the monosynaptic reflex (Araki et al., 1960). It was also suggested that the 1a-
interneurons functioned as another spinal integrative center (Eccles and Lundberg,
1958). Parallel control of alpha and Ia-interneurons for coordinated contraction of
agonists and relaxation of antagonists is a widely debated topic (Lundberg, 1970).
However, it is still not clear how the reciprocal connections at the ankle and elbow
joints can be extended to the wrist flexors and extensors.
Organization and pattern of connection: Ia inhibitory interneurons have three
characteristic features allowing their identification: (1) monosynaptic input from 1a
afferents, (2) projection to motoneurons antagonistic to the muscles from which the 1a
input arises (Tanaka, 1974; Pierrot-Deseilligny et al., 1981; Crone et al., 1987), and
(3) disynaptic recurrent inhibition from those motoneurons that receive the same
monosynaptic 1a excitation (Hultborn et al., 1971b). The Ia-interneurons are
themselves inhibited by ‘opposite’ Ia interneurons (Hultborn et al., 1976a). It was also
59

shown that low-threshold cutaneous afferents excited Ia-interneurons through a short-
latency private pathway (Fedina et al., 1972), however in order to limit the complexity
of the model, we ignored all the cutaneous afferents and their projections. In our
model the projections from the ‘true-antagonist’ heteronymous 1a-interneuron always
inhibited the motoneuron and Ia interneuron, but the ‘partial-synergist’ inhibited only
when it functioned as an antagonist in the type of movement performed.

Figure 2.14: The Reciprocal Ia Inhibitory interneuron and its homonymous and heteronymous
connections (for a single muscle).
60

4. Ib-Inhibition Pathway (Figure 2.15)
The Ib-inhibition pathway is one of the most complex and least understood
pathways in the spinal cord with extensive convergence from different types of
peripheral afferents (Edgley, 2001). The initial opinion was that it subserved an
autogenetic protective reflex during extreme muscle forces, but it was soon replaced
by the view that the tendon organs continuously provided information about the extent
of muscle contraction, similar to what we noticed in our simulations. It was also
suggested that these inhibitory potentials from Ib-interneurons on the motoneurons
might help to limit the firing frequency and/or recruitment of new motoneurons in
order to keep a smooth profile of force development and avoid jerky movements
(Zytnicki et al., 1998). However, the recent finding that, during locomotion, Ib
inhibition is replaced by di- and poly-synaptic excitation has completely altered views
on the functional significance of 1b-pathways (Hultborn et al., 1998; McCrea, 1998;
Gossard et al., 1994; Prochazka et al., 1997a; 1997b).
Organization and pattern of connection: The dominant effects of the Ib-
inhibition pathway are inhibition of homonymous and synergistic motoneurons
through di- and tri-synaptic pathways and excitation of antagonistic motoneurons
through tri-synaptic pathways (Eccles et al., 1957). The Ib-interneurons may also
inhibit gamma motoneurons in parallel to alpha motoneurons, but we ignored these
projections. In addition to the tendon-organs the Ib-interneurons are activated to a
lesser extent by Ia-afferents and through one or two interposed interneurons by group-
61

II afferents from cutaneous, joint and interosseous mechanoreceptors (Harrison et al.,
1985).

Figure 2.15: The Ib Inhibitory interneuron and its homonymous and heteronymous
connections (for a single muscle).

Subthreshold effects from a large number of these afferents thus converged on
a common reflex pathway for stable feedback control during movement (Jankowska et
62

al., 1981). We had to neglect most of these secondary afferent connections in order to
limit the complexity of the model. Based on the above evidence we extended the
known connections of the Ib-pathway to the wrist muscles such that, the ‘true-
antagonist’ always excited the motoneuron, whereas the facilitating ‘partial-synergist’
inhibited and opposing ‘partial-synergist’ excited the motoneurons during the
movement. The facilitating ‘partial-synergist’ also excited the Ib-interneuron (Brink et
al., 1983). The homonymous Ib interneuron formed a synapse at the motoneuron that
was both excitatory and inhibitory to account for both positive and negative feedback,
but for our simulations to date we set it as inhibitory.

5. Recurrent-Inhibition Pathway (Figure 2.16)
In 1941, Renshaw demonstrated that in animals with dorsal roots sectioned,
impulses in the motor axons could evoke a short-latency long-lasting inhibition of the
monosynaptic stretch reflex in homonymous and synergistic motoneurons. The
inhibition was found to be mediated through interneurons, which were named after
Renshaw (Eccles et al., 1954; Windhorst, 1990). It was also found that the relationship
between the amount of injected current and the number of spikes produced by the cells
was sigmoid, similar to our assumption (Hultborn et al., 1979b).

63

Figure 2.16: The Renshaw interneuron and its homonymous and heteronymous connections
(for a single muscle).

Organization and pattern of connection: Convergence onto Renshaw cells
were found to be based on proximity as well as functional factors. Motor axon
collaterals spread a distance of less than 1mm from their parent body, so the excitation
of the Renshaw cells can only be obtained from motor nuclei located in the immediate
64

neighborhood (Cullheim et al., 1978). It was also found that the Renshaw cells were
excited mainly by motoneurons of synergistic muscles (Eccles et al., 1954; 1961b).
Apart from the excitation from motoneurons, the Renshaw cells are both excited
(Ryall et al., 1971) as well as inhibited (Wilson et al., 1964; Fromm et al., 1977) by
cutaneous afferents, which we neglected.
The heteronymous inhibition from Renshaw cells extended to motoneurons of
synergistic muscles, and it was found that there was a striking overlap between the
distribution of Renshaw inhibition and monosynaptic Ia-excitation (Hultborn et al.,
1971a). Renshaw cells inhibited the interneurones mediating disynaptic reciprocal Ia-
inhibition in a strictly parallel fashion to the motoneuron (Hulbornet al., 1979b). This
projection was largely responsible for recurrent-facilitation that was described by
Renshaw in 1941. In our model the axons of heteronymous Renshaw cells of the
‘partial-antagonist’ muscle inhibited the motoneuron when it functioned as a synergist
in the movement. The alpha motoneurons of all synergists excited the corresponding
Renshaw cells during movement. We also modeled mutual inhibition between the
Renshaw cells of ‘true-antagonistic’ muscles (Ryall et al., 1981).
Other Pathways: In order to keep the complexity of the model to realistic
computational limits, we had to neglect the other known pathways and connections in
the spinal cord. The Group-II is one such pathway that was shown to have effects
superimposed on the Group-I interneurons and their connections. The less documented
cutaneous afferents were also neglected but are known to play a significant role in
65

modulating motor behavior through spinal, supraspinal and transcortical pathways
(Pierrot Deseilligny, 2005).
The Integrated model: Based on the information from the literature as
summarized in previous sections, we handcrafted a detailed model of the spinal-cord.
Despite the many experimental studies (De Luca et al., 2002; Koshland et al., 2000;
Aymard et al., 1997; 1995; Wargon et al., 2006; Day et al., 1983; 1984; Raoul et al.,
1999), the structure of the spinal pathways is not yet completely understood for the
wrist muscles, so we had to speculate and extend the connections from the patterns
described mostly for simple synergists and antagonists operating on single degree-of-
freedom hinge-joints. Surprisingly, the integrative action of the interneurons and
reflex pathways did give realistic results during simulations as described in subsequent
chapters.
Due to the large number of connections, the complete model of the spinal-cord
cannot be represented in a single figure. The two subdivisions illustrated as figures
are the connections between the 1) ‘Partial-synergist’ muscles (Figure 2.17) 2) and
‘True-antagonist’ muscles (Figure 2.18), respectively. In the complete model there are
two similar ‘True-antagonist’ type connections and four ‘Partial-synergist’ type
connections between the four muscles. For realistic simulations we delayed the output
from the alpha motoneurons by 20ms to account for the two-way conduction delays
through the spinal circuits. The modeled spinal-cord had a very large number (184) of
control inputs (gains), not unlike the real spinal cord that the brain must learn to
control to produce desired behaviors. We first adjusted these gains intuitively and
66

were encouraged by surprisingly good results (chapter 3). We then systematically
optimized these gains and produced physiologically realistic outputs in a variety of
tasks (chapter 4). All the gains were also systematically analyzed (chapter 5).

67

Figure 2.17: Partial view of the Spinal-Cord model showing the connections between two
‘Partial-Synergist’ muscles. The pathways shown are: Monosynaptic Ia-pathway, (PN)
Propriospinal pathway, (Ia) Reciprocal Ia-inhibitory pathway, (Ib) Ib-Inhibitory pathway, and
(R) Renshaw pathway. The ‘SET’ gains adjusted the background activity in the spinal cord
and the ‘GO’ gains initiated and maintained the transition to a new state.

68

Figure 2.18: Partial view of the Spinal-Cord model showing the connections between ‘True-
Antagonist’ muscles. The pathways shown are: Monosynaptic Ia-pathway, (PN) Propriospinal
pathway, (Ia) Reciprocal Ia-inhibitory pathway, (Ib) Ib-Inhibitory pathway, and (R) Renshaw
pathway. The ‘SET’ gains adjusted the background activity in the spinal cord and the ‘GO’
gains initiated and maintained the transition to a new state.

iv. COMMAND MODEL
The increase in muscle tone in anticipation of movement was first described by
William James (1890) and Sherrington (1906). It is a matter of common observation
69

that readiness, attention, intent and attitude can greatly affect motor responses to
stimuli (Prochazka, 1989). The word ‘SET’ is commonly used by neurophysiologists
and psychologists and is defined as: “a state of readiness to receive a stimulus that has
not yet arrived (perceptual) or a state of readiness to make a movement (motor)”
(Evarts et al., 1984). In our simulations we adjusted the background activity in the
spinal cord, prior to a movement, using the ‘SET’ inputs. They consisted of the gains
at each synapse (white/green circles in Figures 2.17, 2.18) and the bias inputs to the
interneurons and motoneurons. The fusimotor drive to the muscle spindles was also
grouped under the SET commands. We did not optimize the presynaptic inputs but set
them to a constant value. All our simulations had a maximum duration of three
seconds and the background activity in the spinal cord was responsible for maintaining
a steady posture of the hand for a specified initial time (1sec) before the impending
‘GO’ input. The SET gains alone mediated the rapid reflexive response of the hand
during perturbation in the passive postural stabilization task. The ‘GO’ inputs were
simple step inputs and represented the volitional descending commands, which
initiated movement or other state changes and maintained the new state until the end
of the modeled behavior.
The division of labor or the distribution of control between the higher centers
and the spinal cord during voluntary movement is not well understood and is highly
debated. Our general hypothesis is that much of the details associated with a complex
movement are actually generated during execution by the spinal cord and the real task
of the brain is to configure the spinal circuitry. In order to test our hypothesis and in an
70

attempt to explore the capabilities of the integrated spinal cord model, we used the
simplest descending inputs, namely step functions, as ‘GO’ inputs. As mentioned
before the motoneurons received no ‘GO’ inputs directly, but received inputs only
from the interneurons. We found that even with such simple inputs with no
modulation, the spinal cord model produced physiologically realistic and well
described behaviors in the literature.
Input

Figure 2.19: The command model. The ‘SET’ gains adjusted the background activity in the
spinal cord and the ‘GO’ gains initiated and maintained the transition to a new state.

The fusimotor drive to the muscle spindles plays a significant role during the
various kinematic conditions under which the muscles perform (Loeb, 1984). Under
all shortening conditions, primary endings in de-efferented spindles generally
produced zero output; secondaries are also likely to be silenced at higher velocities of
shortening. The co-activation of the alpha and gamma (static) motonurons has been
proposed by Granit (1973) as a means by which the central nervous system can cause
the muscle spindle afferents to act as the error signal for deviations from the desired
trajectory of motion. If the rate of muscle shortening is lower than the rate of intrafusal
71

contraction, the spindle afferents increase their discharge rate, which
monosynaptically increases the level of excitation of the homonymous alpha
motoneurone pool, accelerating the motion. If the rate of muscle shortening is too
high, the spindle afferents will turn off, thereby disfacilitating the motor pool.
Evidence from spindle recordings during active lengthening suggests that concurrent
gamma dynamic activity enhances the sensitivity to velocity changes over a wider
range of properties (Hulliger and Prochazka, 1983). Even under isometric conditions,
the dynamic fusimotor drive to the spindles makes them (primary endings) exquisitely
sensitive to tiny length fluctuations such as are likely to occur during voluntary
attempts to maintain a constant limb position (Hasan and Houk, 1975).
The experimental literature on spindle activity during various behaviors seems
to be more consistent with independent modulation of alpha, gamma static and gamma
dynamic motoneurons that with any simple, fixed rule such as “coactivation” (Loeb,
1985; Prochazka et al., 1988; Taylor et al., 2000a, b). Therefore, we optimized both
gamma static and gamma dynamic efferent inputs (as SET commands) to the muscle
spindle model. Surprisingly, the optimization algorithm picked experimentally
observed and realistic values for the efferent inputs automatically (see results).
However, we did not modulate the gamma drive during the task; instead they were set
at the beginning of the task. This approximation reduced the number of parameters to
be adjusted at the GO phase, by the control algorithms; however, we acknowledge
their significance and recommend this for future work.

72

v. OPTIMIZATION OF RUNTIME
The integrated spinal cord model was developed in Matlab/Simulink
TM
by
using various in-built blocks in the standard library as well as toolboxes such as
SimMechanics
TM
. Each connection in the spinal cord was manually wired and tested
multiple times to ensure accuracy. The final model had hundreds of blocks with many
closed loops between them, resulting in a highly inefficient run-time. A behavioral
task that completed in 3 seconds, took about 45 minutes to compute in simulation.
This was a major bottleneck during optimization of the control inputs as it required
hundreds of simulations to converge and was also particularly annoying when we
wanted to modify individual control parameters to observe their effects in the results.
We reduced the run-time of the model using various techniques as explained
below.
1. The Matlab environment is specifically intolerant of zero-delay input-output
loops. We added Low pass filters (100 Hz) to break the feedback loops. This
also solved the ‘algebraic-loop’ computational error in Matlab.
2. We iterated the simulations with different tolerances and step size parameters
of the solver and picked the best values that provided a trade-off between good
performance and acceptable accuracy.
3. The CMEX compiler in Matlab has adjustable optimization levels. We
transported the spinal cord model from Simulink to C language and set an
adequate compiler optimization level that performed with maximum
efficiency.
73

4. We used Real-Time workshop rapid simulation (RSim) target to recompile the
model in C. The highly optimized RSim target executable ran at an even faster
time without compromising accuracy.
All these enhancements resulted in bringing down the simulation time from 45
minutes to an average of 10 seconds for each task. The model was then interfaced to
m-files in Matlab so that the entire optimization algorithm could run automatically till
it converged to the lowest cost values. Typically, the optimization of each task
required about 400 iterations and thus completed in an average of 1.5 hrs. The
isometric tasks, however, required longer completion times as the tolerances of the
solver had to be reduced to a relatively low value (because of the reduced rate of
change in output parameters).

74

CHAPTER 3: INTUTIVE OPTIMIZATION OF SPINAL
CORD
The general hypothesis of this thesis is that much of the detailed
neuromuscular activity associated with a complex movement is actually generated
during execution by the spinal cord. The spinal interneurons receive direct feedback
via peripheral inputs and are able to respond more rapidly to ongoing changes in
sensory input than the higher centers. Even though the direct inputs from the cortex to
the motoneurons are particularly important for individuated finger movements, it has
been observed that for coordinating large groups of muscles in behaviors such as
reaching and walking, the indirect connections through the spinal interneurons play a
significant role.
The distinct pattern of spinal circuitry and the specific behavior of interneurons
in the spinal cord appear to have explicit roles in facilitating voluntary movement. For
example, the feed-forward connections in the common monsynaptic reflex systems
enhance the effect of the active pathway by suppressing the activity of the other,
opposing pathway (reciprocal innervation as suggested by Sherrington). Afferent
neurons from extensor muscles excite not only the extensor motoneurons, but also
inhibitory neurons that prevent the firing of the motor cells in the opposing flexor
muscles. These organizational features simplify the control of voluntary movements,
for example, they reduce the burden on higher centers to send separate commands to
the opposing muscles. On the other hand the negative feedback inhibition circuits are
75

self-regulating mechanisms. The effect is to dampen activity within the simulated
pathway and prevent it from exceeding a certain critical maximum. For example, the
motoneurons act on inhibitory interneurons (Renshaw interneurons), which feed back
to the motoneurons themselves and thus reduce the probability of firing. Even the
reflexes are not simple repetitions of a stereotyped movement pattern, but they are
modulated by properties of the stimulus or task.
We put our hypothesis to test by trying to replicate four very different but well-
described behaviors in the literature using the spinal cord model: 1) stabilizing
responses to external force perturbations; 2) rapid voluntary movement to a position
target; 3) rapid voluntary output of isometric force to a target level; 4) adaptation to
viscous curl force fields. Surprisingly, the model produced physiologically realistic
outputs in all four behaviors, with very simple unmodulated step input commands. The
modeled system had a very large number (184) of control inputs (gains); the majority
of them adjusted the background activity in the spinal cord (SET) and a few produced
movement or transition to a new state (GO). Even though such a high dimensional
control task is manually intractable, we found that relatively simple and generally
intuitive step-changes in a small subset of descending controls could reproduce each
of the modeled behaviors. Furthermore, details of stability and temporal patterning of
muscle recruitment were surprisingly realistic, even though we made little or no
attempt to optimize the values of many other control signals and we did not modulate
any of the control signals. In fact, these behaviors appear to be robust emergent
properties of the complete set of spinal circuitry as modeled. Such properties would
76

substantially change and generally simplify the learning of motor tasks by the brain.
We later optimized the gains using gradient descent algorithm and it produced even
better results (see chapter 4). The results obtained using intuitive optimization are
described in the following sections.

i. STABILIZING RESPONSE TO FORCE PERTURBATION
The nervous system has multiple mechanisms to counter external perturbations
to the limb. The impedance of the musculoskeletal system created by inertial
properties of the limb (force-acceleration), spring-like (force-length) and viscous-like
(force-velocity) properties of the muscles produces a resistive force against sudden
perturbation called “preflex” (Brown et al., 1995; Brown and Loeb 1999). Unlike
“reflexes” that are initiated only after substantial nerve conduction and synaptic
transmission delays, preflexes are almost instantaneous. In various previous
preparations and behaviors, preflexes appear to provide substantial and immediate
responses to perturbation, which tend to stabilize the system (Humphrey and Reed
1983). The preflexes also reduce the gain required in reflexive loops, thus reducing
instability in the delayed feedback loops (Loeb at al., 1994). We applied an external
force pulse of magnitude 100N and duration 10ms along the direction of wrist flexion
to the isolated biomechanical model detached from the spinal circuitry. The model
produced damped oscillations about the wrist joint typical of a spring-mass-damper
system (Figure 3.1). The time-period of oscillation increased when gravity was
77

removed from the system (Figure 3.1). The force modulations in the passive muscles
showed instantaneous but inadequate response against the large perturbation that was
applied.
While “preflexes” provide the initial resistance and improve stability in the
system, they alone are not sufficient to withstand large perturbations as shown in
Figure 3.1. The contribution of preflexes can be greatly increased by cocontracting
muscles, but that strategy is energetically expensive and usually avoided by human
subjects (see below). Short latency “reflexes” generated by proprioceptive sensors
and transmitted through the circuits in the spinal cord are the next level of defense
against unforeseen as well as expected perturbations. We applied the same external
force perturbation after attaching the spinal circuitry and feedback sensors to the plant.
The hand stabilized back to the resting position in about 600ms (Figure 3.1),
demonstrating the stabilizing property of the spinal cord. The gains in the spinal cord
were mostly left at a low value (as higher gains tend to create instability during large
perturbations) and they required only minimal intuitive adjustment to produce a good
response consistently. The model continued to provide good response even after the
removal of gravity from the system (Figure 3.1). Moreover the kinematic trajectory of
displacement, output of motoneurons and the force modulation of the muscles looked
physiologically realistic (Figure 3.2).
78

Figure 3.1: Rotation of the resting hand about the wrist joint (extension-flexion axis) in
response to a perturbation (100N, 10ms) at one second.

The modulation of muscle force and the output of the alpha motoneurons
showed that the spinal cord responded by a burst in the extensor motoneuron to pull
back the hand to the resting position against the perturbation (Figure 3.2.A). Due to
the magnitude of the initial burst, the hand overshot the initial position at the first
attempt (Figure 3.1), so the spinal cord compensated by adding another burst in the
flexor motoneuron (Figure 3.2.B). A third peak is also noticed in the extensor muscle
force output (Figure 3.2.C). All of these modulations in the output of the motoneurons
and the muscle force were generated by the reflex circuits and resulted in bringing the
hand back to the resting position in about 600ms. The output from the muscle spindles
primaries are shown in Figure 3.3. The Golgi tendon organs (GTO) were relatively
silent (Figure 3.3.C, D) due to the low magnitude of forces involved in this simulation.
The descending inputs to the spinal cord were kept neutral and only the background
activity in the spinal cord was adjusted by a few intuitive changes in the gains. The
79

results demonstrated the inherent stabilizing property of the spinal cord that integrated
the feedback from the spindles and the tendon organs with the passive forces in the
muscles to achieve physiologically realistic results.

Figure 3.2: (A) Output from the motoneuron exciting the extensor muscles, (B) Output from
the motoneuron exciting the flexor muscles, (C) Muscle force modulation of extensors, (D)
Muscle force modulation of flexors.
80

Figure 3.3: (A) Primary output of muscle spindles attached to the extensor muscles, (B)
Primary output of muscle spindles attached to the flexor muscles, (C) Output from GTO
attached to the extensor muscles, (D) Output from GTO attached to the flexor muscles.

The stiffness of a limb about a joint is the sum of the stiffness of all muscles
acting about the joint (Feldman, 1980; Partridge, 1983). Thus, an increase in stiffness
can be brought about with no change in net joint torque using co-contraction, if
agonist and antagonist muscles contract to oppose each other. Co-contraction is
another mechanism to resist expected-perturbation but at the cost of higher metabolic
energy. It takes advantage of the intrinsic force-velocity properties of muscles which
are instantaneous and is the most effective strategy to oppose large and rapid
perturbations. However, in our simulations we limited the initial co-contraction of the
81

muscle to a minimum, in order to take advantage of the properties of the reflex
circuitry in the spinal cord. Even though the reflex pathways are subject to phase lag
due to neural conduction delays, experiments performed by anaesthetizing the
opposing wrist muscles (thus preventing co-contraction), produced comparable results
to our simulations. Hore et al., (1990) anaesthetized the radial nerve to paralyze the
extensors of the wrist. They found task-dependent changes in stiffness even with
paralyzed opposing muscles.
We also tuned the gamma commands to the muscle spindles to improve
response during perturbation. The spindles attached to the stretching muscles
(extensors) were tuned to respond rapidly via the monosynaptic pathway and the
spindles attached to the contracting muscles (flexors) were tuned to remain active and
not fall silent during the perturbation. The fusimotor gain is usually a trade-off
between having a desired range but reduced sensitivity or increased sensitivity with
the risk of saturation. However, by anticipating the range of movements, an optimal
compromise may be reached (Loeb, 1985) as was the case with the improved results in
our simulations after tuning the gamma inputs to the spindles. If the perturbation arises
outside of the events anticipated there are three options (Loeb, 1985): 1) the
perturbation is small and within the range of spindle sensitivity, 2) The perturbation is
transient and thus the response saturates and is not modulated by feedback, and 3) the
perturbation is large and temporally extended. In cases 1) and 2) fusimotor tuning can
be safely ignored but in case 3), which is closer to our scenario, the fusimotor
adjustment has a significant contribution.
82

It should be pointed out that we are not discounting the effects of long-loop
reflexes mediated via the motor cortex and other supraspinal structures. However, the
general conclusion is that the cortical reflexes may be of primary importance in
regulating contractions in muscles where the command itself has to be evolved based
on the needs of the task such as the precise control of the digits. On the other hand for
relatively automatic tasks such as maintaining balance and for producing bodily
movements, the subcortical and spinal reflexes may be responsible for the afferent
regulation of the muscles.

ii. RAPID VOLUNTARY MOVEMENT TO POSITION
TARGET
Cortico-spinal trajectory generation for voluntary reaching movements is
assumed to shift from a feedback position controller to a feedforward trajectory
generator with superimposed dynamic error compensation during fast movements. In
an attempt to find the contribution of the spinal cord in fast movements, we simulated
a rapid, stable wrist extension from neutral to 30
0
in our model (Figure 3.4). The
background activity in the spinal cord was adjusted (SET) to keep the hand steady at
the resting position for the initial one second of the simulation and then we applied
step inputs (GO) to the propriospinal interneurons (Pierrot-Deseilligny, 1996) of the
spinal cord. The inputs were applied only to the interneurons corresponding to the
extensor muscles. We found that based on the magnitude of the step function, the wrist
83

extended to different degrees of rotation. Surprisingly, the details of the movement
such as the overshoot, time to stabilization, velocity profile, muscle force modulation
etc. matched well with physiologically realistic results observed in the literature
(Hoffman et al., 1986; 1999; Koshland et al., 1990; Perlmutter et al., 1998; Maier et
al., 1998; Setin et al., 1988; Levin et al., 1992). The behavior was further tested for
stability by applying a brief torque perturbation (100N, 10ms) during the hold phase of
the movement. The hand stabilized back to the extended position in about 400ms
(Figure 3.4.C).

Figure 3.4: (A) Command input to the spinal cord. The SET input was applied for 1 sec and
then GO inputs were applied to the propriospinal interneurons to produce movement to the
30
0
, (B) External force perturbation (100N, 10ms) applied at 2.3 sec of simulation time (C)
Extension of the wrist to position target of 35
0
and resistance to perturbation.
84

The model produced physiologically realistic results even with minimal hand-
tuning of the feedback gains (SET) and simple step inputs (GO) as the descending
commands to the propriospinal interneurons. The muscles remained silent as the SET
command adjusted biases in the spinal cord (Figure 3.5.C, D). This reflected the
preparatory state before responding to an impending GO command. The GO
commands to the propriospinal interneurons (extensor) initiated the desired wrist
extension to the 30
0
target, which was maintained in the face of the perturbing force
pulse. The actual recruitment of the extensor muscle was complexly modulated by the
ongoing proprioceptive feedback through the propriospinal and other interneurons.
Even though the rapid movements were made under minimal joint stiffness, the force
profiles of the antagonist pair illustrate both reciprocal and co-contraction features
similar to those described experimentally for such tasks (Figure 3.5.C, D). This
reflects the proprioceptive feedback, which is modulated by both the movement and
the muscle forces, and is combined with and effectively modulates the command
signals in the various interneurons.
85

Figure 3.5: (A) Output from the motoneuron exciting the extensor muscles, (B) Output from
the motoneuron exciting the flexor muscles, (C) Muscle force modulation of extensors, (D)
Muscle force modulation of flexors.

Figure 3.6: (A) Primary output of muscle spindles attached to the extensor muscles, (B)
Primary output of muscle spindles attached to the flexor muscles, (C) Output from GTO
attached to the extensor muscles, (D) Output from GTO attached to the flexor muscles.
86

The motoneuron output as well as the muscle force modulation matched well
with the tri-phasic burst pattern (Figure 3.5) observed in EMGs of opponent muscles
during rapid self-terminating limb movements (Bullock et al.,1992; Ghez et al.,1982).
The initial burst in the extensors accelerated the hand to the desired movement
velocity which was followed by a burst in the flexors that was needed to decelerate
and halt the movement. A third burst in the extensors that was smaller in magnitude
and that appeared less reliably prevented the hand from reversing direction and thus
stabilized the wrist at the target position. The alpha motoneurons maintained a
constant level of activation to keep the hand at the new position against gravity and the
passive pulling force of the flexors. The fusimotor drive to the muscle spindles were
adjusted such that the spindles attached to the extensors (stretching muscles during
perturbation) had high dynamic activity and the spindles attached to the flexors
(contracting muscles during perturbation) had high static activity, to prevent the
spindles from going silent during perturbation. However, as can be seen in Figure 3.6
A&B, it is often difficult to manually tune the spindles to facilitate rapid extension as
well as resist perturbation. The muscle force modulation against perturbation also
looks realistic. The gains were adjusted to minimize co-contraction in order to obtain a
rapid response with minimum opposing force from the antagonists.
The simulation results were also very close to similar experimental
observations (Hoffman et al., 1986) as shown in Table 3.1.

87

Parameter Experiment Simulation
Overshoot 10-15 Deg 10-15 Deg
Time to stabilize 800ms 800ms
Time to max
displacement
100ms 100ms
Velocity profile One positive peak before
max displacement
One positive peak before
max displacement
Time to zero velocity
crossing
100ms 100ms
Mean velocity 400deg/sec 350deg/sec
Initial movement
trajectory
Single step Single step

Table 3.1: Comparing results (kinematic parameters) from the simulation with similar
experiment.

Further probing into the discharge patterns of the interneurons and
proprioceptive sensors also revealed surprising similarity with published experimental
results. The interneurons began to discharge on average 50ms before activation of
their target muscle and produced a physiologically realistic initial tonic discharge
followed by a phasic-tonic discharge pattern (Fetz, 1996). The afferent responses from
the muscle spindles showed two to four afferent bursts separated by intervals of 20-
30ms (Figure 3.6.A), which was also reflected in the motoneuron output. (Hagbarth,
1981). The agonist motoneuron initiated each movement with a brief burst of activity
that began approximately 40ms before movement onset followed by the antagonist
88

burst approximately 15ms into the movement (Hoffman, 1990) (Figure 3.5.A). The
characteristic “triphasic burst pattern” associated with rapid voluntary movements was
evident in most cases but the secondary bursts in the agonist appeared only when the
velocity re-crossed zero before the hand reached the peak position.
The various sensorimotor strategies for reaching suggested in the literature
have stressed the significance of fusimotor modulation. One such concept is the ‘dual
control of reaching movement’ (Lan et al., 2005). The dual control suggests that the
muscles are controlled by an alpha command and a continuously modulated gamma
command. The alpha command determines the background activation, while the
gamma command steers the balance of activation in the time course of reaching, and
specifies the final target position corresponding to a steady state value (in line with
equilibrium point hypothesis suggested by Feldman et al., 1986). As mentioned
before, we did not modulate the gamma commands to the muscle spindles but set them
at the beginning of the movement. This was done in order to limit the complexity of
the model, but is suggested as a modification, for future work.

iii. OUTPUT OF ISOMETRIC FORCE TO A TARGET LEVEL
Ghez and Gordon (1987) studied the role of opposing muscles in the
production of isometric force trajectories. While the first burst in the agonist muscle is
present both under isometric and anisometric conditions, the second burst in the
antagonist and the third again in the agonist were reported to be missing under
89

isometric conditions in early experiments on cats (Ghez, 1982). A few years later the
same author (Ghez and Gordon, 1987) studied the role of opposing muscles in the
production of isometric force trajectories in human subjects and found that for brief
force rise times (<120 ms), reciprocal activation of the antagonist muscle occurred
consistently. The authors interpreted this muscle activity as indicative of similar
preprogrammed phasic drive from the motor cortex. In order to increase the force
rapidly without overshooting, it is advantageous to “over-recruit” the agonist muscle
and then cut off its rapidly rising torque with the delayed antagonist burst, suggesting
a learned program in motor cortex. We replicated the experiment in our model and
found similar activity in the antagonists, generated solely by the reflex circuits in the
spinal cord. Moreover, the motoneuron outputs and muscle force modulations matched
well with the results from the experiment.
Experimental setup of Ghez & Gordon: The experimental arrangement is
illustrated schematically in Figure 3.7. Subjects were tested in the sitting position with
the right arm abducted to approximately 70
0
and the elbow flexed to 90
0
. Adjustable
metal restraints were used to immobilize the shoulder, upper arm, forearm and wrist.
A force transducer was coupled to the wrist cuff to measure the isometric force
produced by contraction of muscles at the elbow joint. The target force and force
changes were displayed in a large oscilloscope placed in front of the subject. Another
oscilloscope allowed the subjects to review the results of the trial. In some
experiments, subjects monitored target and response variables through earphones.
90

Figure 3.7: Experimental setup (Ghez and Gordon, 1987)

We replicated the experimental setup in our simulation on the wrist joint. The
joint was constrained using a visco-elastic stop (with a linear opposing torque
function) in both axes. The vector sum of the net forces from the muscles was
calculated to compare with the forces sensed by the strain gauges in the experiment.
As in the experiment, two different scenarios were simulated: 1) pulse force trajectory,
and 2) step force trajectory. The target force levels were set for to require levels of
muscle recruitment comparable to the values in the experiment.
a. Pulse force trajectory
In this scenario the subjects were trained to produce an aimed isometric force
impulse. The task was to generate a single force impulse whose peak amplitude
matched that of a previously presented visual target step, and then allow the force to
91

return passively to baseline. Subjects were specifically instructed to refrain from
correcting their responses once initiated. Additionally, subjects were urged not to
respond to the target step “as soon as possible”, but rather to respond “when ready” so
as to optimally prepare their responses. Responses were initiated from a neutral force
level. Different task conditions were generated, such as 1) Fast condition, where the
subjects were instructed to make the force rise as rapidly as possible, and 2) Matched
trajectory condition, where the subjects had to match both the amplitude as well as the
rise time of target force that varied from 80ms to 400ms.
In our simulations, we manually tuned the spinal cord to produce forces with
brief rise times (<120ms); we then used gradient descent algorithm to track forces
with varying rise times and replicate matched trajectory condition in the experiment
(see chapter 4). In order to produce the force impulse we applied a brief pulse
command to the propriospinal interneurons corresponding to the extensor muscles
(agonists); we did not activate the flexor muscles (antagonists). Surprisingly, force
trajectory, alpha motoneuron output and muscle force modulation matched well with
the corresponding output force and EMG signals observed in the experiment (Figure
3.8, 9).
The antagonist motoneuron bursts appeared consistently in all our simulations,
just like in the experiment. By increasing the fusimotor drive to the muscle spindles to
70% (of the maximum) and by allowing for a slight compliance at the wrist restraints,
the antagonistic bursts appeared consistently in all the simulations (Figure 3.9.D).
According to our calculations, a compliance of less than 0.2cm at the joints in the
92

experiment could have resulted in obtaining the same antagonistic burst as a result of
spindle afferent activity, without any intervention from the higher centers. Such
compliance would be expected in the coupling between the fixation apparatus to the
bones through the skin and other padding in the experimental setup. By reducing the
gamma drive to the spindles, we were also able to suppress the antagonistic bursts,
similar to the scenario in the experiment where the subjects were able to suppress the
reciprocal antagonistic bursts when instructed. This suggests that the sequential
activation of agonist and antagonist may not be an obligatory linkage but depends on
the background activity in the spinal cord, the reflex gains and the fusimotor drive to
the muscle spindles. The subjects in the experiment were given preparatory time
before performing the task in order to condition themselves; this could relate to an
increase in the background activity in the spinal cord and maybe increase in the
fusimotor drive as well, subconsciously by the subjects, similar to our model
conditions in the simulation.

Figure 3.8: (A) Experiment: the target force (60N) and the force impulse generated by the
subject. (B) Simluation: the net force calculated from the force generated by the muscles.

93

Force (N) Motoneuron
o/p
EMG
From Experiment

Figure 3.9: (A) Agonist EMG generated in the experiment for force impulse task (B)
Corresponding Antagonist EMG (C) Motoneuron output in simulation to extensor muscles (D)
Corresponding motoneuron output to flexor muscles (E) Extensor muscle force modulation in
simulation (F) Corresponding Flexor muscle force modulation

94

b. Step force trajectory
In this scenario the subjects were instructed to produce a brief initial impulse,
and then hold the force at the new target level rather than allowing it to return to
baseline. Subjects were also instructed to reach the target level with little or no
overshoot. Like the pulse force trajectory scenario, multiple task conditions were
generated by requesting the subjects to make the movement rapidly as well as to
follow a matched trajectory with a predefined rise time.
We excited the propriospinal interneurons of the extensor muscles with step
commands to generate continuous force at the target level. As before, we did not
activate the flexor propriospinal interneurons, but consistently observed bursts in the
antagonist motoneurons for brief force rise times (Figure 3.10.F). This observation
agrees well with the conclusion in the paper that the antagonist burst did not serve to
return the force to baseline as it was observed with equal magnitudes in both pulse as
well as step force trajectories. However, it contradicts with the notion that the
antagonists bursts are generated via supraspinal commands; in our simulations they
were generated entirely by the reflex circuits in the spinal cord. Moreover, the muscle
force modulations looked almost identical to the output force in the experiment
(Figure 3.10.A&B). We then optimized the gains using gradient descent algorithm to
track step trajectories of different rise times varying from 80 to 500ms (see chapter 4).
In order to achieve the fastest possible rate of rise of force without
overshooting the target, the agonist muscle must be activated at a high level that would
overshoot by itself but is then counteracted by the delayed activation of the antagonist
95

muscle. The antagonist activation thus serves as the braking function in brief rise-time
trajectories. Our simulation results challenge the earlier proposal that the rapidly rising
force trajectories depend on neural commands to opposing muscles that adapts to
constraints imposed by the properties of the neuromuscular plant. Instead it proves that
this phasic pattern can be generated entirely in the spinal cord and is essential to
compensate for the loss of gain and the increased phase lag of muscles at higher
excitation frequencies. These properties will substantially simplify the neural
commands from the brain and assist the higher centers to control the plant efficiently.

96

Figure 3.10: (A) Target force (60N) and force produced by the subject in the experiment (B)
The net force generated in the simulation (60N) (C) EMG generated by agonist muscles in
experiment (D) Corresponding EMG generated by antagonist muscles (E) Motoneuron output
to agonist muscles in simulation (F) Corresponding motoneuron output to antagonist muscles.

97

iv. ADAPTATION TO VISCOUS CURL FORCE FIELDS
In an attempt to find the limits of the spinal cord’s potential role in motor
adaptation, we modeled an experiment in which simple movements must be made in
the face of complex and unusual load conditions. The curl-field task has been widely
accepted as one of the most valuable testbeds for the notion of internal models and the
learning processes involved in their development and adaptation (Kluzik et al., 2008;
Scheidt et al., 2001; 2000; Diedrichsen et al., 2005; Flanagan et al., 1999; Hwang et
al., 2005; Karniel et al., 2002).
Experimental Setup of Scheidt et al. (2001): Subjects with no known
neuromotor disorders participated in the study. They made 20cm reaching movements
with their dominant arm in the horizontal plane while holding the handle of a two-
joint, robotic manipulator (Figure 3.11.A). The robot was comprised of a five-bar
linkage with torque motors controlled by a dedicated PC. Subjects were instructed to
“reach from the beginning target to the ending target in one half second”. A computer
screen placed in the front gave qualitative feedback of the movement duration after
each trial (e.g. too fast, too slow or just right). Subjects were instructed to relax after
each movement while the manipulandum moved the hand slowly back to the
beginning target. The subject’s arms were supported against gravity by a sling
attached to the ceiling and the shoulders were restrained using a torso support. The
position of hand was displayed as a small cursor on the overhead monitor. The robotic
manipulator applied perturbing force fields to the arm during each movement. A
98

perpendicular viscous field was designed to deflect the hand perpendicularly from its
intended path with a force proportional to hand velocity along its path (Figure 3.11.B).

Figure 3.11: (A) Schematic representation of the 2 degree-of-freedom robotic manipulandum
used in the experiment (Scheidt et al., 2001) (B) graphical representation of the
perpendicular force field presented to the subjects (Scheidt et al., 2001).

We replicated the experiment in our simulation; the intended direction of
movement of the hand was defined about the x-axis (flexion-extension) and viscous
perturbing force was applied about the z-axis (radial-ulnar deviation) in proportion to
the velocity in the x-axis. The damping coefficient was selected such as to make the
perturbing force profile comparable to the figures published in the literature (30N peak
force). We then intuitively changed the gains in the spinal cord to adapt to the
perturbing force. A significant amount of adaptation was possible by simple
adjustment of the gains (Figure 3.12.C). The model also showed some after-effects
when the perturbing force was switched off (Figure 3.12.D). The tuning of the
heteronymous gains to counter the nonlinear and disproportionate perturbing force
99

acting along both axes was particularly challenging. We did not pursue this further but
later tuned the gains using gradient descent algorithm to get surprisingly good results
(see chapter 4).

Figure 3.12: (A) Command input to the spinal cord. Initial SET command to set the
background activity and step functions as GO commands to initiate movement (B) Rotation of
the hand in the intended axis (extension) (C) Deflection of the hand due to perturbation
(radial/ulnar axis). Note that after adaptation (manual tuning of gains) the deflection reduced
from 40
0
to less than 7
0
(D) After-effects in the radial/ulnar axis after the removal of
perturbing force.

100

As shown in Figure 3.12.C, the deflection of the hand in the radial-ulnar axis
due to the viscous curl-field perturbation was reduced from 40
0
to 7
0
, by simple
adjustments of the gains. We did not change the step inputs (GO) to the propriospinal
interneurons, but only tuned the SET gains responsible for the reflex sensitivity and
the background activity in the spinal cord. We also adjusted the fusimotor drive to the
muscle spindles to obtain adaptation. The mirror after-effect or the deflection of the
wrist in the opposite direction was also observed when the perturbing force was
removed during wrist extension (Figure 3.12.D). However, the magnitude of the after-
effect was reduced, compared to what was observed in the experiments. We did
program the gains to achieve good after-effects in the absence of perturbation but it
then compromised adaptation in the presence of the force fields. Setting the myriad
gains intuitively to provide both good adaptation as well as physiologically realistic
after-effects proved to be very challenging. We later optimized the gains using
gradient descent algorithm and obtained good adaptation as well as after-effects (see
chapter 4).

v. CONCLUSION
The large number of control inputs in our spinal cord model loomed as a
complicated and unsolvable problem when we first began constructing our models.
Our anxiety was justified from a control engineering standpoint, according to which it
is almost impossible to intuitively solve such a high dimensional problem. However,
101

the model performed surprisingly well and showed realistic results in each of the tasks
we simulated, requiring various adjustments of the control inputs that were intuitively
obvious, at least for experienced biomechanists and neurophysiologists. The next
chapter considers how a naïve nervous system would learn to perform these tasks.

102

CHAPTER 4: MATHEMATICAL OPTIMIZATION OF
SPINAL CORD
Encouraged by the good results from our previous simulations and the ease
with which we could produce realistic results, we tried to replicate and perhaps
improve the results using systematic optimization. We used gradient descent and hill-
climbing algorithms to optimize the gains and minimize deviations from desired
behaviors. A simple kinematic cost function (shown below) was defined to achieve the
desired behavior, effectively penalizing undesirable oscillations or susceptibility to
perturbations.

Algorithm 1 – Random Gradient Descent: Unlike the usual mathematical
framework on which the gradient descent algorithm operates, our model is not
mathematically invertable, so we had to design a custom algorithm to optimize the
cost. The steps in the algorithm are as follows:
1. Random values were assigned to the control inputs initially (Monte Carlo
method). However, we avoided extreme values (less than -0.5 and greater than
0.5) in order to prevent instability in the system.
2. A single control input was picked at random from the pool and optimized
individually.
103

3. Two simulations were required to optimize each control input. The first
simulation perturbed the control input in the positive direction and the second
perturbed it in the negative direction. The value of the control input that
produced the lowest cost in the two simulations was retained.
4. We used an “annealing curve” strategy when perturbing the control inputs, by
starting with a higher value initially ( Δ±0.2) and then reducing it for later
iterations (by successive factors of 2).
5. One iteration of the model was said to be completed when all control inputs in
the model had been optimized once.
6. The whole model was then iterated multiple times until the overall cost
converged to a low value.
Algorithm 2 - SCHLVND: Stochastic Hill Climbing with Learning by Vectors
of Normal Distributions (Rudolf et al., 1996), was another algorithm we used for
optimization. We used this algorithm only for a few tasks as we found that the
gradient descent algorithm performed much better (lower cost) in all the tasks
consistently. The gradient descent algorithm also performed more efficiently and
converged relatively rapidly to the solution.
SCHLVND used Hebbian learning of normal (Gaussian) distributions to
converge on a solution. It was inspired by genetic and stochastic hill climbing
algorithms. A normal distribution was used to represent each vector element (gain)
and vectors of random numbers chosen from these normal distributions were used as
input to the simulation that computed the cost function. For a single vector m, each m
i

104

was the mean of a normal distribution and s – the standard deviation of the normal
distributions. During each iteration, or generation, N random vectors were chosen
from m, where N is the size of the population. These random vectors were used as
inputs to run the simulation and compute the cost function. The best vectors were
stored and m was moved towards the average of the best vectors by the amount m
move
.
The standard deviation s was then reduced by the fixed ratio s
reduce
to limit the search
space. The iterations were repeated till a good performance was obtained.
The random gradient descent algorithm appears to be at least qualitatively
similar to the manner in which the infantile brain learns to use the spinal cord or
individual subjects acquire and refine any new motor skill. The optimization results
led to many intriguing observations that are discussed in detail below: 1) Even though
we started with random control input values initially, the algorithm converged to
produce similar and physiologically realistic outputs in all simulations; however, the
optimized control values were notably different. This indicated that the solution space
had many stable local minima that were good enough to produce acceptable behaviors.
2) Against common intuition, we found that the larger the number of inputs that were
optimized by the algorithm, the faster it converged to better performance. During
optimization we used an “annealing curve” strategy, wherein we perturbed the gains
initially at a larger step-size, namely +/- 0.2, in order to get out of local minima and
when the cost stabilized we reduced the value to lower numbers such as +/- 0.1, +/-
0.05, +/-0.01, etc.
105

i. STABILIZING RESPONSE TO FORCE PERTURBATION
Both “preflexes” and “reflexes” are under the control of the nervous system
and they can also be pre-programmed to deal with expected-perturbations. The
preflexes depend on the activation level of the muscles and depend on the CNS’s
selection of a particular pattern of muscle activation to perform the nominal task
(Brown and Loeb 1996). The reflexes may also be programmed to deal with expected
perturbations by adjusting the bias on the various interneurons that control the gain
between the afferent input and motor output and the unrecruited motoeneurons
themselves. They are thus largely governed by the background activity in the spinal
cord and also by the levels of gamma-static (bias control) and gamma-dynamic
(viscosity control) inputs to the muscle spindles.
It is common to ascribe learned responses to such reflexes, particularly when
they occur where this is feasible. On a similar paradigm, we optimized the gains in the
spinal cord systematically using gradient descent algorithm to resist perturbation in a
specified direction. Only the gains that adjusted the background activity in the spinal
cord (SET gains) and the gamma commands were optimized. The descending inputs
(GO) were fixed to zero. The desired behavior was defined as the initial resting
position of the hand and any deviation from it was penalized in the cost function.
The algorithm converged rapidly (in a single iteration) and stabilized the hand
in 200ms (Figure 4.1). The response was at least three times faster than what was
produced by previous manually tuned simulations (chapter 3) and also unlike previous
simulations, the hand stabilized back to the resting position without any overshoot in
106

the opposite direction. We then trained the model to resist perturbation in four
perpendicular directions (extension, flexion, radial/ulnar deviation) and tested the
converged solution on untrained random directions (for example, extension plus radial
deviation). The model produced similarly rapid and effective responses in all
directions (Figure 4.2).
The multiple iterations of the control inputs and the resulting performance by
the optimization algorithm could be compared to how human subjects successfully
resist perturbations by picking an efficient control strategy if the nature and
probability of perturbation is well perceived in advance. Initially we explored only a
subset of the gains (24 out of 184), confirming that the algorithm did improve
performance as measured by the cost function. However, the optimized trajectory of
the hand had small intrusions and was jerky. We then optimized all 184 gains and
obtained a smoother trajectory, faster recovery to the resting state and no oscillations
(Figure 4.1).
These observations reconfirm the significance and the contribution of all the
pathways and gains in the spinal cord towards achieving good performance. For
example, the Ia afferent feedback increases the closed loop stiffness, while the Ib
afferent feedback decreases it. It has been stated that the system without Ib afference
was found to have higher stiffness, a lower damping factor, and a longer time constant
(Chou and Hannaford, 1997). Van Heijst et al. (1998) noted that independent control
of muscle length and muscle tension was absent when the Renshaw interneurons were
disabled in their model. He et al. (1991) reported that a relatively natural-appearing
107

response to perturbations could only be achieved by including in the performance
criterion the feedback information from all the available modalities of sensors. The
model consistently showed rapid and efficient resistance to perturbation in all our
simulations that converged from multiple starting values for the control inputs. The
model also performed well for this task with the mean gain values that we computed
across all the tasks (see chapter 5).

Figure 4.1: Rotation of the hand about the wrist joint (extension-flexion axis) in response to a
external force perturbation (100N, 10ms) at one second simulation time. After optimization,
the response improved significantly and produced zero overshoot.

108

Figure 4.2: Rotation of the hand about the wrist joint in response to external force perturbation
(100N, 10ms) at one second simulation time about a random axis. (A) Rotation about
extension-flexion axis (B) rotation about radial/ulnar deviation axis
Iterations
1 8

Figure 4.3: Learning curve that shows change in cost with the step size used in annealing
curve. One-iteration constitutes one pass through all the gains. In all simulations the cost
converged to a low value in a single iteration and did not change significantly in subsequent
iterations. This indicates there are many stable local minima in the solution space.
109

As observed in all the other tasks we optimized, the maximum reduction in
cost was obtained in the first iteration that used the biggest step change in gain (Figure
4.3). Further optimization from this point produced only small reductions in cost even
though the gain values changed substantially, indicating that the local minima are
quite broad. The effect of small perturbing values in the gains, used in later
simulations, was almost negligible. However, most importantly, all local minima
appeared quite stable and showed physiologically realistic performance.

ii. RAPID VOLUNTARY MOVEMENT TO POSITION
TARGET
Figure 4.4 illustrates a typical gradient descent solution for a task similar to the
hand-tuned task illustrated in chapter 3 (Figure 3.4). All the SET and GO inputs were
optimized to obtain this response. As shown in Figure 4.4 (A & B), the desired
trajectory was defined as wrist extension to 45
0
with 0
0
deviation in radial/ulnar axis.
When the desired trajectory had zero rise-time, the algorithm anticipated the
movement and started extension before the GO commands initiated the movement, in
order to overcome the slow response of the muscles and thus reduce the cost.
Therefore the desired trajectory was given an initial 100ms rise time to the target. We
also tested the algorithm on different rise times and it produced solutions that followed
the trajectory very closely. It was also interesting to note that all of these ramp outputs
were produced by step inputs (GO), with zero rise-time.
110

The alpha motoneuron activity showed realistic reciprocal bursting for both
the ramp and the reflexive response to the perturbing force pulse (Figure 4.4.C). The
response to perturbation was much faster than the manually tuned model, stabilizing
the hand in about the half the time (similar to the passive perturbation task discussed
before). For initialization to 10 different sets of random starting values, the learning
curves all converged rapidly to stable solutions in the first or second iteration (Figure
4.5). Note the log scaled for cost (y-axis) in Figure 4.5; the initial values (Iteration 1)
differ greatly, while the final values are distinctly different from each other but are all
within typical variability of human performance. We further examined the solutions
themselves to see how they differed from each other. Figure 4.6 compares the final
values (ordinate axis) for 54 of the gains of two solutions from Figure 4.5 that
achieved very similar cost and kinematic details. They differ substantially in many
details, often including both magnitude and sign. This suggests that the state-space
consists of many, widely separated local minima that can be used to perform the task.
111

Figure 4.4: (A) Extension to 45
0
and response to perturbation. The desired trajectory is a ramp
with an input rise time of 100ms. (B) Rotation in the Radial/Ulnar deviation axis. The desired
trajectory is to have zero movement in that axis. (C) Output of the agonist and antagonist
motoneurons exciting the Extensor and Flexor muscles, during the ramp and during
perturbation.
Iterations
1 4

Figure 4.5: Learning curves from multiple starting points in state space (random values for the
control inputs), for 10 extension tasks.
112

Figure 4.6: Comparison of a subset of gains for two equally good solutions

We further analyzed the initial and converged gains by examining the distances
between the starting and final positions in the high-dimensional state-space of gains
(square root of the sum of squared distances in all 184 gain axes), illustrated in the
bar-graphs below, for the 45 possible pairwise comparisons, ordered according to their
distances from each other after optimization (Figure 4.7) and before optimization
(Figure 4.8). There is a weak tendency for solutions that are close to each other in
space to have started closer to each other initially, consistent with the expected
properties of gradient descent. There is no tendency for the normalized cost
differences of the starting positions or optimized solutions to correlate with separation
of the solutions in state-space, consistent with the notion that the local minima are
widely distributed throughout the state-space and not greatly different in local
steepness.
113

Figure 4.7: Pair-wise separations of all 10 solutions in state space, ordered according to their
distance from each other after optimization. Cost difference between the pairs before and after
optimization is also shown.

Figure 4.8: Pair-wise separations of all 10 solutions in state space, ordered according to their
distance from each other before optimization.
114

In order to perform voluntary goal-directed reaching, the motor system must
generate movement commands appropriate for two conditions: 1) internal demands
(target and speed of movement), and 2) external demands (loads and obstacles). We
tested the spinal cord in both these conditions and the model generated the necessary
commands to perform adequately and realistically. Ramps with different rising times
were used as the desired trajectory and the algorithm converged to produce
movements that closely tracked the trajectory. The model recovered quickly and
realistically from external force perturbations. We also increased the mass of the hand
to imitate external loading conditions and the model tracked the trajectory closely
without any overshoot. Surprisingly, simple step input commands were able to track
slow ramp trajectories with a high rise time (1sec). However, we acknowledge that
slow movements may require modulations in the input commands and zero-delay step
inputs may not work in all scenarios.

iii. OUTPUT OF ISOMETRIC FORCE TO A TARGET LEVEL
In this task we looked at the ability of the spinal cord to produce force steps
and pulses with varying speeds even when driven by unmodulated SET and GO
commands. The SET and GO commands were optimized using gradient descent
algorithm to track the desired trajectories with different rise times. The converged
solution produced output force that tracked the trajectories very closely with negligible
overshoot in both step (Figure 4.9.B) and pulse (Figure 4.9.A) scenarios. Similar to the
115

results from the hand tuned model described in chapter 3, bursts in the antagonist
motoneurons appeared consistently for brief rise time force trajectories (Figure
4.11.A). They appeared both in pulse and step scenarios. The antagonist bursts
disappeared when the rise times were increased to more than 120ms. In step force
trajectory an initial phasic command modulated the amplitude and determined the rate
of force change and a later tonic command was responsible for maintain the final
steady state force.

Figure 4.9: (A) Pulse force to the target level without overshoot. (B) Tracking step force
trajectory

116

Figure 4.10: (A) Output from the muscle spindles attached the agonist and antagonist muscles
(B) Output from Golgi tendon organs

Figure 4.11: (A) Output of motoneurons exciting agonist and antagonist muscles for 80ms rise
time pulse force trajectory. Note the conspicuous antagonist burst,(B) motoneuron output for
200ms rise time pulse force trajectory. Note that the antagonist burst is missing for larger rise
time. (C) motoneuron output for 200ms rise time step force trajectory, (D) motoneuron output
for 500ms rise time step force trajectory.
117

Many other results from the simulations matched well with the observations
from the experiment as well. Force trajectories with long rise times (>200ms) were
entirely controlled by the agonist (Figure 4.11.D), intermediate rise times (120 to
200ms) showed a discrete burst in the agonists (Figure 4.11.B, C) and for brief rise
times (<120ms), the reciprocal activation of the antagonist muscle occurred
consistently after a shorter agonist burst (Figure 4.11.A). The antagonist activity was
present with the same magnitude and timing in both force impulses and steps at equal
rise times. The agonist burst preceded the rising phase of force by 30-50ms. The initial
bursts in the agonist terminated abruptly in brief rise time tasks, but the abrupt
termination disappeared in long rise time trajectories. The duration of the silent period
after the initial burst in the agonist was also considerably reduced in step trajectories.
The magnitude of the antagonist burst was inversely proportional to the force risetime,
while the agonist burst was independent of the rise time, but instead scaled with the
force magnitude
The subjects in the experiment were instructed to respond “when ready” so as
to optimally prepare their response, but in a few tasks they were also trained to
completely suppress the antagonistic activity. In some cases they monitored the
response variables through earphones in order to suppress the antagonist activity. We
were also able to suppress the antagonist activity in our simulations by reducing the
fusimotor drive to the muscle spindles. Our assumption is that the subjects were
subconsciously able to reduce the reflex gains and the fusimotor drive to the muscle
spindles in these conditions, although the ability of subjects to consciously modulate
118

fusimotor gain remains controversial at least in the paradigms employed to date (Bent
et al., 2007).
Changes in the motor cortex cell activity during directional isometric force
generation show that the discharge of nearly all cells varied significantly with both
hand location and the direction of isometric force (Sergio, 2003, 2005). The cells in
the motor cortex have been assumed to be associated with “higher-level” goal oriented
details of the task while the spinal circuits presumably contribute to resolving the
challenges presented by the mechanical properties of the muscle and the stereotyped
properties of motor unit recruitment (Perlmutter, 1998). The tension produced during
muscle contraction represents a low-pass filtered transform of the modulation in the
motor neuronal firing rate (Soechting and Roberts 1975). At high frequencies of
modulation the tension generated in the muscles shows an increased phase lag.
Therefore to produce force impulses with short rise times, the controlling mechanism
has to compensate for the loss of gain and increased phase lag in response to the sharp
agonist activation. Our hypothesis is that the short delay reflex circuits in the spinal
cord could provide a way to compensate this effect by breaking the rise in force at the
target level, at the right time. Even if the descending command were preprogrammed
to generate the desired movement or force, the filtering action of the musculo-skeletal
system will distort and delay the effects of the central commands. But, the fast reflex
circuits in the spinal cord may provide a better solution by modifying/adding to the
descending commands, to achieve required performance in tasks with shorter delays.
119

Beyond isometric force modulation, another function for these precisely timed,
stretch-evoked responses that has been suggested is to dampen terminal oscillations
that ensue from rapid displacements of the mass of the limb against elastic forces of
muscle and soft tissue (Ghez et al., 1982). Two distinct strategies were suggested to be
effective in accurate repositioning of the limb and reducing overshoot (Ghez et al.,
1982). 1) Active deceleration through a phasic impulsive command to the antagonist.
2) Increasing joint stiffness at the terminal position through co-activation of antagonist
muscles.

iv. ADAPTATION TO VISCOUS CURL FORCE FIELDS
We repeated the simulations, described in chapter 3 (section IV), to find the
role of the spinal cord in motor adaptation, and optimized the gains using gradient
descent algorithm. Both SET and GO commands were optimized by the algorithm.
The cost function penalized any deviation from 0
0
about the radial-ulnar axis as well
as any deviation from the desired rotation in the intended direction (40
0
Extension).
Even though the model had to generate asymmetrical command inputs that were
required to counter the nonlinear perturbation, the gradient descent algorithm
converged in a single iteration to a very low cost and realistic performance (Figure
4.12). We repeated the simulations with multiple starting points and the algorithm
consistently produced good results, but the degree of after-effects varied between
converged solutions.
120

Figure 4.12: (A) Rotation about the intended direction (40
0
extension) after adaptation. (B)
Negligible deflection in the radial/ulnar axis in the presence of perturbing force. (C) Rotation
about the intended direction (from another random starting gain values). (D) Adaptation with
negligible deflection in radial/ulnar axis (E) After-effects in the intended direction after the
removal of perturbation. (F) After-effects in the radial-ulnar axis after removal of perturbation.

121

Figure 4.13: (A) Output from muscle spindles attached to all the four muscles, (B) Output
from Golgi tendon organs attached to all the four muscles, (C) Motoneuron output exciting all
the four muscles after adaptation, (D) Muscle force modulation (extensors and flexors) after
adaptation. (E) Motoneuron output during after-effects (F) Muscle force modulation during
after-effects.

Figure 4.12 illustrates the surprisingly good performance obtained using a
simple gradient descent solution with un-modulated SET and GO descending
122

commands for gains. The extension movement (Figure 4.12.A, C) follows the desired
trajectory almost as well as for the unopposed extension solution described in the
previous section (Figure 4.4.A); the initial radio-ulnar deviation of about 30° has been
reduced to almost zero (Figure 4.12.B, D). The SLR mechanisms that underlie this
performance appear to involve a combination of asymmetrical fusimotor gains (note
asymmetrical outputs of extensor spindle afferents in Figure 4.13.A) and asymmetrical
activation of propriospinal neurons (note early asymmetries in alpha motoneuron and
muscle force in Figure 4.13.C, D). The after-effects are also surprisingly
physiological when the curl-field is removed from a system that has adapted to it
(Figure 4.12.E, F).
It is possible to interpret these results in the context of “internal models” by
postulating that the transformation of the desired kinematic trajectory into a given
motor output is embedded in the selection of the descending commands to the spinal
cord, but nowhere in that process is there an explicit model of the plant or the load. It
is also possible, perhaps likely, that the actual performance of this task by humans
involves other, more complex calculations than the simplistic SET and GO commands
to which we limited our models, but the burden of proof would seem to lie with those
who have created a construct that may not be necessary to explain the experimental
results.
In order to explore the variability of adaptation across subjects we first
optimized the gains for a few unperturbed flexion-extension tasks that produced many
very different but equally good solutions (i.e. multiple local minima in the state-
123

space). We then used these solutions as starting points for adapting to the curl fields. .
In fact, a human confronted with this unusual load for the first time appears initially to
try to perform the required wrist extension movement according to his/her well-
practiced strategy for making the movement in the unloaded condition. The ensuing
visual and proprioceptive feedback informs the brain of the poor results, leading the
subject to modify the strategy on subsequent trials until desirable performance is
achieved. A classical description of this process is that the subjects must “correct”
their internal model of the manipulandum to incorporate the curl field; some part of
the brain then uses the internal model to convert the desired kinematic trajectory into
the complex pattern of muscle activation actually required to deal with the true load
presented by the manipulandum.
We found these apparently equivalent starting points resulted in substantial
differences for the adaptive learning, specifically, how quickly the model converges to
a solution, how good that solution is, and how large are the after-effects when the curl
field is turned-off. This would be consistent with the general observation that subjects
differ greatly in how quickly they learn to compensate for unusual and complex loads
such as curl fields, how well they perform, and how badly they suffer from
aftereffects.
We hypothesize that the internal model does not exist in any explicit form or
locus in the brain. Instead, the brain learns to adjust its descending commands so that
the spinal cord circuitry compensates for the load. When dealing with one of several
familiar loads, the brain selects the appropriate descending commands from a
124

repertoire of remembered solutions. The subjects in the experiment showed
aftereffects of force adaptation even though they had explicit knowledge that no
external force would be applied to the reaching hand (Kluzik, 2008). This suggests
that force adaptation cannot be switched on or off by cognitive cues alone but may
also involve a spinal level adaptation that takes time to readjust.
The errors produced due to the unexpected dynamic changes are transduced by
the proprioceptive sensors, which in turn evoke reflexes to reduce the error. Our
results suggest that the gains in the spinal cord can be readjusted to eliminate these
errors completely, by the process called ‘error-feedback-learning’, whch is well
described in the literature. The gradient descent algorithm adapts each gain to achieve
an approximate mean effect against the nonlinear sequence of perturbation. We
acknowledge that the model does not account for various other observations in the
experiments such as the generalization of adaption across arms, which happens in the
extrinsic coordinates of the task (Criscimagna-Hemminger et al., 2003). There are
substantial interhemispheral connections, however, as well as uncrossed descending
circuits that may contribute to such generalization, even without an explicit internal
model in extrinsic coordinates.

v. CONCLUSION
We noticed a consistent and prominent characteristic in the gradient descent
process for all tasks, regardless of complexity, and all starting position, regardless of
125

initial performance: The maximal reduction in cost was obtained in the first iteration
that used the biggest step change in gain. Figure 4.14 below shows one illustrative
solution for each of the five tasks that we have modeled to date. Further optimization
from this point produced only very small reductions in the cost even though there were
substantial changes in the control inputs, indicating broad local minima. We have
tried several different annealing curves (patterns of decreasing gain steps explored in
successive iterations), but this finding appears to be robust, suggesting that most local
minima are well-separated from nearby local minima. Of course, under some
circumstance it may be desirable to force systems out of local minima in order to find
global minima for truly optimal performance, as opposed to “good enough”
performance. That may require more complex gradient descent strategies in which
more than one gain is varied at a time. This notion seems to have some subjective
resonance with the strategies used by coaches to force athletes out of “bad habits”, but
that idea will require substantial additional development before it is ready for serious
discussion.
126

Cost
(log scale)
10000
1000
100
10
1
Iterations
01 23 45
Response to force perturbation
Voluntary movement to position target
Isometric STEP force to target level
Isometric PULSE force to target level
Adaptation to curl force fields

Figure 4.14: Learning curve for all the tasks simulated. Note that for every task the maximum
cost reduction is obtained in the first iteration. One-iteration is defined as one pass through all
the gains in the spinal cord.

These characteristics have profound implications for understanding how the
brain learns to control voluntary movements. These results lend support to our general
hypothesis that much of the detail associated with a complex movement is actually
generated during execution by the spinal cord; the real task of the brain is to configure
the spinal circuitry. Our models cannot prove that this is the case, but they can
demonstrate the extent to which this hypothesis is consistent with observed behavior.
One obvious complexity that we have eschewed concerns the temporal modulation of
descending commands during tasks. We have been surprised by the range of dynamic
127

behaviors that we can reproduce using simple SET & GO step functions, but we
expect this to break down for more complex tasks. Recordings of cortical activity
during even simple tasks often show substantial temporal modulation that is more
complex than a step function centered on the task initiation (although those
modulations are typically slower and less consistent than those exhibited by the EMG
of individual muscles). For future analyses of this model, we recommend replacing
the present command step functions with more realistic modulations of the command
signals mimicking such cortical unit recordings,.

128

CHAPTER 5: SYSTEMATIC ANALYSIS OF CONTROL
INPUTS
In the spinal cord model each synaptic connection between any two neurons
had a gain that could be independently set by the optimizing algorithm. Each gain had
a random starting value for every new simulation of the task. In order to systematically
analyze the contribution of each gain for each task, we formulated a method to define
a common starting point for all the tasks. If all the tasks started with the same set of
gains, then by looking at the deviation in the converged solution from this common
point, we could define the importance of each gain for the task.
However, the absolute value of the gain is not relevant in this formulation
because, for example, the gains that are lightly used for a particular task could have a
high starting value (because of random assignment), yet would matter little to the
solution, and instead reflect simply the breadth of the local minima. Therefore we
weighted each gain according to the amount of neural activity actually passing through
the connection as shown in the formula below. We multiplied each gain ‘G’ with the
integral of neural activity ‘I’ through the connection during the simulation time (t = 3
sec). We then normalized this product across all tasks. From our initial pool of all
converged solutions, we picked 5 solution sets for each task. As described in chapters
3 & 4, we simulated 5 different types of tasks in this thesis: 1) Stabilizing response to
external force perturbation, 2) Rapid voluntary movement to a position target, 3)
Rapid voluntary output of isometric force to a target level (STEP), 4) Rapid voluntary
129

output of isometric force to a target level (PULSE), 5) Adaptation to viscous curl force
fields. Therefore we selected a total of 25 solution sets (n=25) for this analysis. Each
gain value was thus weighted and normalized to get an approximate mean value across
all the 25 sets. We then used the computed mean values for each gain as the starting
point for the tasks and re-optimized them to produce good performances in all five
tasks. We iterated each task equal number of times using gradient descent algorithm
for a fair comparison. Therefore by looking at the deviation of each gain value from
the starting point, after optimization, we can infer the significance of that gain for that
task.
∑ ∑
where, n=25, t=3sec
We expected some gains to be dormant and not change across the task.
Surprisingly, each gain had a specific role in each task and the algorithm picked
generally different but realistic strategies for each task. For example, the mean values
did not change much for simple tasks such as response to external force perturbation,
but for more complex tasks such as adaptation to viscous curl force fields, each gain
was significantly changed. The algorithm activated specific interneuronal pathways
for each task to achieve the desired performance.
The results from this analysis could also give us some insight into the role of
the individual interneurons in each task. There have been many hypotheses about the
roles of specific pathways and interneurons in the spinal cord. In the fairly near
130

future, it will probably be possible to “knock-out” individual synaptic receptors and
neurons pharmacologically or genetically. The existence of spinal cord models that
account accurately for a wide range of sensorimotor behaviors provide an opportunity
to test these hypotheses and to anticipate and interpret those experiments. However, it
should be noted that, as explained in previous chapters, all the solutions appear to be
in local minima, and thus there could many other strategies that the spinal cord could
use to achieve the given or even better performance.
The following sections show plots with the deviation of each gain from the
mean value, for all the tasks that we simulated. First the circuitry of the pathway and
our numbering of the gains are shown. Note that each gain has four numbers in the
figure, which corresponds to the four similar gains with the same functional role that
affects the four muscles in our model. Some gains have + and – signs attached to the
numbers; they correspond to selective synapses that the algorithm sets as inhibitory or
excitatory based on the task. The sections are separated based on the different
pathways.
131

i. PROPRIOSPINAL PATHWAY

Figure 5.1: Propriospinal pathway

Figure 5.2: Deviation from mean values for Propriospinal pathway gains in Response to
perturbation task (Mean: 0.0013, Std Deviation: 0.0728)
123
4
567
8
910 11 12 13 14 15
16
17
18
19
20
21
22
23
24
25
26
27 28
29 30
31
32
‐0.5
‐0.3
‐0.1
0.1
0.3
0.5
Deviation from mean values
Propriospinal pathway gains
Task1: Response to perturbation
132

Figure 5.3: Deviation from mean values for Propriospinal pathway gains in Rapid voluntary
movement to position target task (Mean: -0.0361, Std Deviation: 0.0903)

Figure 5.4: Deviation from mean values for Propriospinal pathway gains in Voluntary output
of isometric force to a target level task with Step force profile (Mean: -0.011, Std Deviation:
0.1684)
1 2
3
4
5 6
7
8
910
1112
1314
15
16
17
18
19
20
21
22
23
24
25
26
2728
293031
32
‐0.5
‐0.4
‐0.3
‐0.2
‐0.1
0
0.1
0.2
0.3
0.4
0.5
Deviation from mean values
Propriospinal pathway gains
Task 2: Movement to position target
1 2
3
4
5 6
7
8
910
1112
131415
16
17
18
19
20
21
22
23
24
25
26
2728
293031
32
‐0.5
‐0.4
‐0.3
‐0.2
‐0.1
0
0.1
0.2
0.3
0.4
0.5
Deviation from mean values
Propriospinal pathway gains
Task3(a): Step isomteric force traget
133

Figure 5.5: Deviation from mean values for Propriospinal pathway gains in Voluntary output
of isometric force to a target level task with Pulse force profile (Mean: 0.0013, Std Deviation:
0.140)

Figure 5.6: Deviation from mean values for Propriospinal pathway gains in Adaptation to
viscous curl force fields task (Mean: 0.0486, Std Deviation: 0.2442)
1 2
3
4
5 6
7
8
910
1112
1314
15
16
17
18
19
20
21
22
23
24
25
26
2728
2930
31
32
‐0.5
‐0.4
‐0.3
‐0.2
‐0.1
0
0.1
0.2
0.3
0.4
0.5
Deviation from mean values
Propriospinal pathway gains
Task3(b): Pulse isomteric force target
1 2
3
4
5 6
7
8
9
10
1112
13
14
15
16
17
18
19
20
21
22
23
24
25
26
2728
2930
31
32
‐0.6
‐0.4
‐0.2
0
0.2
0.4
0.6
Deviation from mean values
Propriospinal pathway gains
Task4: Adaptation to curl force fields
134

The most noticeable feature from the above graphs is that the standard
deviations increase proportionally with the complexity of the task. The perturbation
task performs well with the mean gain values, but almost all the gains were changed
completely in the more complicated adaptation to viscous curl force field task. Also,
notice that the symmetrical values of the gains between the flexors and extensors were
preserved in all the tasks automatically, but when adapting to viscous curl force fields,
all gains corresponding to the four muscles have different and asymmetrical values in
order to compensate for the complex perturbation.
In external force perturbation task (Figure 5.2), two major adjustments can be
seen in the heteronymous gains. The algorithm adjusted the selective synapses such
that the two extensors excited each other (partial agonist 1 gains – 17 & 18) while the
extensors inhibited the flexors (partial agonist 2 gains – 29 & 30). This switching
action in the selective synapse was automatically defined by the algorithm in order to
counter the perturbing force (single axis) towards wrist flexion.
For rapid voluntary movement to position target task ((Figure 5.3), the
algorithm boosted the positive spindle feedback from extensors to propriospinal
interneurons to accelerate rotation (extension) to target (gains, 1 &2). It reduced the
drive from the propriospinal interneurons to the flexor motoneurons (gains, 11 & 12)
thus reducing the opposing force via co-contraction, and again helping in accelerating
the rapid movement. It also reduced the heteronymous negative feedback from the
flexors to extensors (gains, 15&16). The heteronymous selective synapses of the
partial agonists were readjusted automatically such that the extensors and flexors were
135

regrouped as synergists, respectively. The selective synapses between the extensors
were configured to be excitatory and the synapses between the extensors and flexors
were configured as inhibitory. All these automatic adjustments by the algorithm
appear to facilitate rapid response to the target position but, as discussed in chapter 4,
the results also closely tracked the rise time in the desired trajectory, suggesting the
role for other pathways.
In isometric force task (Figures 5.4 & 5.5), the algorithm boosted the positive
feedback from muscle spindle as well as the Golgi tendon organ feedback to the flexor
propriospinal interneurons in both step and pulse trajectories (gains, 3&4; 7&8). This
appears to provide the necessary antagonistic braking effect that was required to
stabilize the force at the target level. The algorithm lowered the propriospinal drive to
the antagonist motoneurons in the step task, while it boosted the same for the pulse
task (gains, 11 & 12). For the step task in order to hold the force at the target level,
excess and continuous opposing antagonist force is a disadvantage, while in pulse task
the continuous opposing force is of no concern and may instead help to improve the
braking effect to prevent overshoot. The heteronymous gains appear to be adjusted
quite differently between the pulse and step tasks in order to achieve the desired
performance. The gains in the adaptation to curl force field task (Figure 5.6) appear to
be arranged asymmetrically and is quite complex to analyze intuitively.

136

ii. MONOSYNAPTIC Ia PATHWAY

Figure 5.7: Monosynaptic-Ia Pathway

Figure 5.8: Deviation from mean values for Monosynaptic Ia pathway gains in response to
perturbation task (Mean: 0.0062, Std Deviation: 0.011)
1 2 3
4
5 6 7
8
9
10
11
12
‐0.5
‐0.3
‐0.1
0.1
0.3
0.5
Deviation from mean values
Monosynaptic Ia pathway gains
Task1: Response to perturbation
137

Figure 5.9: Deviation from mean values for Monosynaptic Ia pathway gains in Rapid
voluntary movement to position target task (Mean: 0.039, Std Deviation: 0.081)

Figure 5.10: Deviation from mean values for Monosynaptic Ia pathway gains in Output of
isometric force to a target level task with step force profile (Mean: -0.077, Std Deviation:
0.072)

1 2 3
4
5 6
7
8
9
10
11
12
‐0.5
‐0.4
‐0.3
‐0.2
‐0.1
0
0.1
0.2
0.3
0.4
0.5
Deviation from mean values
Monosynaptic Ia pathway gains
Task 2: Movement to position target
1 2
3
4
5 6
7
8
9
10
11
12
‐0.5
‐0.4
‐0.3
‐0.2
‐0.1
0
0.1
0.2
0.3
0.4
0.5
Deviation from mean values
Monosynaptic Ia pathway gains
Task 3(a): Step isometric force target
138

Figure 5.11: Deviation from mean values for Monosynaptic Ia pathway gains in Output of
isometric force to a target level task with pulse force profile (Mean: -0.010, Std Deviation:
0.132)

Figure 5.12: Deviation from mean values for Monosynaptic Ia pathway gains in Adaptation to
viscous curl force fields task (Mean: -0.007, Std Deviation: 0.193)

1 2
3
4
5 6
7
8
9
10
11
12
‐0.5
‐0.4
‐0.3
‐0.2
‐0.1
0
0.1
0.2
0.3
0.4
0.5
Deviation from mean values
Monosynaptic Ia pathway gains
Task3(b): Pulse isometric force target
1 2
3
4
5
6
7
8
9
10 11
12
‐0.5
‐0.4
‐0.3
‐0.2
‐0.1
0
0.1
0.2
0.3
0.4
0.5
Deviation from mean values
Monosynaptic Ia pathway gains
Task4: Adaptation to curl force fields
139

The standard deviation increases with the complexity of the task as before. The
gain values appear to differ negligibly from the mean values in perturbation (Figure
5.8) task while for the adaptation to curl field task most of the homonymous gains are
changed significantly (Figure 5.12). For the rapid extension to position target task
(Figure 5.9), the monosynaptic Ia gains to the motoneurons (gains, 1 2 3 4) were
boosted for all the four muscles, signifying the role of position and velocity feedback
in this task; the heteronymous gains adjusted such that the circuits connected to
extensor spindles excited each other forming a synergistic pair (gains, 9 & 10).
For isometric force to a target level task, the monosynaptic Ia gains to the
antagonist motoneurons appears to be boosted (gains, 3 & 4) in both step (Figure 5.10)
and pulse (Figure 5.11) scenarios. This observation matches well with our results and
is probably the reason why we saw brief bursts in the output of antagonist motoneuron
for brief force rise time tasks. (Note that fusimotor drive to the muscle spindles is
another mechanism that the algorithm can use to boost this feedback). The
antagonistic burst is critical for the braking effect in order to avoid overshoot above
the desired target force. The heteronymous gains appear to be adjusted in different
ways for both the task. Again, for adaptation to curl force field task (Figure 5.12) the
adjustments are asymmetrical and complicated to analyze intuitively.

140

iii. RECIPROCAL Ia INHIBITION PATHWAY

Figure 5.13: Reciprocal-Ia Inhibition Pathway

Figure 5.14: Deviation from mean values for Reciprocal-Ia Inhibition pathway gains in
response to perturbation task (Mean:-0.087, Std Deviation: 0.0271)
1
2
3
4
5
6
7
8
9
10
11 12 13
14
15
16
17
18
19 2021
22
23
24
25
26
27
28
29 30 31 32
‐0.5
‐0.3
‐0.1
0.1
0.3
0.5
Deviation from mean values
Reciprocal Ia Inhibition pathway gains
Task 1: Response to perturbation
141

Figure 5.15: Deviation from mean values for Reciprocal-Ia Inhibition pathway gains in Rapid
movement to a position target task (Mean: 0.016, Std Deviation: 0.0859)

Figure 5.16: Deviation from mean values for Reciprocal-Ia Inhibition pathway gains in Output
of isometric force to a target task with Step force profile (Mean: -0.015, Std Deviation: 0.153)

1
2
3
4
5
6
7
8
9
10
1112
13
14
15
16
17
18
1920
21
22
23
24
25
26
27
28
29303132
‐0.5
‐0.4
‐0.3
‐0.2
‐0.1
0
0.1
0.2
0.3
0.4
0.5
Deviation from mean values
Reciprocal Ia inhibition gains
Task 2: Movement to position target
1
2
3
4
5
6
7
8
9
10
1112
13
14
15
16
17
18
192021
22
23
24
25
26
27
28
2930
3132
‐0.5
‐0.4
‐0.3
‐0.2
‐0.1
0
0.1
0.2
0.3
0.4
0.5
Deviation from mean values
Reciprocal Ia Inhibition pathway
Task3(a): Step isometric force target
142

Figure 5.17: Deviation from mean values for Reciprocal-Ia Inhibition pathway gains in Output
of isometric force to a target task with Pulse force profile (Mean: -0.065, Std Deviation:
0.118)

Figure 5.18: Deviation from mean values for Reciprocal-Ia Inhibition pathway gains in
adaptation to curl force fields task (Mean: -0.029, Std Deviation: 0.245)

1
2
3
4
5
6
7
8
9
10
111213
14
15
16
17
18
1920
21
22
23
24
25
26
27
28
29303132
‐0.5
‐0.4
‐0.3
‐0.2
‐0.1
0
0.1
0.2
0.3
0.4
0.5
Deviation from mean values
Reciprocal Ia Inhibition pathway
Task3(b): Pulse isometric force target
1
2
3
4
5
6
7
8
9
10
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
2930
3132
‐0.6
‐0.4
‐0.2
‐1E‐15
0.2
0.4
0.6
Deviation from mean values
Reciprocal Ia Inhibition pathway
Task4: Adaptation to curl force fields
143

Similar to the previous pathways, the standard deviation increased with the
complexity of the task. The gains for resistance to perturbation task appear to change
negligibly from the mean values (Figure 5.14). In rapid extension to position target
task, (Figure 5.15) the inhibition to Ia interneurons from the extensors to the flexors
appear to be boosted in order to reduce the reciprocal inhibition from the flexors. This
will help in further accelerating the movement. In the isometric force task, the
contribution from the antagonist motoneurons are boosted by reducing their inhibition
from the reciprocal Ia-interneuron (the activation from the spindle on agonist Ia-
interneuron is reduced, gains, 1&2). This appears to be the case only for the pulse
trajectory (Figure 5.17) task. For step trajectory task (Figure 5.18) the gains from both
extensor and flexor muscle spindles to Ia-interneuron appear to be boosted. The
antagonistic muscle contribution is also increased in step trajectory task, by increasing
the inhibition of agonist Ia interneuron from the antagonist Ia-inerneuron (gains,
9&10).
144

iv. Ib INHIBITION PATHWAY

Figure 5.19: Ib-Inhibition pathway

Figure 5.20: Deviation from mean values for Ib-Inhibition pathway gains in Response to
perturbation task (Mean: -0.0043, Std Deviation: 0.093)
12
3
4
5
6
7
8
9
10
11
12
13 14 15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35 36
‐0.5
‐0.3
‐0.1
0.1
0.3
0.5
Deviation from mean values
Ib Inhibition pathway gains
Task1: Response to perturbation
145

Figure 5.21: Deviation from mean values for Ib-Inhibition pathway gains in Rapid movement
to position target task (Mean: 0.0067, Std Deviation: 0.085)

Figure 5.22: Deviation from mean values for Ib-Inhibition pathway gains Isometric force to
target level task with Step force trajectory (Mean: 0.062, Std Deviation: 0.125)

12
3
4
5
6
7
8
9
10
11
12
13 14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35 36
‐0.5
‐0.4
‐0.3
‐0.2
‐0.1
0
0.1
0.2
0.3
0.4
0.5
Deviation from mean values
Ib Inhibition pathway gains
Task 2: Movement to position target
123
4
5
6
7
8
9
10
11
12
13 14 15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35 36
‐0.5
‐0.4
‐0.3
‐0.2
‐0.1
0
0.1
0.2
0.3
0.4
0.5
Deviation from mean values
Ib Inhibition pathway gains
Task3(a): Step isometric force target
146

Figure 5.23: Deviation from mean values for Ib-Inhibition pathway gains Isometric force to
target level task with Pulse force trajectory (Mean: -0.054, Std Deviation: 0.0109)

Figure 5.24: Deviation from mean values for Ib-Inhibition pathway gains in Adaptation to
viscous curl force field task (Mean: -0.019, Std Deviation: 0.279)

12
3
4
5
6
7
8
9
10
11
12
13 14 15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35 36
‐0.5
‐0.4
‐0.3
‐0.2
‐0.1
0
0.1
0.2
0.3
0.4
0.5
Deviation from mean values
Ib Inhibition pathway gains
Task3(b): Pulse isometric force target
12
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
‐0.6
‐0.4
‐0.2
‐1E‐15
0.2
0.4
0.6
Deviation from mean values
Ib Inhibition pathway gains
Task4: Adaptation to curl force fields
147

The response to perturbation task appears to show some deviation from the
mean values in this pathway (Figure 5.20). However, it is considerably lower than the
standard deviation for adaptive curl force field task. The excitation from tendon organs
to the Ib interneuron appear to be boosted for antagonist muscles (gains 3&4), which
probably will help the agonist in pulling back the hand during perturbation. Significant
changes were seen in the heteronymous connections with selective synapses.
Heteronymous inhibition between the extensors was reduced and the excitation from
flexors to extensors was increased. All these adjustments effectively results in helping
the extensors to pull the hand back to the resting position rapidly.
In rapid response to position target task (Figure 5.21), this pathway appears to
be functioning according to its classic textbook definition, as a negative force
feedback to the motoneurons. Both the excitation of the Ib-interneurons by the Golgi
tendon organs (gains, 1&2) and the inhibition of motoneurons by the Ib-interneurons
(gains, 13&14) appear to be boosted for extensors muscles providing a negative force
feedback. This is critical to stabilize the hand at the target position. The force step
(Figure 5.22) and force pulse (Figure 5.23) scenarios in the isometric output force to
the target task appear to be using completely different strategies. The excitation of the
Ib interneuron by the GTO is boosted for all the muscles in force step task (gains, 1 2
3 4) while the excitation of the interneuron by the antagonist GTO’s alone (gains,
3&4) were reduced in force pulse task. For maintaining the output force at the target
level in the force step task, the force feedback from all the muscles is critical while for
the quick force pulse task, the inhibition of the antagonist motoneuron by the GTOs is
148

probably a disadvantage that will prevent quick rise in force. Similar to other
pathways the gains appear to be arranged in a complicated pattern for the adaptation to
curl force fields task (Figure 5.24).

v. RENSHAW PATHWAY

Figure 5.25: Renshaw pathway

149

Figure 5.26: Deviation from mean values for Renshaw pathway gains in response to
perturbation task (Mean: 0.01918, Std Deviation: 0.068)

Figure 5.27: Deviation from mean values for Renshaw pathway gains in Rapid movement to
position target task (Mean: -0.002, Std Deviation: 0.089)

1
2
3
4
5 6 7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
‐0.5
‐0.4
‐0.3
‐0.2
‐0.1
0
0.1
0.2
0.3
0.4
0.5
Deviation from mean values
Renshaw pathway gains
Task1: Resonse to perturbation
1
2
3
4
5 6 7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
‐0.5
‐0.4
‐0.3
‐0.2
‐0.1
0
0.1
0.2
0.3
0.4
0.5
Deviation from mean values
Renshaw pathway gains
Task2(a): Extension w/o perturbation
150

Figure 5.28: Deviation from mean values for Renshaw pathway gains in Output of isometric
force to a target level task using Step force trajectory (Mean:-0.052, Std Deviation: 0.168)

Figure 5.29: Deviation from mean values for Renshaw pathway gains in Output of isometric
force to a target level task using Pulse force trajectory (Mean: -0.066, Std Deviation: 0.118)

1
2
3
4
5 6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
‐0.5
‐0.4
‐0.3
‐0.2
‐0.1
0
0.1
0.2
0.3
0.4
0.5
Deviation from mean values
Renshaw pathway gains
Task3(a): Step isometric force target
1
2
3
4
5 6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
‐0.5
‐0.4
‐0.3
‐0.2
‐0.1
0
0.1
0.2
0.3
0.4
0.5
Deviation from mean values
Renshaw pathway gains
Task3(b): Pulse isometric force target
151

Figure 5.30: Deviation from mean values for Renshaw pathway gains in Adaptation to viscous
curl force fields task (Mean: -0.25, Std Deviation: 0.313)

As in most of the other pathways, the response to perturbation task had a low
standard deviation (Figure 5.26). The flexor muscles appear to organize as synergists
automatically, by boosting the heteronymous activation of motoneurons to the
corresponding renshaw internurons (gains, 11&12) in the response to perturbation
task. In rapid movement to position target task (Figure 5.27), the homonymous
renshaw inhibition appears to be boosted in the antagonist muscles (gains, 3 & 4). This
reduces the opposing force to the extensors thus helping in rapid movement.
In step isometric force task (Figure 5.28), the recurrent inhibition to the
antagonist motoneurons appear to be considerably reduced by decreasing the
motoneuron activation of the renshaw interneuron (gains, 3&4) as well as decreasing
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
‐0.8
‐0.6
‐0.4
‐0.2
0
0.2
0.4
0.6
0.8
Deviation from mean values
Renshaw pathway gains
Task4: Adaptation to curl force fields
152

the inhibition back to the motoneurons by the renshaw interneuron (gains, 19&20).
The recurrent inhibition to the motoneurons appear to be reduced in antagonistic
muscles in pulse force task (Figure 5.29) as well (gains, 19&20) but the activation of
the renshaw interneuron by the motoneuron is not changed. Unexpectedly the agonist
recurrent inhibition is also boosted in pulse force task (gains 17&18). This feedback is
probably required to match the rise time of force accurately. The gains in adaptation to
viscous curl force field tasks again appear to be set in a complex manner and difficult
to interpret intuitively (Figure 5.30).

vi. CONCLUSION
Information regarding the organization of the spinal circuitry for the wrist
movement is limited, both for human subjects and the cat. We derived our circuitry
from the situation at simple joints, which displays only a rigid and stereotyped motor
behavior. However, the systematic analysis of the gains in the circuits, as used by the
gradient descent algorithm, showed surprising similarity to the classical definitions
and roles of each pathway. The algorithm selectively activated the required pathways
and in general used different strategies for each task. It automatically imposed
symmetry in single axis tasks such as extension, but adjusted each gain differently for
more complicated tasks with nonlinear perturbations, such as the adaptation to the curl
force field task. Moreover, at least qualitatively, each task was performed within the
limits of behavioral variability in humans.
153

The organization of the circuit matched somewhat closely with experimental
observations in the literature. For example there was inhibition between the flexor
radialis and extensor radials Renshaw interneurons in both extension and isometric
force tasks (Figures 5.27, 5.28, 5.29, gains 23&25) similar to the circuit proposed by
Carroll et al. (2005). The algorithm also set the selective synapses such that it was
appropriately defined as excitatory or inhibitory based on the requirements of the task.
By doing so it automatically grouped the ‘partial-agonists’ into synergistic pairs for
the different tasks.
As described before the gradient descent algorithm converged to widely
varying local minima in all iterations and produced different solution sets. Therefore
one such set for each task as used in this analysis may not expose all the strategies
used by the algorithm to perform the task. However, our assumption is that by starting
from a normalized and averaged common point, we would be able to infer the
importance of some gains by looking at their deviation from the corresponding mean
values. The variability in solution sets may also be due to the fact that the algorithm
picks each gain randomly and optimizes it individually. In a high dimensional problem
such as this, we acknowledge that there could be distributed effects across the gains
and the algorithm could be leveraging redundancy in the solution space. All these add
up to the difficulty in analyzing the individual contribution of the gains in the circuit.

154

CHAPTER 6: COMPARISON TO SERVOCONTROL
SCHEME
Our central hypothesis presumes that the actual circuitry of the spinal cord has
developed to confer valuable attributes to the biological system. A counter-hypothesis
is that any sufficiently complex network with these sensors and actuators might have
similar attributes without requiring the specific interneuronal types in the spinal cord.
We substituted the complex spinal cord model with a classical servo-control scheme in
the same simulation environment, to see if a relatively simplistic controller can
replicate the desirable performance and learning characteristics that we observed in
our previous simulations. In the new scheme, each alpha motoneuron received direct
“feed-forward” commands plus homonymous positive feedback from the muscle
spindle (primary) and negative feedback from the Golgi tendon organ (Figure 6.1).
α
TRUE-ANTAGONIST
Ib
Ia
GO
Golgi Tendon Organ
Muscle Spindle
gamma-dynamic
gamma-static

Figure 6.1: Wiring diagram of the servo-controller
155

The only heteronymous projection to each alpha motoneuron was the negative
feedback from the muscle spindle of the antagonist muscle. Unlike the previous
model, all the motoneurons received direct step descending inputs that were set by the
optimization algorithm. The scheme had only 24 control inputs as compared to 184 in
the spinal cord model. We used the same optimization algorithm (as described in
chapter 4) to adjust the control inputs and we trained the model for the same four tasks
that we replicated before, under the same conditions. The details of the results
obtained are explained in the following sections.

i. STABILIZING RESPONSE TO FORCE PERTURBATION
Stability and performance for this task, given a servo-control scheme, depends
on the interplay between three major mechanisms. First, the intrinsic spring-like and
viscous properties of muscle constitute the “preflex” mechanism that acts
instantaneously against changes in posture. Second, appropriate levels of muscle co-
activation can alter the “preflex” response to resist a given perturbation optimally.
Finally, high stretch reflex gains of the muscles opposing the perturbation also provide
a short delay resistance to perturbations.
In most trials, the servo-controller set the stretch reflex gain of the extensors
high with almost no co-activation of antagonist muscles, but performed worse than the
spinal cord model (Figure 4.2 E&F). The best performance was observed when the
control strategies found by gradient descent algorithm involved a balance between all
156

three mechanisms discussed above (Figure 4.2 C&D). In fact, the kinematic
performance of a few trials using this strategy was better than the spinal cord model.
However, all good performances involved unrealistic levels of co-contraction (Figure
4.2 A&B), unlike what was observed in the spinal cord model. A cost function that
penalizes expended metabolic energy would show poor cost performance in these
conditions.

Figure 6.2: (A) Extensor Carpi Ulnaris (ECU) muscle force modulation, (B) Flexor Carpi
Radialis (FCR) muscle force modulation, Note the large co-contraction in the muscles. (C)
The best perturbation response along the x-axis, (D) Corresponding response along the z-
axis.(E) Typical perturbation response along x-axis, (F) Corresponding response along z-axis.
157

Figure 6.2 C&D illustrates the best performance (lowest kinematic cost) by the
servo-control model. The control strategy included co-contraction of the antagonist
Extensor Carpi Ulnaris (ECU) (Figure 6.2.A) and Flexor Carpi Radialis (FCR) (Figure
6.2.B) muscles, occurring prior to the perturbation. At the onset of the perturbation,
two seconds into the simulation, there was a sudden increase in EU force paralleled by
a decrease in FR force. The resistance to the perturbation observed was due to a
combination of the zero-delay response (preflex) due to the intrinsic properties of
muscle in addition to the stretch reflex response, which transiently increased the
extensor motoneuronal output. However, even though the perturbing force vector was
directed entirely in the direction of wrist flexion, a significant deviation appeared in
the ulnar direction (about the z-axis). This was mostly due to the asymmetric
activation and muscle spindle feedback to the extensors, resulting in an unequal
response between the two muscles. A more evenly distributed pattern of activations to
the muscles would obviously overcome this asymmetry but, unlike the spinal cord
model, such solution does not appear to be easily discoverable in the servo-control
scheme through a simple gradient descent algorithm.

ii. RAPID VOLUNTARY MOVEMENT TO A POSITION
TARGET
Gradient descent algorithm found solutions with associated costs that were
higher on average and more variable in comparison to the simulations with the spinal
158

cord model, for this task. Furthermore, the resulting trajectories showed significant
variability of performance between trials. In some cases, there were large deviations
in the resting as well as the movement phases of the trajectory in both x and z-axis.
The general strategy for this task, as found through gradient descent algorithm,
involved mainly co-contraction of antagonist muscles (Figure 6.3 A&B). This
strategy takes advantage of the activation dynamics of muscle. By activating the
extensor muscles slightly more than the flexors, for example, the resulting extensor
torque increases at a higher rate before it reaches the target level, therefore, generating
the force transient required for the short ramp phase of the task. Obviously, this a
highly inefficient strategy due to the enormous metabolic costs involved. Contrary to
this, in the spinal cord model the gains were tuned to minimize the opposing forces
from the antagonist in order to accelerate the movement (see chapter 5 for detailed
analysis) and it took advantage of both homonymous and heteronymous connectivity
in the spinal cord to perform the task with minimal co-contraction.
Even while using co-contraction, the net extensor torque at steady state
continued to rotate the hand at the hold phase even after reaching the target position in
some cases. In one trial (that had the best performance) the algorithm used
proprioceptive feedback effectively to reduce the extensor torque and counter this
effect. As the flexor muscles lengthened, the activity in their muscle spindles inhibited
the alpha motoneurons of the extensors, hence reducing the steady state extensor
torque. The algorithm discovered this strategy only in a single trial, which also
resulted in the best performance (Figure 6.3 C&D).
159

Another anomaly was the lack of symmetry in the converged solutions. In most
of the tasks even though the desired trajectory imposed extension with zero
radial/ulnar deviation, the servo-control scheme produced considerable deviation in
the z-axis (Figure 6.3.F). On the contrary, all the solutions of the spinal cord model
automatically limited any deviation in the z-axis in order to reduce the overall cost.

Figure 6.3: (A) Extensor Carpi Ulnaris (ECU) muscle force modulation, (B) Flexor Carpi
Radialis (FCR) muscle force modulation (C) The best performance: extension (D)
Corresponding Z-axis deviation (E) Typical performance: extension (F) Corresponding Z-axis
deviation

After running several trials of this task, we found that the solution space as
crafted by the servo-control scheme in conjunction with the musculoskeletal structure
160

and proprioceptors, had easily discoverable solutions, but most them were inferior in
performance as compared to the results from the spinal cord model. The quality and
discoverability of the solutions is inherent to the neuronal connectivity that the servo-
controller imposes between the muscles and the descending commands. The direct
descending input to the alpha motoneurons, for example, allows gradient descent to
discover the crude strategy of co-contraction easily. It appears that the strategy lies in
local minima from which it is difficult for the algorithm to escape in order to find the
right combination of proprioceptive feedback that would further optimize the cost.

iii. OUTPUT OF ISOMETRIC FORCE TO A TARGET LEVEL
a. Force step trajectory
The servo-controller adopted various strategies for this task, with the most
common being high activation of an extensor muscle (Figure 4.4.B) with negligible
proprioceptive feedback (the associated gains were tuned to very low values). This
was sometimes accompanied by minor activation of the flexors that effectively
brought down the steady state level of the force while affecting the rapid rise time of
the force slightly. The strategy that produced the best performance involved over-
activation of extensor muscles and use of inhibitory propiocepive feedback, namely
homonymous Ib feedback and heteronymous muscle spindle feedback (Figure 6.4.C).
This feedback allowed the force to rise near the desired rate while trimming down the
late portion of the activation step input, hence reducing the steady state force level
161

during the hold phase. This is a major contrast with the strategy employed by the
spinal cord model, especially in brief rise time force trajectories, that actively used the
opposing force from the flexors for the braking effect required to avoid overshooting
the target.
In general the servo-control scheme performed much worse than the spinal
cord model. Most of the time it produced considerable overshoot, and it failed to
mimic the desired rise time of force trajectory. The rise times in most of the trials were
considerably longer than requested; for brief rise time trajectories, the force overshot
the target consistently and sometimes even produced oscillations in the hold state of
the task (Figure 6.4.D).

Figure 6.4: (A) Output of the motoneuron to extensor muscle, (B) Extensor muscle force
modulation (C) Best solution (D) Typical solution for brief rise time force trajectory

162

b. Force pulse trajectory
The strategy that produced the best performance in this task consisted of a high
input pulse to the extensor motoneurons, followed by homonymous GTO feedback
that trimmed down the late portion of the activation pulse. The controller took
advantage of the muscle activation dynamics to match the rise time. It also adjusted
the magnitude of the pulse to the motoneurons to avoid overshoot (Figure 4.5A).
However, there were many differences from the strategy used by the spinal
cord model: 1) the flexors were mostly not activated as compared to the spinal cord
model, which used the flexors for braking effect in order to avoid overshoot above the
target, especially in brief rise time trajectories, 2) because the net output force was
computed as the vector sum of forces from all the muscle, the servo-controller
activated the muscles asymmetrically even while matching the desired force
trajectory; for example, in one of the best performance trials only one of the extensors
were activated to match the net force while in contrast the spinal cord model equally
distributed the activation between the two extensors that added up to track the force
trajectory, 3) Surprisingly, the spinal cord model tracked not only the rise time of
force trajectory but also the fall time. The servo-controller performed terribly at
controlling the decaying rate of force. The de-activation dynamics of the muscles are
comparatively slower than activation, so it was impossible for the servo-controller to
match both the rise time as well as the fall time of the desired pulse trajectory when
using only the extensor muscles for the task. This suggests the significance of the role
of the proprioceptive feedback and the opposing muscles in the spinal cord model.
163

Figure 6.5: (A) Output of the motoneuron to extensor muscle, (B) Extensor muscle force
modulation (C) Best solution (D) Typical solution for brief rise time force trajectory

iv. ADAPTATION TO VISCOUS CURL FORCE FIELDS
This is obviously the most complicated of all tasks and the servo-controller
performed poorly in most trials. Only two trials showed acceptable performance,
qualitatively (Figure 6.6. C&D). The rest consisted of significant deviations from both
the x and z axis target trajectories. Specifically, the strategies that were used did not
compensate enough for the curl field, leading to significant ulnar (z-axis) deviation.
Another common observation within most trials was a significant discrepancy in the
extension direction during the initial rest phase of the movement (Figure 6.6.E). Most
of the solutions (50%) resulted in significant oscillations. As before, the most
164

successful and commonly used strategy was co-contraction (Figure 6.6. A&B). The
best performance trial co-contracted the extensor radialis and flexor ulnaris muscles
and did not take advantage of any proprioceptive feedback. Contrary to these results,
the spinal cord model used a more efficient strategy (little co-contraction), by
effectively using the proprioceptive feedback and by activating the gains
asymmetrically to achieve surprisingly good performance and after-effects.

Figure 6.6: (A) Extensor Carpi Ulnaris (ECU) muscle force modulation, (B) Flexor Carpi
Radialis (FCR) muscle force modulation (C) The best performance: extension (D)
Corresponding Z-axis deviation (E) Typical performance: extension (F) Corresponding Z-axis
deviation

165

v. CONCLUSION
We repeated all the tasks that we replicated with the realistic spinal cord model
using the servocontrol scheme, under the same conditions and on the same
biomechanical plant. In general the mean performance cost of the servo-controller was
significantly higher in every task as compared to the performance of the spinal cord
model (Figure 6.7). The performance was highly sensitive to the random starting
values of the gains and the initial step of the annealing curve. With the same annealing
curve used to optimize the spinal cord model, the gradient descent algorithm
converged to poor solutions frequently. However, we found that increasing the initial
step size and steepness of the annealing curve resulted in significantly better
performance in some tasks, although still generally worse and more variable than that
produced by the spinal cord model (Figure 6.7).
As illustrated in the examples before, the common strategy used by the servo-
controller across all the tasks was co-contraction. Most of the trials that produced good
results had unphysiological amounts of co-contraction. This strategy is highly
inefficient as it requires high metabolic energy and is usually employed by human
subjects only in the first few trials of a new task if they are confused by its difficulty.
A cost function that penalizes both kinematic deviations as well the expended
metabolic energy will specifically differentiate co-contraction strategy from others.
Improving the cost function is suggested for future work.
Another significant shortcoming in the performance of the servo-controller was
the lack of the use of proprioceptive feedback in the tasks. The spinal cord model
166

relied heavily on proprioceptive feedbacks to improve performance in all the tasks,
and their significance was particularly observed by the large gains of the feedback
loops in the converged solutions as set by the optimizing algorithm. On the contrary
the servo-controller relied on co-contraction. These observations correspond well with
experimental findings in the literature, where deafferented primates relied heavily on
co-contraction as the strategy to accomplish tasks (Bizzi et al., 1976; Polit and Bizzi,
1978, 1979).

Figure 6.7: Comparison of the performance between the Spinal cord model and the simple
servo-control scheme for all the tasks. The variability in experimental data from literature
(Ghez et al., 1987; Wierzzbicka et al., 1991; Liles et al., 1985) is also shown. Two sets of
experimental data are shown for the Force Pulse task 1) with least amount of co-contraction,
2) with significant co-contraction (dotted).

Figure 6.7 shows the cost comparisons between the two schemes. Five trials of
each task using the servo-control scheme are compared against five trials of similar
tasks performed by the spinal cord model. The spinal cord model produced lower cost
167

solutions, on average, across all the tasks (note log scale for cost). The standard
deviation of the solutions for each task also appears to be considerably low for the
spinal cord model solutions. However, in perturbation tasks a few solutions from the
servo-control scheme appear to perform under the same low cost as the spinal cord
model. We checked the performance of these lower cost solutions and found that all of
them used high co-contraction as the control strategy to reduce the cost. As mentioned
before, a more realistic cost function that includes metabolic cost would render these
solutions as inferior. All the solutions of the spinal cord model in more difficult tasks
such as the isometric force output and adaptation to curl force fields, appear to be
considerably superior to most of the solutions produced by the servo-control scheme.
Moreover, the performance of the spinal cord model appears to be well within the
experimentally observed variability for similar tasks on humans and primates.
It is also worth noting that similar to the spinal cord model, the gradient
descent algorithm converged to solutions rapidly. The results indicate that the servo-
controller can mediate acceptable performance, but the corresponding solutions are far
from physiological and are more difficult to find relative to the spinal cord model.
Experimenting with different annealing curves and random starting values for the
gains suggested that the solution space is relatively coarse, consisting of many local
minima that are undesirable. The local minima appear to be very broad as well and it
was often quite difficult to escape from poor minima even with multiple iterations.
The results discussed in this chapter thus support the value of the unique and complex
circuitry of the spinal cord.
168

CHAPTER 7: CONCLUSION AND FUTURE WORK
i. CONCLUSION
If all of the elements comprising a system are known, then the emergent
behavior of a model of the system should replicate the observed behavior of the real
system. We have developed an explicit model of the spinal cord circuitry in this thesis.
All the well known pathways described in the literature were modeled. We then
programmed this model to emulate actual performances of a wide range of tasks as
described in human subjects and nonhuman primates. A simplified model of a two-
dimensional wrist operated by four realistic muscles accounted surprisingly well for
the details of all the tasks we replicated.
While a great deal is known about the spinal interneuronal circuitry, the
information is certainly incomplete. Much of the details in our model were
extrapolated from literature based on simple circuits mediating movements in less
complicated joints such as the elbow joint. However, this model appears to be the first
to incorporate all the known pathways of the spinal cord and also to deal with the
shifting patterns of synergistic and antagonistic muscle activity that accompanies
multimuscle, multi-degree-of-freedom systems.
The model had a large number of input parameters not unlike the real spinal
cord. Surprisingly, against common intuition, it was not difficult to obtain
physiologically realistic performance in all the tasks we modeled with just simple
intuitive adjustments of these parameters. Moreover, the other details of the task such
169

as the temporal patterning of the muscle recruitment, motoneuron output, etc. looked
physiologically realistic. We then optimized the control inputs using a gradient
descent algorithm that, at least qualitatively, resembles how subjects learn and refine
new motor skills. The algorithm converged rapidly to provide physiologically realistic
results. The most salient emergent property of the whole system is that despite the
complexity of the available circuitry, many different combinations of command
signals result in generally stable and desirable output behaviors. All the local minima
in the solution space were surprisingly stable and provided physiologically realistic
results.
We then systematically analyzed the converged solutions of each task and
found close resemblance to the classical definitions of the role of the pathways in the
spinal cord. We found that the algorithm selectively activated the necessary
connections and pathways to perform each of the tasks. The distribution of gains also
suggested the significance of both the homonymous and heteronymous connections in
the spinal cord. We repeated all the tasks using a simple servo-controller scheme.
Even though the servo-controller produced good results for a few trials in each task, in
general, the solution space had many local minima that had substantial deviations from
physiologically realistic behavior. Even when the performance matched the results
from the spinal cord model, the co-contraction strategies used would have been highly
inefficient in terms of energy consumption.
These characteristics have profound implications for understanding how the
brain learns to control voluntary movements. Our general hypothesis is that most of
170

the details associated with a complex movement are actually generated during
execution by the spinal cord; the real task of the brain is to configure the spinal
circuitry. Our models cannot prove that this is the case, but they can demonstrate the
extent to which this hypothesis is consistent with observed behavior.

ii. FUTURE WORK
a. Improvements in the current model
One obvious complexity that we have eschewed to date concerns the temporal
modulation of descending commands during tasks. We have been surprised by the
range of dynamic behaviors that we can reproduce using simple SET & GO step
functions, but we expect this to break down for more complex tasks. Most obviously
it cannot account for cyclical tasks such as locomotion, but this raises a question that
has come up many times in theories of motor control. How many separately
controlled phases are there in a task that appears to consist of sequential output states?
As we have seen in the results, sequences of activation in individual muscles (e.g.
triphasic agonist-antagonist bursts) may reflect the emergent reverberations of spinal
circuits that converge uniphasic descending commands with ongoing sensory
feedback. These emergent phases were able to be modified individually without
invoking similarly phased descending commands, simply by training the system to
make slower movements (always using step functions for the commands).
Nevertheless, recordings of cortical activity during even simple tasks often show
171

substantial temporal modulation that is more complex than a step function centered on
the task initiation (although those modulations are typically slower and less consistent
than those exhibited by the EMG of individual muscles). We suggest replacing the
command step functions with more realistic modulations of the command signals
mimicking such cortical unit recordings. We also recommend modulating the
fusimotor inputs to the model, as is known to occur during locomotion and other tasks
(Loeb, 1985). The performance of such modeling and physiological experiments is
outside the scope of this thesis, however.
Another suggested addition is to improve the cost function used by the gradient
descent algorithm for optimization. We used a simple kinematic cost function for all
our tasks. Surprisingly, all solutions converged to qualitatively realistic performances
rapidly. Even though humans behave sub-optimally in most tasks, a cost function that
penalizes metabolic energy would be more in tune with the realistic computations
made by the human brain. It would, for example, reject strategies that impose high co-
contraction of the muscles while performing the task. The Virtual Muscle model thus
has to be modified to include energy consumption calculations.
In most of our simulations the control inputs were optimized for individual
tasks separately. However, generating a set of inputs that can generalize across a few
or all the tasks might be more physiologically correct. This imposed limitation may
also perhaps reduce the number of well performing local minima in the solution space.
As explained before we did not model the direct corticospinal projections to the
motoneurons in our model. It should be noted that we do not rule out the possibility of
172

adding the direct pathway in the future. Adding the direct pathway to our model and
re-optimizing a previously converged solution set without the direct pathway might
provide hints on its role in our model. We suggest that for more complex
biomechanical models such as the digits of the hand, the indirect pathway as we
modeled might be insufficient and they would probably require the addition of the
direct cortiospinal projections to the motoneurons to achieve good performance.
In order to reduce the computational complexity of our model, we had to
approximate our neuron models as simple, time-invariant, stateless functions. Using
advanced computational technologies such as distributed computed and parallel
processing, these neurons may be modeled as more accurate functions. In fact, each
interneuron can be modeled according to its distinct characteristics. A few other
afferent feedbacks such as cutaneous receptors and joint receptors, which we left out,
could also be added to the model with the availability of improved computational
technology. Another option is to use programmable gate arrays to run these complex
models in hardware.
We limited the complexity of the biomechanical plant in order to concentrate
more on the spinal circuits. Our model essentially addresses the primary problem of
controlling a two degree-of-freedom joint; however, a more realistic model of the
human hand controlled by the five primary muscles with realistic moment arms and
force vectors may be useful in comparing the results of the model with experimental
observations. This is however, beyond the scope of this thesis. We used a reasonably
accurate model of the muscle which is a substantial difference from the other previous
173

models in the literature. We suggest repeating the simulations with a less accurate
model of the muscle, which could provide some hints on the significance and the
contribution of the muscle model in obtaining good results. Even though we analyzed
the control inputs of the model systematically, we also suggest doing a principle
component analysis (PCA) of the converged solutions sets, which could provide
information on the co-variance between the parameters in the solution set.
b. Model of shoulder-elbow motion
The spinal circuitry performed surprisingly well for the two degrees-of-
freedom wrist joint and the next obvious step is to extend the circuitry to control the
movements of the entire arm. The common experimental configuration of the arm
model is to use separate coplanar hinge-like elbow and shoulder joints in tasks
involving planar arm motion. Each joint is operated by a pair of antagonist muscles
that provide flexion and extension torque. The model must also include two
biarticular muscles, one providing flexion and the other extension torques across both
joints. A similar model configuration was used by Brown and Loeb (2000c) to
identify the importance of preflexes in highly dynamic tasks.
Construction of the spinal circuitry model will involve two phases. The first
phase will be the identification of antagonist or synergist muscle relationships among
all muscle pair combinations in the set. The second phase is building a spinal circuitry
network that embodies these relationships as well as determining the number of
parameters that need to be optimized to produce specific behaviors. The second phase
174

will involve building the network of synergist and antagonist circuits of the five
classical pathways: propriospinal, Ia-inhibitory, Ib-inhibitory, Renshaw and stretch
reflex. In complex biomechanical models such as the digits of the hand, the muscle
synergies cannot be derived clearly; in such cases we suggest classifying the muscles
as “partial synergists”. This will require modeling both the synergist and antagonist
circuitry for those muscles, but at the cost of doubling the control parameters.
The groundwork for this project has already been completed (Tsianos et al.,
2009). Active torque analysis throughout reaching movements (Graham et al., 2003)
and EMG analysis (Karst and Hasan 1991) were used to gain insights into the
relationships between monoarticular muscles crossing different joints and the two
biarticular muscles in the model. The projected parameter distribution and
representative synergistic and antagonistic circuits of the Renshaw pathway are shown
in the figures below.

Figure 7

Fig
7.1: Represent
gure 7.2: Para

tative Rensha
(T
ameter distrib
(T
aw pathway in
Tsianos et al.,

bution in the s
Tsianos et al.,
n the spinal ci
2009)
spinal circuitr
2009)

ircuitry for th
ry for the arm

he arm model
m model
175

176

c. Clinical implications
A better understanding of the normal role of the spinal cord seems likely to
improve the diagnosis and treatment of many disorders of sensorimotor function,
including paralysis and spasticity. For example, neural prosthetic modulation of
function is likely to benefit from biomimetic strategies for control, particularly when
those prostheses involve reanimation of paralyzed musculoskeletal systems by
electrical stimulation of their motoneurons. Wireless multi-channel stimulators such
as the BION can be injected into muscles and near nerves to deliver pulsed neural
stimulation (Loeb et al., 2000). These novel interfaces can apply such stimulation
reliably and efficiently to large numbers of muscles (Loeb et al., 2006) and can also be
used to record posture and movement data similar to that provided by biological
proprioceptors (Tan and Loeb, 2007; Sachs and Loeb, 2007). Other researchers are
working to record command signals directly from the cerebral cortex of patients
suffering from paralysis and amputations (Hochberg et al., 2006).
In separate research, we developed a real time portable platform for running
spinal-like regulators (Figure 7.3). The platform had interfaces to control many
injectable BION stimulators that can provide Functional Electrical Stimulation (FES)
to the muscles of the paralyzed limb. The platform had dual core architecture with a
32-bit ARM processor and an FPGA and it used a real-time operating system to run
complex multi-threaded applications such as the spinal cord circuitry models.

177

Figure 7.3: Block diagram of the BION controller.

178

Thus, all of the major input-output signals of the spinal cord are now amenable
to replacement by neural prosthetic interfaces. Prosthetic alteration or replacement of
the functionality of the spinal cord itself depends on a theory of computation for
functionality that could be embedded in the computer algorithms or silicon circuitry of
such prostheses. This thesis seeks to provide such a theory of computation.

179

BIBLIOGRAPHY
Abbas J.J. Neural networks for control of posture and locomotion, Proceedings of the
American Control Conference. 1997.
Abraham L.D, Marks W.B, Loeb G.E. The distal hindlimb musculature of the cat:
Cutaneous reflexes during locomotion. Exp Brain Res, 58:594-603, 1985
Ajemian R, Green A, Bullock D, Sergio L, Kalaska J, Grossberg S. Assessing the
function of motor contex: Simgle-neuron models of how neural response is modulated
by limb biomechanics. Neuron, 58:414-428, 2008.
Alstermark B, Lundberg A, Norrsell U, Sybirska E. Integration in descending
motor pathways controlling the forelimb in the cat .9. Differential behavioral defects
after spinal cord lesions interrupting defined pathways from higher centers to
motoneurons. Experimental Brain Research, 42:299-318, 1981b.
Alstermark, B, Sasaki, S. Integration in descending motor pathways controlling the
forelimb in the cat. 13. Corticospinal effects in shoulder, elbow, wrist, and the digit
motoneurons. Exp Brain Res, 59:353-364, 1985.
Alstermark B, Lundberg A. The C3-C4 propriospinal system: target-reaching and
food-taking. Muscle Afferents and Spinal Control of Movement, Oxford: Pergamon
Press, 327-354, 1992.
Alstermark B, Isa T, Pettersson L.G, Sasaki S. The C3-C4 propriospinal system in
the cat and monkey: a spinal pre-motoneuronal centre for voluntary motor control.
Acta Physiol, 189:123-140, 2007.
Araki T, Eccles J.C, Ito M. Correlation of their inhibitory post-synaptic potential of
motoneurones with the latency and time course of inhibition of monosynaptic reflexes.
Journal of Physiology, 154: 354-377, 1960.
Ashby P, Labelle K. Effects of extensor and flexor group I afferent volleys on the
excitability of individual soleus motoneurones in man. Journal of Neurology,
Neurosurgery and Psychiatry, 40:910-919, 1977.
180

Aymard C, Chia L, Katz R, Lafitte C, Penicaud A. Reciprocal inhibition between
wrist flexors and extensors in man: a new set of interneurons? Journal of Physiology,
487.1:221-235, 1995.
Aymard C, Decchi B, Katz R, Lafitte C, Penicaud A, Raoul S, Rossi A. Recurrent
inhibition between motor nuclei innervating opposing wrist muscles in the human
upper limb. Journal of Physiology, 499.1:267-282, 1997.
Bashor D.P. A large-scale model of some spinal reflex circuits. Biological
Cybernetics, 78(2):147-157, 1998.
Bent L.R, Bolton P.S, Macefield V.G. Vestibular inputs do not influence the
fusimotor system in relaxed muscles of the human leg. Exp Brain Res, 180:97-103,
2007.
Bizzi E, Polit A, Morasso P. Mechanisms underlying achievement of final head
position. J Neurophysiol, 39:435-444, 1976.
Brink E, Jankowska E, McCrea D, Skoog B. Inhibitory interactions between
interneurones in reflex pathways from group 1a and group Ib afferents in the cat.
Journal of Physiology, 343:361-379, 1983.
Brown I.E, Scott S.H, Loeb G.E. “Preflexes”-programmable, high-gain, zero-delay
intrinsic responses of perturbed musculoskeletal systems. Sco Neuros Abstr, 21:1433,
1995.
Brown I.E, Linama T.L, Loeb G.E. Relationships between range of motion and
passive force in five strap-like muscles of the feline hind limb. Journal of
Morphology, 230:69-77, 1996
Brown I.E, Scott S.H, Loeb G.E. Mechanics of feline soleus: II. Design and
validation of a mathematical model. Journal Muscle Research and Cell Motility,
17:221-233, 1996.
Brown I.E, Loeb G.E Post-activation potentiation – a clue for simplifying models of
muscle dynamics. Amer Zool, 38:743-754, 1998.
181

Brown I.E, Kim D.H, Loeb G.E. The effect of sarcomere length on triad location in
intact feline caudofemoralis muscle fibers. Journal Muscle Res Cell Motility, 19:473-
477, 1998.
Brown I.E, Cheng E.J, Loeb G.E. Measured and modeled properties of mammalian
skeletal muscle: II. The effects of stimulus frequency on force-length and force-
velocity relationships. Journal Muscle Res Cell Motility, 20:627-643, 1999.
Brown I.E, Loeb G.E. Measured and modeled properties of mammalian skeletal
muscle: III. The effects of stimulus frequency on stretch-induced force enhancement
and shortening-induced force depression. Journal Muscle Res Cell Motility, 21:21-31,
2000a.
Brown I.E, Loeb G.E. Measured and modeled properties of mammalian skeletal
muscle: IV. The dynamics of activation and deactivation. Journal Muscle Res Cell
Motility, 21:33-47, 2000b.
Brown I.E, Loeb G.E. A reductionist approach to creating and using
neuromusculoskeletal models. Chapter in: Biomechanics and Neuro-Control of
Posture and Movement, (Eds.) J. Winters and P. Crago, Springer-Verlag New York,
148-163, 2000c.
Bullock D, Grossberg S. Emergence of tri-phasic muscle activation from the
nonlinear interactions of central and spinal neural network circuits. Human Movement
Science ii, 157-167, 1992.
Bullock D, Grossberg S, Guenther F.H. A self-organizing neural model of motor
equivalent reaching and tool use by a multijoint arm. Journal of Cognitive
Neuroscience, 5:408-435, 1993.
Bullock D, Contreras-Vidal JL, Grossberg S. Equilibria and dynamics of a neural
network model for opponent muscle control. In G. Bekey and K. Goldberg (Eds.),
Neural Networks in Robotics, Norwell, MA: Kluwer Academic, 439-457, 1993.
Burke D, Gracies J.M, Mazevet D, Meunier S, Pierrot-Deseilligny E. Convergence
of descending and various peripheral inputs onto common propriospinal-like neurons
in man. Journal of Physiology, 449:655-671, 1992a.
182

Burke R.E. Sir Charles Sherrington’s The integrative action of the nervous system: a
centenary appreciation. Brain, 130:887-894, 2007.
Burkitt A.N, Meffin H, Grayden D.B. Study of neuronal gain in a conductance-
based leaky integrate-and-fire neuron model with balanced excitatory and inhibitory
synaptic input. Biological Cybernetics, 89:119-125, 2003.
Buys E.J, Lemon R.N, Mantel G.W.H, Muir R.B. Selective facilitation of different
hand muscles by single corticospinal neurons in the conscious monkey. Journal of
Physiology (London), 381:529-549, 1989.
Carroll T.J, Baldwin E.R.L, Collins D.F. Task dependent gain regulation of spinal
circuits projecting to the human flexor carpi radialis. Experimental Brain Research,
161:299-306, 2005.
Cavallari P, Fournier E, Katz R, Pierrot-Deseillingny E, Shindo M. Changes in
reciprocal Ia inhibition from wrist extensors to wrist flexors during voluntary
movement in man. Experimental Brain Research, 56:574-576, 1984.
Cavallari P, Katz R, Penicaud A. Pattern of projections of group-1 afferents from
elbow muscles to motoneurons supplying wrist muscles in man. Experimental Brain
Research, 91:311-319, 1992.
Chalmers G.R, Bawa P. Synaptic connections from large afferents of wrist flexor and
extensor muscles to synergistic motoneurons in man. Experimental Brain Research,
116:351-358, 1997.
Chanaud C.M, Pratt C.A, Loeb G.E. Functionally complex muscles of the cat
hindlimb. II. Mechanical and architectural heterogeneity within a parallel-fibered
muscle. Experimental Brain Research, 85:257-270, 1991.
Chan S.S, Moran D.W. Computational model of a primate arm: from hand position
to joint angles, joint torques and muscle forces. Journal of Neural Eng, 3:327-337,
2006.
Cheng E.J. Brown I.E, Loeb G.E. Virtual Muscle: A computational approach to
understanding the effects of muscle properties on motor control. Journal of
Neuroscience Method, 101: 117-130, 2000.
183

Chou C, Hannaford B. Study of human forearm posture maintenance with a
physiologically based robotic arm and spinal level neural controller. Biological
Cybernetics, 76:285-298, 1997.
Chrstodoulou C, Bugmann G, Clarkson T.G. A spiking neuron model: applications
and learning. Neural Networks, 2002.
Cisek P, Grossberg S, Bullock D. A Cortico-spinal model of reaching and
proprioception under multiple task constraints. Journal of Cognitive Neuroscience, 10-
4:425-444, 1998.
Cisi R.R.L, Kohn A.F. Simulation system of spinal cord motor nuclei and associated
nerves and muscles, in a web-based architecture. Journal Computational
Neuroscience, 2008.
Corneil B.D, Olivier E, Richmond F.J.R, Loeb G.E, Munoz D.P. Neck muscles in
the rhesus monkey. II. Electromyographic patterns of activation underlying postures
and movements. Journal of Neurophysiology, 86:1729-1749, 2001.
Crago P.E, Houk J.C, Rymer W.Z. Sampling of total muscle force by tendon
organs. Journal of Neurophysiology, Vol.47, No.6, 1982.
Criscimagna-Hemminger S.E, Donchin O, Gazzaniga M.S, Shadmehr R. Learned
dynamics of reaching movements generalize from dominant to nondominant arm. J of
Neurophysiology, 89(1):168-76, 2003.
Crone C, Hultborn H, Jespersen B. Reciprocal Ia inhibition between ankle flexors
and extensors in man. Journal of Physiology, 389:163-185, 1987.
Cullheim S, Kellerth J.P. A morphological study of the axons and recurrent axon
collaterals of cat alpha-motoneurones supplying different hind-limb muscles. Journal
of Physiology, 281:285-299, 1978.
Davoodi R, Kleiman D, Murakata T, Loeb G.E. A Biomechanical Model of the
Elbow, Forearm and Wrist. IV World Conference of Biomechanics, 2002.
Davoodi R, Loeb G.E. A Biomimetic strategy for control of FES reaching. Proc 25rd
Annual International Conference of the IEEE Engineering in Medicine and Biology
Society, 17-21, 2003.
184

Davoodi R, Brown I.E, Loeb G.E. Advanced modeling environment for developing
and testing FES control systems. Medical Engineering & Physics, 25:3-9, 2003.
Davoodi R, Urata C, Hauschild M, Khachani M, Loeb G.E. Model-based
development of neural prostheses for movement. IEEE Trans Biomedical Engineering,
54(11):1909-1918, 2007.
Day B.L, Rothwell J.C, Marsden C.D. Transmission in the spinal ia reciprocal
inhibitory pathway preceding willed movements of the human wrist. Neuroscience,
37: 245-250, 1983.
Day B.L, Marsden C.D, Obeso J.A, Rothwell J.C. Reciprocal inhibition between
the muscles of the human forearm. Journal of Physiology, 349:519-534, 1984.
DeLuca C.J, Erim Z. Common drive in motor units of a synergistic muscle pair.
Journal Neurophysiology, 87:2200-2204, 2002.
Diedrichsen J, Hashambhoy Y, Rane T, Shadmehr R. Neural correlates of reach
errors. Journal of Neuroscience, 25(43):9919-9931, 2005.
Duenas S.H, Loeb G.E, Marks W.B. Monosynaptic and dorsal root reflexes during
locomotion in normal and thalamic cats. Journal of Neurophysiology, 63:1467-1476,
1990.
Duysens J, Loeb G.E. Modulation of ipsi- and contralateral reflex responses in
unrestrained walking cats. Journal of Neurophysiology, 44:1024-1037, 1980.
Eccles J.C, Fatt P, Koketsu K. Cholinergic and inhibitory synapses in a pathway
from motor-axon collaterals to motoneurones. Journal of Physiology, 126:524-562,
1954.
Eccles J.C, Fatt P, Landgren S. The central pathway for the direct inhibitory action
of impulses in the largest afferent nerve fibers to muscle. Journal of Neurophysiology,
19: 75-98, 1956.
Eccles J.C, Eccles R.M, Lundberg A. Synaptic actions of motoneurones caused by
impulses in Golgi tendon organ afferents. Journal of Physiology, 138:227-252, 1957.
185

Eccles R.M, Lundberg A. Integrative pattern of Ia synaptic actions of motoneurons
of hip and knee muscles. Journal of Physiology, 144: 271-298, 1958.
Eccles R.M, Lundberg A. Supraspinal control of interneurons mediating spinal
reflexes. Journal of Physiology, 147:565-584, 1959.
Eccles J.C, Eccles R.M, Iggo A, Lundberg A. Electrophysiological investigations of
renshaw cells. Journal of Physiology, 159:461-478, 1961b
Edgley S.A. Organization of inputs to spinal interneuron populations. Journal of
Physiology, 533.1:51-56, 2001.
Evarts E.V, Shinoda Y, Wise S.P. Neurophysiological approaches to higher brain
functions. John Wiley, 198, 1984.
Fagg A.H, Shah A, Barto A.G. A computational model of muscle recruitment for
wrist movement. Journal of Neurophysiology, 88:3348-3358, 2002.
Fedina L, Hultborn H. Facilitation from ipsilateral primary afferents of interneuronal
transmission in the Ia inhibitory pathway to motoneurones. Acta Physiologica
Scandinavica, 94:198-221, 1972.
Feldman A.G. Functional tuning of the nervous system with control of movement or
maintenance of a steady posture. II. Controllable parameters of the muscles. Biophys
Journal, 11:766-775, 1966.
Feldman A.G. Superposition of motor programs. II. Rapid forearm flexion in man.
Neuroscience, 5:91-95, 1980.
Feldman A.G. Once more on the equilibrium-point hypothesis (lambda-model) for
motor control. Journal Motor Behavior, 18:17-54, 1986.
Feldman A.G, Ostry D.J, Levin M.F, Gribble P.L, Mitnitski A.B. Recent tests of
the equilibrium-point hypothesis (lambda model). Motor Control, 2:26-42, 1998.
Fetz E.E, Finnochio D.V. Correlations between activity of motor cortical cells and
arm muscles during operantly conditioned response patterns. Experimental Brain
Research, 23:217-240, 1975.
186

Fetz E.E, Cheney P.D. Postspike facilitation of forelimb muscle activity by
populations of corticomotoneuronal and rubromotoneuronal cells. Journal of
Neurophysiology, 44:751, 1980.
Fetz E.E, Cheney P.D, Mewes K, Palmer S. Control of forelimb muscle activity by
populations of corticomotoneuronal and rubromotoneuronal cells. Prog Brain Res,
80:437-449, 1989.
Fetz E.E, Shupe L.E, Murthy V.N. Neural networks controlling wrist movements.
Neural Networks,Vol 2, 675-679, 1990.
Fetz E.E. Are movement parameters recognizably coded in the activity of single
neurons? Behavioral and Brain Sciences, 15:679-690, 1992.
Fetz E.E, Perimutter S.I, Maier M.A, Flament D, Fortier P.A. Response patterns
and postspike effects of premotor neurons in cervical spinal cord of behaving
monkeys. Can J Physiol Pharmacol, 74:531-546, 1996.
Fetz E.E, Perlmutter S.I, Prut Y, Seki K, Votaw S. Roles of primate spinal
interneurons in preparation and execution of voluntary hand movement. Brain
Research Reviews, 40:53-65, 2002.
Flanagan J.R, Nakano E, Imamizu H, Osu R, Yoshioka T, Kawato M.
Composition an ddecomposition of internal models in motor learning under altered
kinematic and dynamic environments. Journal of Neuroscience, Vol.1, 1999.
Fleshman J.W, Segev I, Burke R.E. Electronic architecture of type-identified alpha-
motoneurons in the cat spinal cord. Journal of Neurophysiology, 60:60-85, 1988.
Fromm C, Haase J, Wolf E. Depression of recurrent inhibition of extensor
motoneurones by the action of group II afferents. Brain Research, 120:351-354, 1977.
Garner B.A, Pandy M.G. The obstacle-set method for representing muscle paths in
musculoskeletal models. Computer methods in biomechanics and biomedical
engineering, Vol.3, 1-30, 2000.
Garner B.A, Pandy M.G. Musculoskeletal model of the upper limb based on the
visible human male dataset. Computer methods in biomechanics and biomedical
engineering, Vol.4, 93-126, 2001.
187

Gelfan, S. Neuronal interdependence. Progress in Brain Research, 11:238-60, 1964.
Georgopoulos A.P, Schwartz A.B, Kettner R.E. Neuronal population coding of
movement direction. Science, 233:1416-1419, 1986.
Ghafouri M, Feldman A.G. The timing of control signals underlying fast point-to-
point arm movements. Exp Brain Res, 137:411-423, 2001.
Ghez C, Martin J.H. The control of rapid limb movement in the cat III. Agonist –
Antagonist coupling. Experimental Brain Research, 45:115-125, 1982.
Ghez C, Gordon J. Trajectory control in targeted force impulses I. Role of opposing
muscles. Experimental Brain Research, 67:225-240, 1987.
Gillard D,M. Yakovenko S, Cameron T, Prochazka A. Torque-Angle Relationships
of Human Wrist Muscles. Proceedings of the First Joint BMES/EMBS Conference,
1999.
Gonzalez R.V, Buchanan T.S, Delp S.L. How muscle architecture and moment arms
affect wrist flexion-extension moments. Journal of Biomechanics,Vol.30, No.7, 705-
712, 1997.
Gossard J.P, Brownstone R.M, Barjon I, Hultborn H. Transmission in a
locomotor-related group Ib pathway from hindlimb extensor muscles in the cat.
Experimental Brain Research, 98:213-228, 1994.
Gracies J.M, Meunier S, Pierrot-Deseilligny E. Pattern of propriospinal-like
excitation to different species of human upper limb motoneurons. Journal of
Physiology, 434:151-167, 1991.
Graham K.M, Moore K.D, Cabel D.W, Gribble P.L, Cisek P, Scott S.H.
Kinematics and kinetics of multijoint reaching in nonhuman primates. Journal of
Neurophysiology, 89:2667-2677, 2003.
Granit R, Burke R. The control of movement and posture. Conference report- Brain
Res, 53:1–28,1973.
188

Gribble P.L, Ostry D.J, Sanguineti V, Laboissiere R. Are complex control signals
required for human arm movement. Journal of Neurophysiology, 79:1409-1424,
1998.
Hagbarth K.E, Hagglund J.V, Wallin E.U, Young R.R. Grouped spindle and
electromyographic responses to abrupt wrist extension movements in man. Journal of
Physiology, 312:81-96, 1981.
Harrison P.J, Jankowska E. Source of input to interneurones mediating group 1 non-
reciprocal inhibition of motoneurones in the cat. Journal of Physiology, 361:379-401,
1985.
Hasan Z, Houk J.C. Transition in sensitivity of spindle receptors that occurs when
muscle is stretched more than a fraction of a millimeter. J of Neurophysiology,
38(3):663-672, 1975.
He J, Levine W.S, Loeb G.E. Feedback gains for correcting small perturbations to
standing posture. IEEE Transactions on Automatic Control,Vol 36, No.3, 0300-0322,
1991.
He J.P, Weber D.J. Cortical control of arm movement. Proceedings of ACCC, 2323-
2328, 2002.
Heckman C.J, Weytjens J.L.F, Loeb G.E. Effect of velocity and mechanical history
on the forces of motor units in the cat medial gastrocnemius muscle. Journal of
Neurophysiology, 68:1503-1515, 1992.
Herrmann A.M, Delp S.L. Moment Arm and Force-Generating Capacity of the
Extensor Carpi Ulnaris After Transfer to the Extensor Carpi Radialis Brevis. Journal
of Hand Surgery, Vol.24A, No.5, 1996.
Hochberg L.R, Serruya M.F, Friehs G.M, Mukand J.A, Saleh M, Caplan A.H,
Branner A, chen D, Penn R.D, Donoghue J.P. Neuronal ensemble control of
prosthetic devices by a human with tetraplegia. Nature, 442(7099):164-171, 2006.
Hoffer, J. A. Central control and reflex regulation of mechanical impedance: the basis
for a unified motor control scheme. Behavioral and Brain Sciences, 1990.
189

Hoffman D.S, Strick P.L. Step-tracking movements of the wrist in humans.-1,
Kinematics Analysis. The Journal of Neuroscience,Vol.6, 3309-3318, 1986.
Hoffman D.S, Strick P.L. Step-tracking movements of the wrist in humans.-II, EMG
Analysis. The Journal of Neuroscience,10:142-152, 1990.
Hoffman D.S, Strick P.L. Step-tracking movements of the wrist in humans.-1V,
Muscle activity associated with movements in different directions. The Journal of
Neurophysiology, 81:319-333, 1999.
Hogan N. An organizing principle for a class of voluntary movements. Journal of
Neuroscience, Vol.4, No.11, 2745-2754, 1984.
Hogan N. The mechanics of multi-joint posture and movement control. Journal of
Neuroscience, 1991.
Hore J, McCloskey D.I, Taylor J.L. Task-dependent changes in gain of the reflex
response to imperceptible perturbations of joint position in man. J of Physiology,
429:309-321, 1990.
Hulliger M, Prochazka A. A new simulation method to deduce fusimotor activity
from afferent discharge recorded in freely moving cats. J Neurosci Methods, 8:197–
204, 1983.
Hultborn H, Jankowska E, Lindstorm S. Recurrent inhibition from motor axon
collaterals of transmission in the Ia inhibitory pathway to motoneurones. Journal of
Physiology, 215:591-612, 1971a.
Hultborn H, Jankowska E, Lindstrom S. Recurrent inhibition of interneurones
monsynaptically activated from group Ia afferents. Journal of Physiology, 215:613-
636, 1971b.
Hultborn H, Jankowska E, Lindstorm S. Relative contribution from different
nerves to recurrent depression of Ia IPSPs in motoneurones. Journal of Physiology,
215:637-664, 1971b.
Hultborn H, Illert M, Santini M. Convergence on interneurones mediating the
reciprocal Ia inhibition of motoneurons I. Disynaptic Ia inhibition of Ia inhibitory
interneurones. Acta Physiologica Scandinavica, 96:193-201, 1976a.
190

Hultborn H, Pierrot-Deseilligny E, Wigstrom H. Recurrent inhibition and after-
hyperpolarization following motoneuronal discharge in the cat. Journal of Physiology,
297:253-266, 1979b.
Hultborn H, Conway B.A, Gossard J.P. How do we approach the locomotor
network in the mammalian spinal cord. Annals of New York academy of sciences,
860:70-82, 1998.
Humphrey D.R, Reed D.J. Separate cortical systems for control of joint movement
and joint stiffness: reciprocal activation and coactivation of antagonist muscles. Motor
Control Mechanisms in Health and Disease. Desmedt JE (Ed) Raven Press, New York,
347-372, 1983.
Huruno M, Wolpert D.M. Optimal control of redundant muscles in step-tracking
wrist movements. Journal of Neurophysiology, 94:4244-4255, 2005.
Hwang E.J, Shadmehr R. Internal models of limb dynamics and the encoding of
limb state. Journal of Neural Engineering, 266-278, 2005.
Illert M, Lundberg A, Tanaka R. Integration in descending motor pathways
controlling the forelimb in the cat. 3. Convergence on propriospinal neurons
transmitting disynaptic excitation from the corticospinal tract and other descending
tracts. Experimental Brain Research, 29:323-346, 1977.
Illert M, Lundberg A, Padel Y, Tanaka R. Integration in descending motor
pathways controlling the forelimb in the cat. 5. Properties of and monosynaptic
excitatory convergence on C3-C4 propriospinal neurons. Experimental Brain
Research, 33:101-130, 1978.
Illert M, Wiedemann E. Pyramidal actions in identified radial motor nuclei of the
cat. Pflugers Arch, 41:132-142, 1984.
Isa T, Ohki Y, Alstermark B, Pettersson l.G, Sasaki S. Direct and indirect cortico-
motoneuronal pathways and control of hand/arm movements. Physiology 22:145-152,
2007.
Ivashko, D.G, Prilutsky B. I, Markin S. N, Chapin J. K, Rybak, I. A. Modeling the
spinal cord neural circuitry controlling cat hindlimb movement during locomotion.
Neurocomputing, 52(54): 621-629, 2003.
191

Jankowska E, Lundberg A. Interneurones in the spinal cord. Trends in
Neuroscience, 4:230-233, 1981.
Jankowska E. Interneuronal relay in spinal pathways from proprioceptors. Progress
in Neurobiology, 38:335-378, 1992.
Jones K.E, Antonia F, Hamilton C, Wolpert D.M. Sources of signal-dependent
noise during isometric force production. Journal of Neurophysiology, 88:1533-1544,
2002.
Kandell E.R, Schwartz J.H, Jessell T.M. Principles of neural science. Edition-4,
McGrawHill, 1991.
Karniel A, Mussa-Ivaldi F.A. Does the motor control system use multiple models
and context switching to cope with a variable environment. Experimental Brain
Research, 143:520-524, 2002.
Karst G.M, Hasan Z. Timing and magnitude of electromyographic activity for two-
joint arm movements in different directions. Journal of Neurophysiology, 66:1594-
1604, 1991.
Kawato M. Internal models for motor control and trajectory planning. Curr Opin
Neurobiol, 9(6):718-727, 1999.
Kitazawa S, Ohki Y, Sasaki M, Xi M.C, Hongo T. Candidate premotor neurons of
skin reflex pathway to T1 forelimb motoneurons of the cat. Exp. Brain Res, 95:291-
307, 1993.
Kluzik J, Diedrichsen J, Shadmehr R, Bastian A.J. Reach adaptation: What
determines whether we learn an internal model of the tool or adapt the model of our
arm? Journal of Neurophysiology, 100:1455-1464, 2008.
Knight W.B. Dynamics of encoding in a population of neurons. The Journal of
General Physiology, Vol.59, 734-766, 1972.
Koshland G.F, Gerilovsky L, Hasan Z. Activity of wrist muscles elicited during
imposed or voluntary movements about the elbow joint. Journal of Motor Behavior,
1990.
192

Koshland G.F, Galloway J.C, Nevoret-Bell C.J. Control of the wrist in three-joint
arm movements to multiple directions in the horizontal plane. The Journal of
Neurophysiology,Vol.83, No.5, 3188-3195, 2000.
Krieger D.N, Berger T.W, Sclabassi R.J. Instantaneous characterization of time-
varying nonlinear systems. IEEE Transactions in Biomedical Engineering, 39:420-
424, 1992.
Lan N, Crago P.E. Feedback control methods for task regulation by electrical
stimulation of muscles. IEEE Transactions on Biomedical Engineering, Vol.38,
No.12, 1991.
Lan N, Feng H.Q, Crago P.E. Neural network generation of muscle stimulation
patterns for control of arm movements. IEEE Trans on Rehab Eng, Vol2, No4, 213-
223, 1994.
Lan N, Crago P.E. Optimal control of antagonistic muscle stiffness during voluntary
movements. Biological Cybernetics, 71:123-135, 1994.
Lan N. Analysis of an optimal control model of multi-joint arm movements.
Biological Cybernetics, 76:107-117, 1997.
Lan N, Song D, Mileusnic M, Gordon J. Modeling spinal sensorimotor control for
reach task. Proceedings of the IEEE Engineering in Medicine and Biology, 2005.
Lan N, Li Y, Sun Y, Yang F.S. Reflex regulation of antagonist muscles for control of
joint equilibrium position. IEEE Transactions on Neural Systems and Rehabilitation
Engineering, Vol.13, No.1, 2005.
Latash M.L, Gottlieb G.L. An equilibrium-point model for fast single-joint
movement. I. Emergence of strategy-dependent EMG patterns. Journal of Motor
Behavior, 23:163-178, 1991
Latash M.L, Gottlieb G.L. An equilibrium-point model for fast single-joint
movement. II. Similarity of single-joint isometric and isotonic desending commands.
Journal of Motor Behavior, 23:179-191, 1991.
Latash M.L. Virtual trajectories of single-joint movements performed under two
basic strategies. Neurscience, 47: 357-365, 1992.
193

Lavoie S, Mcfadyen B, Drew T. A kinematic and kinetic analysis of locomotion
during voluntary gait modification in the cat. Exp Brain Res, 106:39-56.
Lemay M.A, Crago P.E. A dynamic model for simulating movements of the elbow,
forearm, and wrist. Journal of Biomechanics, Vol.29,No.10, 1319-1330, 1996.
Lemon R.N, Kirkwood P.A, Maier M.A, Nakajima K, Nathan P. Direct and
indirect pathways for conticospinal control of upper limb motoneurons in the primate.
Prog Brain Res, 143:263-279, 2004.
Lemon R.N, Griffiths J. Comparing the function of the corticospinal system in
different species: Organizational differences for motor specialization. Muscle &
Nerve, 32:261-279, 2005.
Levin M.F, Feldman A.G, Milner T.E, Lamarre Y. Reciprocal and coactivation
commands for fast wrist movements. Experimental Brain Research, 89:669-677, 1992.
Liles S.L. Activity of neurons in putamen during active and passive movements of
wrist. J. of Neurophysiology, Vol.53, No.1, 1985.
Lloyd D.P.C. Integrative pattern of excitation and inhibition in two neuron reflex arcs.
Journal of Neurophysiology, 9:409-426, 1946.
Loeb G.E, Bak M.J, Duysens J. Long-term unit recording from somatosensory
neurons in the spinal ganglia of the freely walking cat. Science, 197:1192-1194, 1977.
Loeb G.E, Duysens J. Activity patterns in individual hindlimb primary and secondary
muscle spindle afferents during normal movements in unrestrained cats. Journal of
Neurophysiology, 42:420-440, 1979.
Loeb G.E. The control of responses of mammalian spindles during normally executed
motor tasks. Exercise and Sports Sciences Reviews, 12:157-204, 1984.
Loeb G.E. Motoneurone task groups: coping with kinematic heterogeneity. J. of Exp
Biol, 115:137-146, 1985.
Loeb G.E, Gans C. Electromyography for Experimentalists. Chicago. IL: The
University of Chicago Press, 1986.
194

Loeb G.E, Marks W.B, Hoffer J.A. Cat hindlimb motoneurons during locomotion:
IV. Participation in cutaneous reflexes. Journal of Neurophysiology, 57:563-573,
1987.
Loeb G.E, He J, Levine W.S. Spinal cord circuits: are they mirrors of
musculoskeletal mechanics? Journal of Motor Behavior, 21:473-491, 1989.
Loeb G.E , Levine W.S, He J. Understanding sensorimotor feedback through optimal
control. Cold Spring Harbor Symposia on Quantitative Biology, Vol LV, 1990.
Loeb G.E. Past the equilibrium point. Behavioral and Brain Sciences, 15:774-775,
1992.
Loeb G.E. The distal hindlimb musculature of the cat: Interanimal variability of
locomotor activity and cutaneous reflexes. Experimental Brain Research, 96:125-140,
1993.
Loeb G.E, Richmond F.J.R. Architectural features of multiarticular muscles. Hum.
Mov. Sci, 13:545-556, 1994.
Loeb G.E, Brown I.E, Cheng E.J. A hierarchical foundation for models of
sensorimotor control. Experimental Brain Research, 126:1-18, 1999.
Loeb, G.E, Richmond F.J.R. BION Implants for Therapeutic and Functional
Electrical Stimulation”, chapter in “Neural Prosthesis for Restoration of Sensory and
Motor Function. CRC Press, Boca Raton, 75-99, 2000.
Loeb G.E. Learning from the spinal cord. J. of Physiology, 533.1:111-117, 2001.
Loeb G.E, Richmond F.J.R, Baker L.L. The BION devices: Injectable interfaces
with peripheral nerves and muscles. Neurosurgical Focus, 20:1-9, May, 2006.
Loren G.J, Shoemaker S.D, Burkholder T.J, Jacobson M.D, Friden J, Lieber
R.L. Human wrist motors: biomechanical design and application to tendon transfers.
Journal of Biomechanics,Vol.29, No.3, 331-342, 1995.
Lundberg A. The excitatory control of the Ia inhibitory pathway. Excitatory synaptic
mechanisms, 333-340, 1970.
195

Lundberg A. To what extent are brain commands for movements mediated by spinal
interneurons? Behavioral and Brain Sciences,15:775, 1992.
Maier M.A, Perlmutter S.I, Fetz E.E. Response patterns and force relations of
monkey spinal interneurons during active wrist movement. The Journal of
Neurophysiology, Vol.80, No.5, 2495-2513, 1998.
Maier M.A, Kirkwood P.A, Nielsen J, Lemon R.N. Does a C3-C4 propriospinal
system transmit corticospinal excitation in the primate? An investigation in the
macaque monkey. Journal of Physiology (London), 511:191-212, 1998.
Maier M.A, Shupe L.E, Fetz E.E. Dynamic neural network models of the
premotoneuronal circuitry controlling wrist movements in primates. Journal of
Computational Neuroscience, 19:125-146, 2005.
Malmgren K, Pierrot-Deseilligny E. Evidence for non-monosynaptic Ia excitation of
wrist flexor motoneurons, possibly via propriospinal neurons. Journal of Physiology,
405:747-764, 1988a.
Matthews P.B.C. Mammalian muscle spindles and their central action.
London:Arnold ,630, 1972.
McCrea D.A. Spinal cord circuitry and motor reflexes. Exerc Sport Sci, 14:105-141,
1986.
McCrea D.A. Neuronal basis of afferent-evoked enhancement of locomotor activity.
Annals of New York Academy of Sciences, 802:216-225, 1998.
McCrea D.A, Rybak I.A. Organization of mammalian locomotor rhythm and pattern
generation. Brain Res, 57:134-146, 2008.
McCulloch W, Pitts W. A logical calculus of the ideas immanent in nervous activity.
Bulletin of Mathematical Biophysics, 7:115-133, 1943.
Meunier S, Pierrot-Deseilligny E, Simonetta M. Pattern of monosynaptic
heteronymous Ia connections in the human lower limb. Experimental Brain Research,
96: 533-544, 1993.
196

Mileusnic M.P, Brown I.E, Lan N, Loeb G.E. Mathematical models of
proprioceptors. I. control and transduction in the muscle spindle. Journal of
Neurophysiology, 96:1772-1788, 2006.
Mileusnic M.P, Loeb G.E. Force estimation from ensembles of Golgi tendon
organs. J. Neural Eng, 6:1-15, 2009.
Ostry D.J, Feldman A.G. A critical evaluation of the force control hypothesis in
motor control. Experimental Brain Research, 221:275-288, 2003.
Partridge L.D, Benton L.A Muscle, the motor. In: Brooks VB (ed) Motor Control.
Handbook of Physiology, The nervous system, Sect 1, Vol II. Arm Physiol Soc,
Bethesda, 43-106, 1983.
Perlmutter S.I, Maier M.A, Fetz E.E. Activity of spinal interneurons and their
effects on forearm muscles during voluntary wrist movements in the monkey. The
Journal of Neurophysiology, Vol.80, No.5, 2475-2494, 1998.
Pierrot-Deseilligny E, Morin C, Bergego C, Tankov N. Pattern of group 1 fibre
projections from ankle flexor and extensor muscles in man. Experimental Brain
Research, 42:337-350, 1981.
Pierrot-Deseillingy E. Transmission of the cortical command for human voluntary
movement through cervical propriospinal premotoneurons. Progress in Neurobiology,
48(4-5):489-517, 1996.
Pierrot-Deseillingy E. Propriospinal transmission of part of the corticospinal
excitation in humans. Muscle and Nerve, 26:155-172, 2002.
Pierrot Deseilligny E , Burke D. The circuitry of the human spinal cord, its role in
motor control and movement disorders” Cambridge Univ Press, 2005.
Polit A, Bizzi E. Processes controlling arm movements in monkeys. Sci, 201
(4362):1235-1237, 1978.
Polit A, Bizzi E. Characteristics of motor programs underlying arm movements in
monkeys. J.Neurophysiol, 42:183-194, 1979.
197

Pratt C.A, Chanaud C.M, Loeb G.E. Functionally complex muscles of the cat
hindlimb. IV. Intramuscular distribution of movement command signals and cutaneous
reflexes in broad bifunctional thigh muscles. Experimental Brain Research, 85:281-
299, 1991.
Prochazka A, Trend P.S. Instability in human forearm movements studied with feed-
back-controlled muscle vibration. The J of Physio, 402:421-442, 1988.
Prochazka A. Sensorimotor gain control: a basic strategy of motor systems? Journal
of Neurobiology, Vol.33, 281-307, 1989.
Prochazka A, Gillard D, Bennett D.J. Positive force feedback control of muscles.
Journal of Neurophysiology, Vol.77. No.6, 3226-3236, 1997a.
Prochazka A, Gillard D, Bennett D.J. Implications of positive feedback in the
control of movement. Journal of Neurophysiology, Vol.77. No.6, 3237-3251, 1997b.
Prut Y, Fetz E.E. Primate spinal interneurons show pre-movement instructed delay
activity. Nature, 401:590-594, 1999.
Rall W, Burke R.E, Holmes W.R, Jack J.J.B, Redman S.J. Matching dendritic
neuron models to experimental-data. Physiological Reviews, 72:S159-S186, 1992.
Raoul S, Aymard C, Katz R, Lafitte C, Lo E, Penicaud A. Differential changes in
Ia reciprocal inhibition at wrist level during voluntary contractions of flexor and
extensor muscles in healthy subjects and hemiplegic patients. Clinical
Neurophysiology, 110 Suppl.1, S.138, 1999.
Rathelot J.A, Strick P.L. Subdivisions of primary motor cortex based on cortico-
motoneuronal cells. Proc. Natl Acad Sci USA, 106:918-923, 2009.
Richmond F.J.R, Thomson D.B, Loeb G.E. Electromyographic studies of neck
muscles in the intact cat: I. Patterns of recruitment underlying posture and movement
during natural behaviors. Experimental Brain Research, 88:41-58, 1992.
Rindos A.J, Loeb G.E, Levitan H. Conduction velocity changes along lumbar
primary afferents in cast. Exp Neurol, 88:208-226, 1984.
198

Rudolf S, Koppen M. Stochastic hill climbing with learning by vectors of normal
distributions. First On-line Workshop on Soft Computing, 1996.
Ryall R.W, Piercey M.F. Excitation and inhibition of renshaw cells by impulses in
peripheral afferent nerve fibers. Journal of Neurophysiology, 34:242-251, 1971.
Ryall R.W. Patterns of recurrent excitation and mutual inhibition of cat renshaw cells.
Journal of Physiology, 316:439-452, 1981.
Sachs N.A, Loeb G.E. Development of a BIONic muscle spindle for prosthetic
proprioception. IEEE Transactions on Biomedical Engineering, 54(6):1031-1041,
2007.
Sasaki S, Isa T, Pettersson L.G. Dexterous finger movements in primate without
monosynaptic corticomotoneuronal excitation. Journal of Neurophysiology, 92:3142-
3147, 2004.
Schaal S. Arm and hand movement control. The handbook of brain theory and neural
networks, 2
nd
Edition, 110-113, 2002.
Scheidt R.A, Reinkensmeyer D.J, Conditt M.A, Rymer W.Z, Mussa-Ivaldi F.A.
Persistence of motor adaptation during constrained, multi-joint, arm movements.
Journal of Neurophysiology, 84:853-862, 2000.
Scheidt R.A, Dingwell J.B, Mussa-Ivaldi F.A. Learning to move amid uncertainty.
Journal of Neurophysiology, 86:971-985, 2001.
Schieber M.H, Thach W.T. Trained slow tracking-I, Muscular production of wrist
movement. Journal of Neurophysiology, Vol.54, No.5, 1985.
Scott S.H, Loeb G.E. The computation of position sense from spindles in mono-and
multiarticular muscles. Journal of. Neuroscience, 14:7529-7540, 1994.
Scott S.H, Loeb G.E. The mechanical properties of the aponeurosis and tendon of the
cat soleus muscle during whole-muscle isometric contractions. Journal of Morph,
224:73-86, 1995.
199

Scott S.H, Kalaska J.F. Changes in motor cortex activity during reaching movements
with similar hand paths but different arm postures. Journal of Neurophysiology,
73(6):2563-2567, 1995.
Scott S.H, Brown I.R, Loeb G.E. Mechanics of feline soleus: I. Effect of fascicle
length and velocity on force output. Journal of Muscle Research and Cell Motility,
17:221-233, 1996.
Scott S.H. Optimal feedback control and the neural basis of volitional motor control.
Nature Rev Neuroscience, 5(7):532-546, 2004.
Sergio L.E, Kalaska J.F. Systematic changes in motor cortex cell activity with arm
posture during directional isometric force generation. Journal of Neurophysiology, 89:
212-228, 2003.
Sergio L.E, Hamel-Paquet C, Kalaska J.F. Motor cortex neural correlates of output
kinematics and kinetics during isometric-force and arm-reaching tasks. Journal of
Neurophysiology, 94:2353-2378, 2005.
Sherrington C. The integrative action of the nervous system. New Haven CT: Yale
university press, 1906.
Sherrington C. Flexion-reflex of the limb, crossed extension-reflex, and reflex
stepping and standing. Journal of Physiology, 40:28-121, 1910.
Shimansky Y, Cai X.Y, Weber D.J, He J. Adaptation of motor cortical cell activity
to perturbations of reaching movement as revealed by chronic multi-electrode
recording in monkeys. Proc. IEEE Eng. Med. Biol, 1766-1769, 2003.
Soechting J.F, Roberts W.J. Transfer characteristics between EMG activity and
muscle tension under isometric conditions in man. Journal of Physiology, 70:779-793,
1975.
Song Dan, Raphael G, Lan N, Loeb G.E. Computationally efficient models of
neuromuscular recruitment and mechanics. Journal Neural Engineering, 5:175-184,
2008.
Stein R.B, Cody F.W.J, Capaday C. The trajectory of human wrist movements.
Journal of Neurophysiology,Vol.59, No.6, 1988.
200

Tan W, Loeb G.E. Feasibility of prosthetic posture sensing via injectable electronic
modules. IEEE Transactions on Neural Systems & Rehabilitation Engineering,
15(2):295-309, 2007.
Tanaka R. Reciprocal Ia inhibition during voluntary movements in man.
Experimental Brain Research, 2:529-540, 1974.
Taylor C.L, Schwarz R.J. The anatomy and mechanics of the human hand. Artificial
limbs, Vol.2-2:22-35, 1955.
Taylor A, Durbaba R, Ellaway P.H, Rawlinson S. Patterns of fusimotor activity
during locomotion in the decerebrate cat deduced from recordings from hindlimb
muscle spindles. J of Physiology, 522.3:515-532, 2000.
Taylor A, Ellaway P.H, Durbaba R, Rawlinson S. Distinctive patterns of static and
dynamic gamma motor activity during locomotion in the decerebrate cat. J of
Physiology, 529.3:825-836, 2000.
Tibshirani R. Regression shrinkage and selection via the Lasso. Journal of the Royal
Statistical Society Series B (Methodological), 58:267-288, 1996.
Todorov E. Direct cortical control of muscle activation in voluntary arm movements:
a model. Nature America Inc ,Vol3, 391-397, 2000.
Todorov E. Stochastic optimal control and estimation methods adapted to the noise
characteristics of the sensorimotor system. Neural Computation, 17(5), 1084-1108,
2005.
Tsianos G.A, Raphael G, Loeb G.E. Neural circuit models to study the role of the
spinal cord in control of arm movement. Neuroscience, Abstr. 2009.
Van Heijst J.J, Vos J. E. Development in a biologically inspired spinal neural
network for movement control. Neural Networks, 11:1305-1316, 1998.
Wargon I, Lamy J.C, Baret M, Ghanim, Z, Aymard C, Penicaud A, Katz R. The
disynaptic group 1 inhibition between wrist flexor and extensor muscles revisited in
humans. Experimental Brain Research, 168:203-217, 2006.
201

Westwick D.T, Pohlmeyer E.A, Solla S.A, Miller L.E, Perreault E.J. Identification
of multiple-input systems with highly coupled inputs: application to EMG prediction
from multiple intracortical electrodes. Neural Comput, Vol. 18, 329-55, 2006.
Wierzbicka M.M, Wiegner A.W, Logigian E.L, Young R.R. Abnormal most-rapid
isometric contractions in patients with Parkinson’s disease. J. Neurol. Neurosurg
Psychiatry, 54:210-216, 1991.
Wilson V.J, Talbot W.H, Kato M. Inhibitory convergence upon renshaw cells.
Journal of Neurophysiology, 27:1063-1079, 1964.
Windhorst. Activation of Renshaw Cells” Progress in Neurobiology, Vol.35:135-
179, 1990.
Wolpaw J.R, Braitman D.J, Seegal R.F. Adaptive plasticity in primate spinal stretch
reflex: initial development. Journal of Neurophysiolgy, 50:1296-1311, 1983.
Wolpaw J.R, Carp J.S. Adaptive plasticity in spinal cord. Adv Neurol, 59:163-174,
1993.
Young R.P, Scott S.H, Loeb G.E. The distal hindlimb musculature of the cat:
Multiaxis moment arms at the ankle joint. Experimental Brain Research, 96:141-151,
1993.
Zajac F.E. Muscle and Tendon: properties, modelsm scaling and application to
biomechanics and motor control. CRC Critical Reviews in Biomedical Engineering,
1988.
Zajac F.E, Gordon M.E. Determining muscle’s force and action in multi-articular
movement. Sport Science Review, 17:187-230, 1989.
Zytnicki D, Jami L. Presynaptic inhibition can act as a filter of input from tendon
organs during muscle contraction” Presynaptic Inhibition and Neural Control, Oxford
Univ Press, 303-314, 1998.

Abstract (if available)

Abstract The performance of motor tasks requires the coordinated control and continuous adjustment of myriad individual muscles. The basic commands for the successful performance of a sensorimotor task originate in “higher” centers such as the motor cortex, but the actual muscle activation and resulting torques and motion are considerably shaped by the integrative function of the spinal interneurons. The relative contributions of brain and spinal cord are less clear for reaching movements than for automatic tasks such as locomotion. We have modeled a two-axis, four-muscle wrist joint with realistic musculoskeletal mechanics and proprioceptors and a network of spinal circuitry based on the classical types of interneurons (propriospinal, monosynaptic Ia- excitatory, reciprocal Ia-inhibitory, Renshaw inhibitory and Ib-inhibitory pathways) and their supraspinal control (via biasing activity, presynaptic inhibition and fusimotor gain). The modeled system has a very large number of control inputs, not unlike the real spinal cord that the brain must learn to control to produce desired behaviors. We then programmed this model to emulate actual performance in four very different but well-described behaviors: 1) stabilizing responses to force perturbations

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

Investigating the role of muscle physiology and spinal circuitry in sensorimotor control

PDF

Model-based studies of control strategies for noisy, redundant musculoskeletal systems

PDF

A non-invasive and intuitive command source for upper limb prostheses

PDF

insideOut: Estimating joint angles in tendon-driven robots using Artificial Neural Networks and non-collocated sensors

PDF

Decoding limb movement from posterior parietal cortex in a realistic task

PDF

Neuromuscular dynamics in the context of motor redundancy

PDF

A percutaneously implantable wireless neurostimulator for treatment of stress urinary incontinence

PDF

Dynamical representation learning for multiscale brain activity

PDF

Electromyography of spinal cord injured rodents trained by neuromuscular electrical stimulation timed to robotic treadmill training

PDF

Hierarchical tactile manipulation on a haptic manipulation platform

PDF

Iterative path integral stochastic optimal control: theory and applications to motor control

PDF

Feasibility theory

PDF

Design and use of a biomimetic tactile microvibration sensor with human-like sensitivity and its application in texture discrimination using Bayesian exploration

PDF

The representation, learning, and control of dexterous motor skills in humans and humanoid robots

PDF

On the electrophysiology of multielectrode recordings of the basal ganglia and thalamus to improve DBS therapy for children with secondary dystonia

PDF

Model-based approaches to objective inference during steady-state and adaptive locomotor control

Asset Metadata

Creator Raphael, Giby (author)

Core Title Spinal-like regulator for control of multiple degree-of-freedom limbs

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Biomedical Engineering

Publication Date 07/30/2009

Defense Date 06/29/2009

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

Tag motor control,neural networks,OAI-PMH Harvest,robotics,spinal cord

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Loeb, Gerald E. (committee chair), Gordon, James (committee member), Valero-Cuevas, Francisco J. (committee member)

Creator Email giby.raphael@gmail.com,graphael@b-alert.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m2436

Unique identifier UC1495345

Identifier etd-Raphael-3151 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-575766 (legacy record id),usctheses-m2436 (legacy record id)

Legacy Identifier etd-Raphael-3151.pdf

Dmrecord 575766

Document Type Dissertation

Rights Raphael, Giby

Type texts

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

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu