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Virtual surgeries as a tool for studying motor learning

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Virtual surgeries as a tool for studying motor learning

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Virtual Surgeries as a Tool for Studying Motor Learning

by

Victor Ramon Barradas Patino

A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(Biomedical Engineering)

December 2019

ii

Acknowledgements

I am indescribably grateful to my parents, Cristina and Victor, who from an early age fostered in
me the curiosity and the thirst of knowledge that is necessary for the pursuit of science, and
supported every decision I made to achieve my goals.
I would of course like to specially thank my advisor Dr. Nicolas Schweighofer for his guidance
and support throughout the journey that was my PhD. From him I learned to think about problems
through the lens of science, which complemented my engineering-focused view, and to
communicate my ideas in a clearer and more succinct way.
I would also like to thank Dr. Yasuharu Koike for his very generous support, allowing me to learn
new research techniques and perform research in his lab at the Tokyo Institute of Technology.
Without his support the data and research that I present here would not have been possible.
I would like to acknowledge my qualifying exam and defense committee members: Dr. Jason
Kutch, Dr. Terence Sanger, Dr. James Gordon, and Dr. James Finley. Their input was essential to
shape my work into its final form.
I also thank Dr. Toshihiro Kawase and Woorim Cho for their invaluable efforts in data collection.
I would like to again thank Dr. James Finley for generously allowing me to use his lab equipment.
I would also like to thank Natalia Sanchez, Rini Varghese, and Yu-Chen Chung for their support
in experiment preparation.
Thanks to current and former CNRL members: Chunji, Sujin, Kangwoo, Yannick, Nadir and Vince
for their positive outlook, their camaraderie, and the discussions and feedback that they provided.
iii

I would also like to thank my friend and flatmate, Adam, for his friendship and patience.
A special thanks goes to Aiko, for our never-endingly fun conversations, her unconditional support
from the other side of Earth, and everything that she has taught me.
Finally, a special thanks goes also to my sister, Maria Luisa, for constantly making me laugh and
cheering me up no matter the distance between us.

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Table of Contents

Acknowledgements ......................................................................................................................... ii
Abstract ......................................................................................................................................... vii
Chapter 1. Introduction ................................................................................................................... 1
Motivation ................................................................................................................................... 1
Background ................................................................................................................................. 3
Muscle synergy hypothesis ...................................................................................................... 3
Virtual surgeries ...................................................................................................................... 4
Overview of dissertation ............................................................................................................. 5
Chapter 2. When 90% of the variance is not enough: residual EMG from muscle synergy
extraction influences task performance .......................................................................................... 7
Abstract ....................................................................................................................................... 7
Introduction ................................................................................................................................. 8
Materials and Methods .............................................................................................................. 12
Data analysis ............................................................................................................................. 22
Results ....................................................................................................................................... 27
Discussion ................................................................................................................................. 33
Chapter 3. Differences in learning rates after virtual surgeries are determined by the statistics of
post-surgery movements ............................................................................................................... 40
Abstract ..................................................................................................................................... 40
Introduction ............................................................................................................................... 41
Methods ..................................................................................................................................... 44
Results ....................................................................................................................................... 55
Discussion ................................................................................................................................. 60
Appendix ................................................................................................................................... 65
Muscle tuning curves: experimental vs minimum effort simulation ..................................... 65
Maximum learning rate predictions and their relationship with failure of learning .............. 66
Chapter 4. Gradual and abrupt virtual surgeries elicit different muscle activation strategies ...... 68
Abstract ..................................................................................................................................... 68
Introduction ............................................................................................................................... 69
Materials and Methods .............................................................................................................. 71
v

Results ....................................................................................................................................... 79
Discussion ................................................................................................................................. 83
Chapter 5. Concluding remarks and future work .......................................................................... 87
References ..................................................................................................................................... 90

vi

List of figures
Figure 2.1. Experimental schedule and virtual surgery construction............................................ 15
Figure 2.2. Example of cursor trajectories during the EMG control task by a representative
subject. .......................................................................................................................................... 28
Figure 2.3. Comparison between actual and estimated directional errors. ................................... 29
Figure 2.4. Residual forces can explain the differences in initial direction error at surgery onset
between the easy and hard surgery conditions. ............................................................................. 30
Figure 2.5. Estimated errors in initial direction with easy and hard virtual surgeries based on
synergy sets with N = 1,…,10. ...................................................................................................... 33
Figure 3.1. Structure of the computational model. ....................................................................... 45
Figure 3.2. The shape of the cost function influences the convergence rate to the minimum in
gradient descent. ........................................................................................................................... 48
Figure 3.3. Process for initializing the inverse model f
-1
. ............................................................. 52
Figure 3.4. Forces at each target before, at the onset and at the end of training during the
incompatible and compatible virtual surgeries for a representative simulated subject. ............... 56
Figure 3.5. Motor and performance error during the virtual surgery tasks. .................................. 57
Figure 3.6. Probability density distributions of the eigenvalue ratio before the surgeries and at the
onset of both surgeries. ................................................................................................................. 58
Figure 3.7. Predicted maximum learning rates vs estimated learning rates.................................. 59
Figure 3.8. Experimental and simulated muscle tuning curves for ten muscles. .......................... 66
Figure 4.1. Experimental schedule and virtual surgery construction............................................ 73
Figure 4.2. Mean initial angular error for abrupt and gradual virtual surgeries. .......................... 80
Figure 4.3. Mean IMCJ in shoulder and elbow joints during the abrupt and gradual virtual
surgeries. ....................................................................................................................................... 82
Figure 4.4. Good performance in the virtual surgery task is associated with low levels of joint co-
contraction..................................................................................................................................... 83

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Abstract

Stroke is one of the leading causes of long-term disability in the world. One of its possible
consequences is upper limb hemiparesis, which is characterized by an abnormal coupling of the
movements of the elbow and shoulder joints, resulting in a significant loss of motor coordination
and a high impact on the quality of life of stroke survivors. This abnormal coupling is associated
with the abnormal co-activation of elbow flexor and shoulder abductor and adductor muscles.
Currently, most rehabilitation treatments do not directly target the recovery of motor coordination,
but rather, the recovery of function, which promotes the use of compensatory strategies to
circumvent the impaired coordination. Therefore, there is a need to develop new rehabilitation
treatments that directly promote the improvement of motor coordination. A promising approach is
the use of electromyography (EMG) -computer interfaces, which allow interaction with a virtual
environment through a mapping from muscle activations, as measured by EMG, to variables in the
virtual environment. Virtual surgeries are tasks in an EMG-computer interface that map EMG onto
forces in the virtual space, and mimic tendon transfer surgeries by assigning each muscle a pulling
force in the virtual environment that is different from its actual pulling force. This kind of task has
been used to induce learning of new patterns of muscle activation in non-impaired subjects,
suggesting a potential use in stroke rehabilitation to abolish the abnormal coupling of elbow and
shoulder muscles. Our long-term goal is to assess the feasibility of using virtual surgeries as a
component of rehabilitation treatments for hemiparesis after stroke. However, it is first necessary
to study the effects of virtual surgeries on motor learning in non-impaired subjects thoroughly. We
present three studies to this effect. The first study shows the importance of residual muscle
activation, which is often ignored, in the execution of the virtual surgery task. The second study
proposes that differences in learning rates observed when learning compatible and incompatible
viii

virtual surgeries are due to the statistics of the produced virtual forces. Finally, the third study
characterizes motor learning during the gradual and abrupt introduction of virtual surgeries. We
hope that this set of studies can be used as a starting point for building a framework to use virtual
surgeries in the rehabilitation of hemiparesis after stroke.

1

Chapter 1. Introduction

Motivation

Stroke is one of the leading causes of long-term disability in adults in the world. One of its possible
consequences is hemiparesis, manifested as a diminished ability to move one of the lateral halves
of the body. More specifically, the disability of the upper limb is of particular importance due to
its direct impact on the quality of life of stroke survivors. One of the hallmarks of upper limb
disability is the emergence of stereotypic movement patterns, characterized by a coupling of the
movements of the elbow and shoulder joints (Brunnstrom 1970).
Previous studies have shown that this movement coupling is likely due to abnormal descending
motor commands (Beer et al. 2007), and is caused by deficits in motor control independent from
weakness (Sukal et al. 2007). Moreover, it has been found that during isometric force production,
the impaired limb shows co-activation of elbow flexor and shoulder abductor/adductor muscles
not present in the unimpaired arm (Dewald et al. 1995), which may help explain the coupling
between the elbow and shoulder joints. Further studies have found differences in the muscle co-
activation patterns of unimpaired and impaired limbs using muscle synergy analyses (Cheung et
al. 2012; Roh et al. 2013).
Current rehabilitation treatments for the upper limb post-stroke mainly target the functional aspect
of the disability, that is, the difficulties in performing daily life activities. These treatments aim for
the acquisition of compensatory strategies to allow for the recovery of independence in daily life
activities. However, improvements in function are not necessarily associated with reductions in
the impairment aspect of the disability, such as the coupling of joints, because the metrics used to
quantify deficits in function cannot reliably measure impairment. Furthermore, these rehabilitation

2

interventions are only able to achieve modest gains even in the recovery of function (Krakauer and
Carmichael 2017). Therefore, there is a need to develop new rehabilitation protocols that aim to
reduce impairment, which could potentially promote recovery through restoration of pre-stroke
motor behaviors.
A promising approach to develop impairment-focused treatments is the use of electromyography
(EMG) -computer interfaces, which allow the control of objects in a virtual environment through
the activation of muscles, as measured by EMG (Berger and d'Avella 2017). Studies with healthy
subjects have used virtual surgeries, arbitrary mappings from muscle activity to the position of a
cursor in an EMG-computer interface, to show that muscle activation patterns can change at the
muscle synergy level during adaptation to a novel virtual surgery (Berger et al. 2013). This opens
the possibility of designing virtual surgeries that promote learning of arbitrary muscle co-
activation patterns, or muscle synergies, in a directed way (Wright et al. 2014).
These results converge to the fact that abnormal muscle activation patterns and the mechanisms
behind them are key to understanding the impairment of the upper limb after stroke and potentially
developing interventions that directly target impairment. Our long-term goal is to investigate
whether abnormal synergies can be reduced by gradually normalizing the activity of multiple
muscles via virtual surgeries. However, it is necessary to first study the effects of the virtual
surgeries on motor learning in healthy subjects to determine whether they would be beneficial in
a rehabilitation setting. Therefore, we present here three studies with which we aim to determine
some of the basic properties of motor learning during virtual surgery tasks.

3

Background

Muscle synergy hypothesis

One of the most salient problems that the central nervous system (CNS) faces when generating
movements is how to deal with the redundancy of the motor system (Bernstein 1967). This
redundancy spans the length of the causal chain of motor control: from neurons, to muscles, to
joints. As a result, the CNS could generate, in theory, an infinity of different motor commands to
produce the same movement. Yet, movements are highly stereotypical. The muscle synergy
hypothesis posits that the CNS groups the control of functionally similar muscles into modules
called muscle synergies (Lee 1984). This would reduce the number of variables that the CNS needs
to control to produce a movement, decreasing the complexity of the computations necessary for
motor control (Bizzi et al. 1991).
Direct evidence for the muscle synergy hypothesis comes from experiments in animal models
(Bizzi et al. 1991; Caggiano et al. 2016; Mussa-Ivaldi et al. 1994; Tresch et al. 1999). These show
that simultaneous stimulation of different groups of motor neurons elicits force fields at the limbs
that correspond to the superposition of the force fields obtained by stimulating each group of
neurons separately (Bizzi et al. 1991; Caggiano et al. 2016; Mussa-Ivaldi et al. 1994). However,
most of the supporting evidence in humans is indirect and comes from measurements of EMG
from multiple muscles during a variety of motor tasks (Berger et al. 2013; Cheung et al. 2012;
d'Avella et al. 2006; Ivanenko et al. 2005; Roh et al. 2012). Dimensionality reduction techniques,
such as non-negative matrix factorization, show that different muscles tend to co-activate in
reliable patterns during task execution (Tresch et al. 2006). One interpretation of these results is
that they reveal functional muscle synergies (Bizzi and Cheung 2013; Cheung et al. 2012; Roh et

4

al. 2012; Torres-Oviedo et al. 2006). An alternative interpretation, however, is that the co-
activation patterns arise because of biomechanical constraints imposed by the task (Burkholder
and van Antwerp 2013; Kutch and Valero-Cuevas 2012; Steele et al. 2015).

Virtual surgeries

In an EMG-computer interface, the activations of different muscles are mapped to arbitrary
variables in a virtual environment, such as the position of a cursor. This allows users to control
objects in the virtual environment through voluntary muscle contractions. During isometric force
production, the relationship between the activations of a set of muscles and the forces produced at
the end-point of the limb is approximately linear. Therefore, estimates of end-point force can be
obtained by measuring EMG, and used as virtual forces applied to objects in the virtual
environment. This provides an intuitive mapping between muscle activations and the controlled
object, in which each muscle is assigned a pulling force.
A virtual surgery redefines the pulling forces of arm muscles in the intuitive mapping from EMG
to two-dimensional isometric force at the wrist, which affects the overall virtual force applied on
the object, and in consequence, the performance during a task. This has an effect on force
production that is similar to an actual tendon transfer surgery. Therefore, recovering the initial
performance level requires motor learning to adjust the generated muscle activation patterns to
other patterns suitable for the new mapping.
Virtual surgeries can be defined arbitrarily, but families of virtual surgeries can be defined by
considering their effects on muscle synergies extracted through matrix factorization methods. The
EMG-force mapping can be simplified into a synergy-force mapping by combining the pulling

5

forces for each arm muscle according to a set of previously identified muscle synergies. As a result,
a virtual surgery also alters the extracted synergy-force mapping.
The family of incompatible surgeries are a class of virtual surgeries that specifically alter the
synergy-force mapping such that forces associated with synergy activity are constrained along only
one direction on the plane, impeding force production in the orthogonal direction (Berger et al.
2013). In contrast, the family of compatible surgeries allows the forces associated with synergy
activity to span the force space. Thus, in theory, learning an incompatible surgery requires the
acquisition of new patterns of muscle activations, or muscle synergies, whereas learning a
compatible surgery requires only the recombination of the existing muscle synergies. Therefore, it
is reasonable to hypothesize that learning an incompatible surgery is slower than learning a
compatible surgery.

Overview of dissertation

In this dissertation we present three studies in which we aim to determine some basic properties
of motor control and motor learning under the virtual surgeries.
The first study exploits the existence of virtual surgeries that are identical in the synergy-force
space, but differ in the muscle-force space. We found that incompatible virtual surgeries with
identical synergy-force mappings but different EMG-force mappings have a differential effect on
task performance. This suggests that muscle synergies extracted through matrix factorization
methods are not sufficient to explain the differences in performance. We showed that residual
muscle activity following muscle synergy identification can have a large systematic effect on the
virtual surgery task, even when the number of synergies approaches the number of muscles.

6

Therefore, current synergy extraction techniques must be updated to identify true physiological
synergies.
In the second study we used a computational model to replicate experimental results of learning
under compatible and incompatible virtual surgeries, in which learning is faster for the compatible
virtual surgery. We showed that this result can be explained by models that do not contain
explicitly defined muscle synergies. We demonstrated that differences in the statistics of the virtual
forces produced at the onset of the surgery can explain the learning rate differences, without taking
into account the formation of new muscle synergies.
In the third study we determined the effects of introducing a virtual surgery gradually as opposed
to abruptly. We found that introducing the surgery gradually did not significantly reduce the final
performance during the task. However, a gradual virtual surgery is more likely to induce higher
co-contraction during the task, indicating a difference in the muscle activation strategies when
facing a perturbation gradually vs abruptly.

7

Chapter 2. When 90% of the variance is not enough:
residual EMG from muscle synergy extraction
influences task performance

Authors: Victor R. Barradas, Jason Kutch, Toshihiro Kawase, Yasuharu Koike and Nicolas
Schweighofer

Abstract

Muscle synergies are usually identified via dimensionality reduction techniques, such that the
identified synergies reconstruct the muscle activity to a level of accuracy defined heuristically,
often set at explaining 90% of the variance. Here, we question the assumption that the residual
muscle activity that is not explained by the synergies is due to noise. We hypothesize instead that
the residual activity is not random and can influence the execution of a motor task. Young healthy
subjects performed an isometric reaching task in which the surface electromyography of 10 arm
muscles was mapped onto a two-dimensional force used to control a cursor. Three to five synergies
explained 90% of the variance in muscle activity. We altered the muscle-force mapping via “hard”
and “easy” virtual surgeries. Whereas in both surgeries the forces associated with synergies
spanned the same dimension of the virtual environment, the muscle-force mapping was as close
as possible to the initial mapping in the easy surgery; in contrast, it was as far as possible in the
hard surgery. This design maximized potential differences in reaching errors attributable to residual
activity. Results show that the easy surgery produced smaller directional errors than the hard
surgery. Additionally, error estimations for easy and hard surgeries constructed with 1 to 10
synergies show that the errors differ significantly for up to 8 synergies, which explain 98% of the
variance on average. Our study indicates the need for cautious interpretations of results derived

8

from synergy extraction techniques based on heuristics with lenient accuracy levels.

Introduction

One of the most salient problems that the central nervous system (CNS) faces when generating
movements is how to deal with the redundancy of the motor system (Bernstein 1967). This
redundancy spans the length of the causal chain of motor control: from neurons, to muscles, to
joints. As a result, the CNS could generate, in theory, an infinity of different motor commands to
produce the same movement. Yet, movements are highly stereotypical. The muscle synergy
hypothesis posits that the CNS groups the control of functionally similar muscles into modules
called muscle synergies (Lee 1984). This would reduce the number of variables that the CNS needs
to control to produce a movement, decreasing the complexity of the computations necessary for
motor control (Bizzi et al. 1991).
Direct evidence for the muscle synergy hypothesis comes from experiments in animal models
(Bizzi et al. 1991; Caggiano et al. 2016; Mussa-Ivaldi et al. 1994; Tresch et al. 1999). These show
that simultaneous stimulation of different groups of motor neurons elicits force fields at the limbs
that correspond to the superposition of the force fields obtained by stimulating each group of
neurons separately (Bizzi et al. 1991; Caggiano et al. 2016; Mussa-Ivaldi et al. 1994). However,
most of the supporting evidence in humans is indirect and comes from measurements of
electromyography (EMG) from multiple muscles during a variety of motor tasks (Berger et al.
2013; Cheung et al. 2012; d'Avella et al. 2006; Ivanenko et al. 2005; Roh et al. 2012).
Dimensionality reduction techniques, such as non-negative matrix factorization, show that
different muscles tend to co-activate in reliable patterns during task execution (Tresch et al. 2006).

9

One interpretation of these results is that they reveal functional muscle synergies (Bizzi and
Cheung 2013; Cheung et al. 2012; Roh et al. 2012; Torres-Oviedo et al. 2006). An alternative
interpretation, however, is that the co-activation patterns arise because of biomechanical
constraints imposed by the task (Burkholder and van Antwerp 2013; Kutch and Valero-Cuevas
2012; Steele et al. 2015).
This controversy notwithstanding (Tresch and Jarc 2009), dimensionality reduction techniques for
the extraction of muscle synergies rely on the ability of the extracted synergies to accurately
reconstruct the originally measured EMG signals (Berret et al. 2019). That is, the extracted
synergies must capture a high proportion of the variability in the recorded EMG. Unfortunately,
the dimensionality reduction techniques for synergy extraction consist of unsupervised learning
methods that require the number of muscle synergies to be defined beforehand, which is not
possible because the ground truth is unknown. As a result, the choice of the number of muscle
synergies is left as a free parameter. Although a number of heuristics have been proposed (Cheung
et al. 2005; Clark et al. 2010; Santuz et al. 2017), a widely used rule of thumb is to set the number
of synergies to the minimum number that accounts for at least 90% of the variability in the EMG.
Such heuristic rules carry the implicit assumption that the unexplained portion of the variability
arises from measurement noise and process noise. Indeed, surface EMG measurements are highly
sensitive to skin condition, electrode placement and movement artifacts, making them vulnerable
to measurement noise (Merletti et al. 2010). Additionally, it is well established that the processes
of movement and force generation are stochastic in nature (Harris and Wolpert 1998), which arises
in part from noise in motor execution (Schmidt et al. 1979; van Beers et al. 2004). The inherent
presence of noise in the EMG has therefore motivated the development of heuristic procedures for
determining the number of muscle synergies that define the point at which the extracted synergies

10

begin to fit the noise in the signal rather than capturing the underlying structure of the data
(d'Avella and Lacquaniti 2013; d'Avella et al. 2003; Torres-Oviedo and Ting 2010). Furthermore,
models of muscle activation based on the muscle synergy hypothesis define muscle activity as
linear combinations of non-negative synergies contaminated by noise, to test the effectiveness of
a variety of synergy extraction algorithms on simulated datasets (Delis et al. 2013; Steele et al.
2013; Tresch et al. 2006). These methodologies indicate that muscle activity unaccounted for by
the extracted muscle synergies is often attributed to noise.
In addition, synergy extraction methods that aim to exclusively reconstruct the measured muscle
activity neglect the fundamental role of muscle synergies as building blocks of movement, as they
ignore the ability of the extracted muscle synergies to reconstruct the observed movement
(Alessandro et al. 2013; Berret et al. 2019; de Rugy et al. 2013). This is problematic, as it has
notably been shown that the ability of muscle synergies to reconstruct measured forces in an
isometric task at the wrist becomes largely degraded as the number of considered muscle synergies
decreases (de Rugy et al. 2013). This is true even when the extracted synergies capture an
acceptable portion of the variability in the EMG signals according to the defined heuristics. This
suggests that the portion of EMG variability that is not captured by the extracted muscle synergies
is important for a full description of the motor action.
The study by (de Rugy et al. 2013) shows, based on an offline analysis, that a large number of
synergies is required for a reliable reconstruction of the forces. However, a further study indicates
that the failure to reconstruct forces reliably based on synergy control in de Rugy et al. 2013 could
be biased by the presence of online feedback only during the EMG control task, which was not
present in the synergy reconstruction (Berger and d'Avella 2014). Here, we therefore aimed to
complement this view by introducing a novel method that separates the effects of synergy and

11

residual muscle activity in an online task. This component separation allowed us to determine the
importance of the residual EMG in the execution of a virtual motor task, which would not be
possible in a non-virtual setting. We tested the null hypothesis that following extraction of muscle
synergies with non-negative matrix factorization and using the 90% of explained variance rule to
select the number of synergies, the residual muscle activity is exclusively due to noise. Therefore,
if our experimental data failed to support this hypothesis, it would suggest that the residuals contain
muscle activity relevant to motor performance.
To this end, we used the virtual surgery paradigm, which simulates tendon transfer surgeries
(Berger et al. 2013). The virtual surgery alters the pulling forces of arm muscles in a virtual
mapping from EMG to two-dimensional isometric force at the wrist, which affects performance
during the reaching task. Because the EMG-force mapping can be simplified into a synergy-force
mapping by combining the pulling forces for each arm muscle according to a set of previously
identified muscle synergies, a virtual surgery also alters the resulting synergy-force mapping.
Incompatible surgeries are a class of virtual surgeries that specifically alter the synergy-force
mapping such that forces associated with synergy activity are constrained along only one direction
on the plane, impeding force production in the orthogonal direction (Berger et al. 2013). Given
that the number of muscles is necessarily larger than the number of extracted synergies, it is
possible to build virtual surgeries that produce identical synergy-force mappings, but different
EMG-force mappings. We exploited this property by designing incompatible virtual surgeries that
modified the EMG-force mapping to two opposite extremes, while producing the same synergy-
force mapping: “easy” and “hard” incompatible surgeries.
The “easy” surgery modified the EMG-force mapping as little as possible with respect to the
baseline mapping; in contrast, the “hard” surgery modified the mapping as much as possible. These

12

two virtual surgeries were designed based on the extracted muscle synergies that account for at
least 90% of the variability in the EMG, and therefore have the same effect on the synergy
component of the EMG. Consequently, the incompatible virtual surgeries allowed us to isolate the
effects of the residual portion of the EMG from the portion of the EMG attributed to muscle
synergies, leading to possible differences in the effects of the easy and hard surgeries on task
variables. If the EMG residuals are attributable to noise, then both surgeries should produce similar
errors in the direction of reaching when introduced suddenly. Alternatively, if the EMG residuals
are non-randomly organized, then both surgeries should have a differential effect on the residuals
and on the error in the direction of reaching. We found that the sudden introduction of both kinds
of virtual surgeries produced largely different errors, supporting the existence of an underlying
organization in the EMG residuals, suggesting that current methods for muscle synergy extraction
do not capture all task-relevant muscle activity.

Materials and Methods

Subjects. Fifteen right-handed subjects (mean age, 27.9 years (SD 8.8); thirteen males)
participated in the study after providing written informed consent. All procedures were approved
by the Ethical Review Board of the Tokyo Institute of Technology.
Experimental setup. Each participant sat on a racecar seat while gripping a handle located at the
height of the base of their sternum with their right hand. The arm posture corresponded to an elbow
flexion of around 90° and the elbow was supported on a stand at approximately the same height as
the hand. A splint was used to immobilize the hand, wrist and forearm. Participants were instructed
to lean on the back of the seat for the duration of the experiment to avoid the application of trunk

13

forces on the load cell. The experimenter (the first and/or the third authors) constantly monitored
the participants, and instructed them to lean back again if they moved forward. The base of the
handle was attached to a six axis force transducer (Dyn Pick; Wacoh-Tech Inc.) used to measure
isometric forces. The force transducer was mounted on a 2-D sliding rail to allow for an adjustable
configuration for each participant. A virtual environment was displayed on a computer screen
placed at the height of the participants’ eyes at a distance of around 1 m. The virtual environment
consisted of a circular red cursor (1 cm diameter), and several ring-shaped white targets (force
control: 3.5 cm diameter; EMG control: 2 cm diameter) displayed on a black background.
We recorded surface EMG activity from 10 muscles crossing the shoulder and elbow joints:
pronator teres, brachioradialis, biceps brachii long head, triceps brachii lateral head, triceps brachii
long head, anterior deltoid, middle deltoid, posterior deltoid, pectoralis major, and middle
trapezius. Active bipolar electrodes (DE 2.1; Delsys) were used to record EMG activity. EMG
signals were band-pass filtered (20-450 Hz) and amplified (gain 1000, Bagnoli-16; Delsys). Force
and EMG recordings were digitized at 2 kHz using a USB analog-to-digital converter (USB-6259;
National Instruments).
To reduce random oscillations of the cursor caused by the stochastic nature of EMG signals, a
mass-spring-damper dynamics filtered the EMG signals further (Berger et al. 2013). The mass-
spring-damper dynamics governed the movement of the cursor according to:
𝐩 ̈ = −
b
m
𝐩 ̇ −
k
m
𝐩 + F(t) (2.1)
where p is a vector containing the x and y positions of the cursor on the screen and its derivatives
are indicated in dot notation, m is the system’s mass, k is the stiffness, and b is the damping
coefficient (m = 0.05 kg, b = 100 kg/s). F(t) is the force recorded by the force transducer (during

14

force control) or the estimated force by the EMG-force mapping (during EMG control; see next
section). The stiffness k was calculated as a function of the maximum voluntary force (MVF), so
that targets at equal percentages of MVF required the same cursor displacement across
participants.
Experimental protocol. In all phases of the experiment, participants used their right arm to
perform a number of isometric force tasks. These tasks required the displacement of a cursor on a
visual display from a center position to one of eight targets uniformly distributed around the center.
Participants first performed a force control task and then an EMG control task (Fig 1a). In the force
control task, the cursor was controlled via forces applied by the arm on a load cell (force control).
In the EMG control task, the cursor was controlled by a linear approximation of the force derived
from EMG measurements of 10 arm muscles (EMG control).

15

Figure 2.1. Experimental schedule and virtual surgery construction.
a. Experimental schedule. Following a maximum voluntary force (MVF) block, participants performed the
force control task. Simultaneous recording of EMG and force data in this task were analyzed to extract
muscle synergies, to produce the baseline EMG-force mapping, and to construct the easy and hard
incompatible virtual surgeries. Participants then performed the EMG control tasks, starting with a
familiarization block, followed by baseline, and then one of the two virtual surgeries (easy or hard). In this
cross-over study, participants then performed the other virtual surgery following a new baseline. Note that
here, we only analyzed the data from the first block of each of the two virtual surgery procedures. b–d.
Virtual surgery construction; example from one participant. c. EMG-force mapping extracted after the force
control task. Top: Each arrow represents the estimated force on the horizontal plane that a single muscle
would produce when fully activated in isolation from the other muscles (columns of the 2 x 10 M matrix
representing the EMG to force mapping). Bottom: Forces produced by each of the muscle synergies
extracted after the force control task (columns of 2 x N MS matrix, where N is the number of synergies).
Before applying any virtual surgery, these forces span the 2-dimensional plane. b. Hard incompatible
surgery. We designed the hard incompatible surgery by defining a matrix T H that rotates the synergy force
vectors so that they became collinear at an angle of 135° degrees, while maximizing the angles between the
original and transformed EMG to force mapping (M and MT H respectively). d. The easy incompatible
surgery is similar but minimizes the angles between the column vectors of MT and MT E. We constrained
the synergy-force mapping associated with the easy surgery MT ES and the hard surgery MT HS to be equal,
and thus produce the same synergy-related force. Note that because the virtual surgeries are built based on
S, their intended effects are only expressed on the synergy component of force, and the effect on the residual
force is not explicitly specified.

16

The force control task started with a maximum voluntary force (MVF) block, in which participants
were instructed to produce a maximum voluntary force in each of eight directions spanning the
horizontal plane, with two trials for each direction. The mean MVF was calculated as the mean of
the maximum forces recorded across all trials. For each muscle, the value at the 95 percentile of
the recorded EMG signal across all trials was used to normalize the values of EMG from the
corresponding muscle in all subsequent tasks.
Participants then performed an isometric reaching task by applying force to reach targets in the
virtual environment. The recorded force and EMG signals during this task were processed to
compute the EMG-force mapping, extract muscle synergies, and construct the virtual surgeries.
Targets were arranged radially in eight directions and required 5, 10, 15 or 20% of MVF to be
reached. Each trial started by displaying the target at the central position. The central position
corresponded to the position of the cursor when no forces were applied. After placing the cursor
inside the central target for two seconds, the central target disappeared and one of the radial targets
appeared. After reaching each target, both the cursor and the target disappeared from the screen
and participants were asked to hold the applied force as steadily as possible for two seconds. Next,
the cursor and the central target reappeared and participants were asked to move the cursor back
to the center. After this, another trial began. Each target was presented three times, with a total of
96 trials. Targets were presented in a randomized order. Trials were repeated if participants failed
to reach a target.
Next, cursor control was switched to EMG control without the participants’ knowledge. The first
EMG control block was a familiarization block, and was followed by one type of incompatible
surgery, easy or hard, followed by the other in a cross-over design (Figure 2.1a). The order of the
easy and hard surgeries was pseudo-randomized such that 7 participants started with the easy

17

surgery. Participants rested for 5 minutes between surgery types. Each surgery condition consisted
of three phases: baseline, virtual surgery, and washout, which consisted of 6, 12, and 6 blocks,
respectively. Each block consisted of 24 trials: three trials for each of the eight targets at a
magnitude of 10% MVF randomized within target sets containing each one of the eight targets.
The level of baseline noise in each EMG signal was measured at the start of every block while the
participant was relaxed. This baseline noise was subtracted from the EMG signals measured during
the corresponding block. Note that in this study, we only analyze data recorded during the first set
of eight targets following the onset of each virtual surgery (analysis of the following blocks will
be covered in a separate manuscript).
EMG-force mapping. Force produced at the hand with the arm in a static posture can be
approximated as a linear function of the activations of muscles that actuate the shoulder and elbow
(Berger et al. 2013; Valero-Cuevas 2009), especially when low forces are involved (Zhou and
Rymer 2004):
𝐟 = 𝐌𝐦 (2.2)
where f is a two-dimensional force vector produced on the horizontal plane, m is a ten-dimensional
vector of muscle activations, composed by normalized EMG signals recorded from ten muscles
simultaneously, and M is a 2 × 10 matrix that maps muscle activations to forces, as represented in
the upper panel of Figure 2.1c. M was determined via linear regression of 10 EMG signals against
2D forces recorded during every trial of the main force control subtask. Before performing the
regression, forces were low-pass filtered (second-order Butterworth; 1 Hz cutoff) and EMG signals
were high-passed and low-passed filtered (second-order Butterworth; high-pass: 20 Hz, low-pass:
5 Hz), rectified, and normalized. The signals were recorded from the time of target go to the end
of target hold.

18

Synergy extraction and number of synergies. We used non-negative matrix factorization (NMF)
(Lee and Seung 1999) to extract muscle synergies from the EMG signals collected during the main
force control subtask:
𝐦 = 𝐒 𝐜 (2.3)
where S is a 10 × N matrix that contains the identified synergies in its columns with N being the
number of synergies, and c is an N-dimensional vector of synergy activations. Equation 2.3
assumes perfect matrix factorization (no residual EMG activity).
EMG signals collected during the main force control subtask were processed in the same way as
described in the EMG-force mapping section. The synergy extraction procedure closely followed
a previously described method (Berger et al. 2013). Synergies were extracted for all N from 1 to
10. For each case, the synergy extraction algorithm was run 100 times, and the result with the
highest reconstruction quality R
2
of the original EMG signals was kept. Two criteria were required
to select N. The first was to set N as the minimum number of synergies necessary to explain at
least 90% of the EMG data variance. The second involved calculating the changes in slope in the
R
2
curve as a function of N. Linear regressions were performed on sections of the curve between
N and 10. N was selected as the smallest value for which the mean squared error of the linear
regression was < 10
-4
(d'Avella et al. 2006). If the two criteria did not match, N was selected as the
case in which the extracted synergies had the smallest number of similar preferred directions
(number of adjacent directions separated by less than 20°). This occurred for seven of the
participants.

19

Construction of easy and hard incompatible surgeries. As in a previous study, a virtual surgery
modifies the EMG-force mapping (M) by applying a linear transformation in muscle space (Berger
et al. 2013):
𝐌 ′ = 𝐌𝐓 (2.4)
where T is a 10 × 10 matrix that defines the virtual surgery.
Incompatible virtual surgeries are designed such that muscle activations m produced by synergy
combinations Sc are restricted to generate forces along only one dimension of the force space
(Figure 2.1b, d), while the resulting EMG-force mapping M’ spans the whole force space.
It is important to note that the set of incompatible surgeries is infinite. This is because the number
of muscles used in the virtual mapping is larger than the number of muscle activity patterns found
using muscle synergy analysis. A previous study (Berger et al. 2013) combined randomness and
difficulty matching to select compatible and incompatible virtual surgeries. In contrast, here we
specified a series of constraints to yield only two possible virtual surgeries.
Specifically, we built hard TH and easy TE incompatible surgeries such that they were equivalent
in the force space spanned by each participant’s extracted muscle synergies (Figure 2.1b and d).
We first note that, assuming that muscle activations are only generated by combinations of
synergies, according to equations 2.2, 2.3 and 2.4, forces produced during the surgery are given
by:
𝐟 = 𝐌 ′
𝐦 = 𝐌𝐓𝐒𝐜 (2.5)
This equation shows that surgery T can alternatively be thought to transform the extracted
synergies S into a new set of synergies S’:

20

𝐟 = 𝐌𝐒 ′𝐜 (2.6)
In order to build an incompatible surgery, it is necessary to find S’ such that the matrix MS’ is
rank deficient. This guarantees that forces produced by this mapping lie in a single dimension.
Geometrically, this means that the forces associated with each individual synergy from S’ are
collinear (Figure 2.1b and d).
Easy surgeries were built such that the angles between the column vectors of the original M
mapping and of the transformed mapping M’ were as small as possible (Figure 2.1d). In contrast,
hard surgeries were built by making these angles as large as possible (Figure 2.1b). For this, we
used a two-step optimization procedure: first, we obtained a transformed set of synergies S’, and
second, we computed the incompatible surgery T.
In the first step, we constrained S’ to be equal for both the easy and hard incompatible surgeries.
This ensured that the only difference between both virtual surgeries is the transformed mapping
MT. We chose a configuration such that the individual force vectors associated to each synergy in
S were rotated onto a line that bisected the plane at an angle of 135° with the x-axis. Therefore,
each force vector conserved its magnitude, and its direction was assigned to the direction of the
bisecting line that was closest to it: 135° or -45°. If we include each of these forces as a column of
a 2 × N matrix Fdes, where N is the number of extracted synergies, this operation can be represented
as a system of equations in which the elements of S’ are the unknowns:
𝐌𝐒 ′ = 𝐅 𝐝𝐞𝐬 (2.7)
Because this problem has 10N unknowns and only 2N equations, we introduced an optimization
objective to arrive to a unique solution. A reasonable objective is to minimize the sum of the
squares of the elements of S’, as this creates a sparse set of synergies. Additionally the elements

21

of S are required to be non-negative. This optimization problem can be posed as a quadratic
program, which involves the minimization of a quadratic function subject to linear equality and
inequality constraints (Gill and Wong 2015):
min
𝐬 ∑ ∑ s
ij
2 N
j=1
10
i=1

s. t. {
𝐌 𝐒 ′
= 𝐅 𝐝𝐞𝐬 s
ij
≥ 0 for i = 1, … ,10 j = 1, … , N
(2.8)
We transcribed this quadratic program into its canonical form and solved it using the quadprog
function in Matlab. The optimization procedure was initialized with a zero S’ matrix.
In the second step, after obtaining S’, we computed the incompatible hard and easy surgeries by
first noting that for a surgery T,
𝐒 ′ = 𝐓𝐒 (2.9)
This is a system of equations where the elements of T are the unknowns. We note that T is a 10 ×
10 matrix, so in this case there are 100 unknowns and 10N equations. The system is
underdetermined in all cases where N < 10, which in our case is guaranteed.
To find the easy virtual surgery, we minimized the dot product between the respective columns of
M and M’, which is proportional to the angle between the original and transformed muscle pulling
directions:
max
𝐓 ∑ 𝐡 𝐢 ∙ 𝐡 𝐢 ′ 𝟏𝟎
𝐢 =𝟏 (2.10)
where hi and hi’ are the column vectors of M and M’, respectively. Because the column vectors
hi’ are a function of T (equation 2.4), the optimization is performed in the space of the elements

22

of T. As this optimization objective is not bounded, we additionally constrained the magnitude of
the resulting h’ vectors:
‖𝐡 𝐢 ′
‖ ≤ 1.5‖𝐡 𝐢 ‖ (2.11)
This problem can be posed as a linear program with quadratic constraints, with equation 2.10 as
the objective, and equations 2.9 and 2.11 as equality and inequality constraints, respectively. The
result of this optimization procedure yields TE, the easy incompatible virtual surgery.
The procedure is the same for the hard incompatible TH, with the only difference that the
optimization objective is minimized instead of maximized. In turn, this maximizes the angles
between hi and hi’:
min
𝐓 ∑ 𝐡 𝐢 ∙ 𝐡 𝐢 ′ 𝟏𝟎
𝐢 =𝟏 (2.12)
Both linear programs with quadratic constraints were solved using the fmincon function in Matlab.
The optimization procedure was initialized with a zero T matrix.

Data analysis
Task performance metric. We used the initial angular error as a metric to quantify task
performance during the experiment. The initial angular error was calculated for each trial as |θtarget
- θcursor|. θtarget is the direction of the target. θcursor is defined as the direction of the line segment that
joins the point at which the cursor exits a 2 cm diameter circumference at the center of the screen
and the position of the cursor 100 ms after exiting the circumference. We averaged the initial
angular error for the targets within sets of eight trials. We only took into account targets that were
not aligned with the line of action of the surgery. That is, targets other than those at 135° and -45°

23

from the horizontal on the screen. Note that the initial angular error more closely represents the
output of the issued feedforward motor command than a metric such as target accuracy, in which
feedback mechanisms are more likely to be engaged to complete the task and obscure the effect of
the residual component of muscle activation.

EMG residual analysis. Whereas the easy and hard surgeries are built based on S, and therefore
should have the same effects on the synergy component of the EMG, we expected to observe a
differential effect on the residual components due to the large differences in the MTH and MTE
mappings.
Therefore, we analyzed the residual EMG signals obtained after reconstructing the measured EMG
signals based on the extracted muscle synergies. After synergy extraction using the NMF
algorithm, and extending equation 2.3, muscle activations can be represented as
𝐦 = 𝐒𝐜 + 𝐫 = 𝐦 𝐬𝐲𝐧 + 𝐫 (2.13)
where msyn is the synergy component of muscle activation, and r is the residual component of
muscle activation that cannot be accounted for by the extracted synergies. Consequently, the forces
associated with the EMG signals have both a synergy and a residual component:
𝐅 = 𝐌𝐦 = 𝐌 (𝐒𝐜 + 𝐫 ) = 𝐌 𝐦 𝐬𝐲𝐧 + 𝐌𝐫 = 𝐅 𝐬𝐲𝐧 + 𝐅 𝐫𝐞𝐬 (2.14)
where Fsyn and Fres are the synergy and residual components of force, respectively.
In order to decompose a given EMG sample m into its synergy and residual components (msyn and
r), we first computed msyn via non-negative least squared regression of S and m, which yielded c.

24

This algorithm optimizes the same cost function as the NMF algorithm (equation 2.3). The residual
component r is found by subtracting msyn from m.
We then analyzed the effects of the easy and hard surgeries on both the synergy and residual
components of EMG. For this, we used the EMG activity that participants produced when they
acquired each target during the first baseline phase of the experiment. We then separated the
average EMG activity m of each subject and target into msyn and r, with which we estimated both
force components Fsyn and Fres produced for each target at the onset of the easy and hard virtual
surgeries by substituting M by M’ in equation 2.14. We then compared the estimated force
direction to the intended direction for each target to obtain an estimate of the error that subjects
would produce at the onset of each virtual surgery.
Plausibility of null hypothesis. Because the easy and hard surgeries transform the forces
generated by individual muscles in very different ways, it could be argued that if the residual
muscle activation were exclusively noise, there could be differences in the initial direction error
after applying the surgeries, and not necessarily indicate task-relevant activity in the residuals.
Indeed, differences in the initial angular error could arise from a differential effect of each kind of
surgery on the baseline distribution of noise. To test whether this may be the case, we ran a Monte
Carlo simulation in which ground truth data was generated from a model based on muscle synergy
control with additive noise.
We first used the EMG-force mapping M and the muscle synergies S extracted from each
experimental subject to create a corresponding simulated subject. Next, for each simulated subject,
we found appropriate synergy activation coefficients c, that in combination with M and S produce
forces Ft that correspond to the eight targets in the experimental task. The force magnitude required
to reach a target was set to 10% of the MVF measured in the experiment for the corresponding

25

subject. We found the appropriate synergy activation coefficient c by considering an optimal
control strategy to exactly reach each of the eight targets based on effort minimization. We defined
effort as the sum of squared synergy activation coefficients c:
u = ∑ c
i
2 N
i
(2.15)
The optimization procedure was implemented by solving a quadratic program with the effort term
as the minimization objective (equation 2.15). Equations 2.2 and 2.3 were used as equality
constraints, with forces equal to Ft, and the non-negativity of synergy activations ci ≥ 0 as
inequality constraints.
We contaminated the muscle activations m obtained from the optimal synergy activation
coefficients c (equation 2.3) with signal-dependent noise. For each simulated subject and target
we computed the forces F generated by 1000 instances of contaminating the optimal muscle
activation with noise and calculated the error in the direction of F with respect to the direction of
the corresponding target Ft under each surgery condition. We then calculated the mean direction
error for each simulated subject. In separate simulations, we used Gaussian signal-dependent noise
with zero mean and a standard deviation that scales linearly with m, with a coefficient of variation
ranging from 0.1 to 0.5 for each separate simulation (the 0.1 - 0.25 range corresponds to empirical
observations of the firing of motor neurons (Harris and Wolpert 1998)). We then computed the
mean error in the force direction for each subject and noise condition for statistical comparison.
Shuffling of EMG residuals. We then performed a Monte Carlo simulation to determine the
effects of shuffling the measured residual muscle activations among samples on task performance.
Shuffling the residual component of different EMG signal samples creates random residual
components with the same statistical properties as the original residuals. If the residual EMG

26

activity can be disregarded as noise, then shuffling the residuals should have no significant effect
on the estimated forces with respect to pre-shuffling. On the contrary, if the residuals have a non-
random organization, shuffling the residuals would destroy this organization. Consequently, the
force estimates would most likely be different from the pre-shuffling estimates. We therefore
shuffled the residual components of the EMG samples that we used to estimate forces, and re-
estimated the resulting total forces at the onset of the easy and hard virtual surgeries. We averaged
the results of 1000 different shuffling instances for each target, which should be close to the real
mean performance when shuffling the residuals, as established by the Law of Large Numbers and
the Central Limit Theorem.
Statistical analysis. The main outcome of the experiment was the average initial angular error
across subjects in both the easy and hard virtual surgery conditions. We tested the null hypothesis
that there would be no difference in the average initial angular error between conditions by means
of a paired t-test. We also tested the same hypothesis in the simulations with the manipulated
residual component of the muscle activations with paired t-tests. We used Bonferroni corrections
for multiple comparisons when testing the outcomes of residual manipulation considering different
numbers of muscle synergies. All analyses were performed in Matlab R2016b. The significance
threshold was set at p = 0.05.

27

Results

Hard incompatible virtual surgeries produced larger initial angular errors than
easy incompatible surgeries
A linear model was used to estimate the relationship between EMG recorded from 10 arm muscles
and forces produced at the wrist on a horizontal plane. We found that the assumption of a linear
model adequately describes the relation between EMG and force (R
2
= 0.77 (SD 0.04)). This is in
agreement with a previous study (Berger et al. 2013), which showed a comparable quality of the
linear model (R
2
= 0.81 (SD 0.05)) while recording EMG from 13 muscles. The number of
extracted muscle synergies N for all subjects ranged from three to five (N = 3, 1 subjects; N = 4,
11 subjects; N = 5, 3 subjects).
During the force control task, the mean error in initial direction was smaller than during the
baseline blocks of EMG control (force control: 8.3° (SD 3.0), EMG control: 13.1° (SD 7.8)). This
indicates that the EMG control task was more difficult than the force control task. However, this
decrease in the quality of control is expected, and is comparable to previous studies (Berger et al.
2013).
Figure 2.2 shows sample cursor trajectories before and after the onset of the virtual surgeries. Both
surgeries produced a bias in the cursor movement along the designed direction as predicted,
although cursor movements were not perfectly constrained to this direction. Overall, deviations
from the line of action of the surgery were closer to the intended target during the easy surgery
than during the hard surgery (Figure 2.2b).

28

Figure 2.2. Example of cursor trajectories during the EMG control task by a representative subject.
a. Sample cursor trajectories. These trajectories correspond to the last target set of the baseline with EMG
control, and the first target set after the onset of the hard and easy incompatible virtual surgeries. The
trajectories tended to fall along the line of action of the virtual surgery, notably in the hard surgery. b.
Comparison of initial directions of cursor movement between the onset of the easy and hard virtual
surgeries. Solid lines correspond to the initial directions during the easy surgery onset and dotted lines
correspond to the hard surgery onset. This subject produced larger initial errors at the onset of the hard
virtual surgery than at the onset of the easy surgery (see targets at 45° and 90°).

Over all 15 participants, the mean error for the first set of targets after the onset of the surgery was
clearly larger for the hard surgery than for the easy surgery (hard surgery: 81.4° (SD 14.7), easy
surgery: 54.5° (SD 17.9), p < 0.001, paired t-test; see Figure 2.3b, experiment). This difference in
errors may appear surprising at first, given that the easy and hard surgeries had the same effect on
the synergy component of the force. That is, they restricted the forces associated with the synergies
along one dimension. However, the synergies only accounted for 90% of the variance in EMG.
Therefore, the EMG residuals appeared to generate an additional component of force.

29

Figure 2.3. Comparison between actual and estimated directional errors.
a. Correlation between the estimated average angular error after applying the surgery and the actual average
error during the first target set of the virtual surgery for all participants. b. Actual and estimated errors in
initial direction of force at the onset of the easy and hard virtual surgeries. In the experiment, the hard
surgery produced a larger initial error than the easy surgery. The null residual estimate produced error
estimates that showed no difference between the hard and the easy virtual surgeries. The EMG signals with
shuffled residuals produced error estimates that were similar to those obtained using only the synergy
component of the EMG signal. Error bars indicate the standard error.

Initial angular error was determined by effect of surgery on residual EMG
activity
To verify the effect of residuals on movement error, we decomposed the recorded EMG signals
into their synergy and residual components (equation 2.14). Figure 2.4 shows the estimated forces
corresponding to the total EMG activity F (top), and the synergy (Fsyn) and residual (Fres)
components of force (middle and bottom, respectively) at each target for a representative subject
(equation 2.14). The center column shows F, Fsyn and Fres before the onset of the surgeries. The
left and right columns show F, Fsyn and Fres after applying the hard and easy surgeries, respectively.
The incompatible design of the surgery can be appreciated on Fsyn, as these forces lie on the 135°
line of action of the virtual surgery (Figure 2.4, middle row).

30

Figure 2.4. Residual forces can explain the differences in initial direction error at surgery onset
between the easy and hard surgery conditions.
Top row: Estimated forces F at each target before and after applying the hard and easy virtual surgeries.
We indicate the average estimated error across targets for each virtual surgery. Middle row: Estimated
synergy components of force F syn at each target. Bottom row: Estimated residual components of force F res
at each target. We indicate the average rotation of F res after each virtual surgery with respect to F res before
the surgery. Middle column: F, F syn and F res before the surgery. Left and right columns: F, F syn and F res
after applying the hard and easy surgeries, respectively. Colors represent the eight targets in the task as
indicated in the middle bottom diagram. Data shown in this figure corresponds to the same representative
subject as in Figure 2.2.

Given that the EMG signals that we used to estimate forces were representative of the subjects’
forces during baseline, and assuming that subjects produced these EMG signals when suddenly
exposed to the virtual surgeries, the directions of the estimated forces after applying the virtual
surgery (equation 2.5) also provided an estimate of the cursor error to each target (Figure 2.4, top
row). These initial error estimates were consistently higher for the hard surgery than for the easy
surgery (hard surgery: 82.8° (SD 16.2), easy surgery: 45.6° (SD 11.7), p < 0.001, paired t-test),

31

and qualitatively matched the experimental results of the cursor error (robust regression, slope =
0.47 ± 0.15 s.e., p = 0.004, R
2
= 0.25) (Figure 2.3a).
Errors following the easy and hard surgeries can be explained by the residual’s structure (Figure
2.4, bottom row). The hard surgery produced a mean rotation of Fres with respect to baseline that
was much larger than that produced by the easy surgery (hard surgery: 113.6° (SD 39.0), easy
surgery: 4.4° (interquartile range 2.4), p < 0.001, paired t-test). Note that although we did not
specify the effect of the virtual surgery on the residual component of force, the hard surgery
produced much larger rotations of Fres than the easy surgery, showing systematic differences in
the effect of surgeries on the residual component of EMG.

Differences in initial error are not due to the statistical properties of the output
of virtual surgeries
We tested in simulation whether differences in the initial direction error between the hard and the
easy surgeries could arise if the residual muscle activation is fully composed of noise (see
Methods), which would be a confound for our hypothesis. The simulation assumed that forces are
generated based on a muscle synergy controller whose output is contaminated with signal-
dependent noise. The differences in the simulated initial direction error between the easy and hard
surgeries were not significant for any tested value of the coefficient of variation in the noise, which
included the physiological range (η = 0.1, p = 0.39; η = 0.15, p = 0.34; η = 0.2, p = 0.24; η = 0.25,
p = 0.21; η = 0.3, p = 0.21; η = 0.4, p = 0.15; η = 0.5, p = 0.15).

32

Shuffling residual EMG activity revealed organization in the residuals
We then shuffled the residual EMG components among trials to all targets to demonstrate a
possible underlying organization. Initial error estimates based on the shuffled signals did not
indicate a significant difference in average initial error between the easy and hard virtual surgeries
(hard surgery: 72.4° (SD 10.7), easy surgery: 66.4° (SD 9.0), paired t-test, p = 0.10) (Figure 2.3b,
shuffled residuals). Furthermore, the magnitude of this error lied at an intermediate level between
the errors observed experimentally for the easy and hard surgeries. Importantly, the means of the
estimates produced by shuffled signals were indistinguishable from estimates produced based on
a null residual condition, that is, exclusively using the synergy component of the EMG to produce
estimates (easy surgery, paired t-test, p = 0.45; hard surgery, paired t-test, p = 0.68) (Figure 2.3b,
null and shuffled residuals).

Estimated differences between errors for easy and hard surgeries remained
significant for high-dimensional synergy sets
We then tested whether building virtual surgeries based on synergy sets with a larger N would
abolish the differences in initial direction error observed in the experiment. For each participant
we built easy and hard surgeries based on surgeries considering N = 1, …, 10 and applied the
newly constructed surgeries to the same set of EMG signals that we used to estimate errors after
the introduction of the surgery. This allowed us to simulate the errors that participants would
produce in an experimental setting under the new set of virtual surgeries.
We found that the surgeries produced estimated differences in initial direction errors that were
maximal for N = 1, and gradually decreased until disappearing at N = 10 (as expected, since activity

33

from 10 muscles was recorded; Figure 2.5). The estimated error differences remained significant
up to N = 8 (p = 0.001, paired t-test). This indicates that the residual components of EMG produced
a differential effect on the estimated error even for high-dimensional synergy sets that explained a
portion of the variance that largely exceeded the heuristic rule requirements (R
2
= 0.98
(interquartile range 0.01) at N = 8) were used to build the virtual surgeries.

Figure 2.5. Estimated errors in initial direction with easy and hard virtual surgeries based on synergy
sets with N = 1,…,10.
Differences in estimated errors between easy and hard surgeries were significant up to N = 8 (p = 0.001,
paired t-test). Bars represent the mean estimated error across the 15 participants and error bars represent
the standard error of the estimated error. The significance of the difference between the estimated errors in
the hard and easy surgeries is indicated with asterisks on top of each pair of bars. ***: p < 0.0001, **: p <
0.001, and *: p < 0.005 The solid black line shows the mean across participants of R
2
, the reconstruction
quality of the baseline EMG signals used for the error estimation when considering N = 1,…,10.

Discussion

Muscle synergy extraction techniques require that combinations of the identified synergies
reconstruct the measured muscle activity to a heuristically defined level of accuracy, such as
accounting for at least 90% of the variance in the EMG. These techniques therefore attribute the

34

residual muscle activity not reconstructed by the identified synergies to noise. Here we studied the
importance of residual EMG activity in the execution of a virtual motor task. We designed the
virtual task based on a virtual surgery (Berger et al. 2013; de Rugy et al. 2012) and exploited the
property that virtual surgeries can produce equivalent muscle synergy-force mappings while
resulting in different individual muscle-force mappings. We tested two different virtual surgeries
that shared a common muscle synergy-force mapping, but differed maximally in their individual
muscle-force mappings (easy and hard virtual surgeries). The surgeries had the desired effect only
on the portion of the EMG signals explained by the extracted muscle synergies, defined to account
for at least 90% of the variability in the signal. Therefore, the effect on the residual EMG variability
was unspecified, allowing for a possible differential effect on the performance of the task.
We found that participants produced larger errors at the onset of the hard surgery than at the onset
of the easy surgery. We were able to predict this result qualitatively (Figure 2.3a) by estimating
the forces and errors that would be produced during each virtual surgery by using representative
EMG signals recorded during the baseline phase of the experiment and transforming the estimated
forces using the virtual surgeries. Importantly, this procedure also allowed us to separate the
recorded EMG signals and the estimated forces into their synergy and residual components (Figure
2.4). The virtual surgeries produced the expected effects on the synergy component of the EMG.
However, the easy surgery barely produced any changes on the direction of the forces associated
with the residual component, whereas the hard surgery produced large changes in the direction of
these forces (Figure 2.4). Given that the total force is equal to the sum of the synergy and residual
components, any difference between both virtual surgeries in the estimated force and error must
arise from the difference in the residual components. This provides evidence that the residual

35

component of the EMG is essential for accounting for our experimental results, suggesting an often
ignored organization in the residuals.
We also considered the alternative case in which the residual EMG activity is composed of noise.
In this situation, we posited that there would be no differential effect of the easy and hard surgeries
on the initial error, or that this effect would be small. To test this, we used the previously
decomposed EMG signals and shuffled the residual components among all these EMG samples.
This effectively destroyed any potential organization in the residual component, as they became
randomized. We found that the easy and hard surgeries did not produce significant differential
effects in the estimated initial error across participants when applied to the shuffled EMG signals.
This analysis suggests that the residual component of the EMG cannot be disregarded as purely
noise, and therefore demonstrates an underlying organization in the residuals.
Dimensionality reduction techniques such as NMF are useful for extracting patterns from high-
dimensional data sets, such as EMG recordings from multiple muscles. These techniques are
usually able to extract as many patterns or synergies as individual muscles. However, because
these are unsupervised techniques, there is no objective means for selecting the number of
synergies of interest a priori given the exploratory nature of the analysis and the lack of a ground
truth. Therefore, heuristic rules, such as selecting the number of synergies based on predefined
goodness of reconstruction criteria are needed (i.e., reconstructing the data to a given level of
accuracy, or finding an elbow in the goodness of reconstruction curve). These heuristic rules are
necessarily ad hoc, and are tailored to produce useful results in the domain of the studied problem
(James et al. 2013). Importantly, most of these heuristic rules result in the selection of a number
of synergies that is smaller than the number of synergies for which we found that the influence of
the residual muscle activity on the task becomes small (Figure 2.5). This suggests that some

36

important components of muscle activity are ignored in conventional muscle synergy analyses.
This is further illustrated by the fact that the number of analyzed muscles (Steele et al. 2013), and
the pre-processing of the EMG signals, particularly the selection of cut-off frequencies for the low
and high-pass filtering, can influence the selection of the number of muscle synergies in matrix
factorization (Hug 2011; Santuz et al. 2017; Shuman et al. 2017). This implies that relevant
residual activity could start being ignored as early as in the signal processing stage.
In addition, these heuristic rules ignore the role of muscle synergies in the generation of movement.
That is, muscle synergy extraction has mainly focused on describing muscle activity in the input
space, but has neglected the reconstruction of forces and movements in the task space (Alessandro
et al. 2013; Berret et al. 2019). To address this issue, a number of studies have incorporated task-
relevant constraints, such as force reconstruction, in the dimensionality reduction procedure (Ting
and Macpherson 2005; Torres-Oviedo et al. 2006). However, in these studies, assumptions of
linearity were made between muscle activations and forces, limiting generalizability. (Neptune et
al. 2009) took a simulation approach by using muscle synergy activity derived experimentally as
input to a computational biomechanical model to assess the goodness of the resulting movement
reconstruction. However, tuning of muscle activations during the simulations was necessary to
obtain favorable results. Further difficulties in the use of computational biomechanical models to
test for reconstruction of task space variables could stem from the difficulty of measuring EMG
from all muscles involved in a movement and of building sufficiently accurate musculoskeletal
models.
An alternative approach for studying the influence of extracted synergies on task-space variables
consists in using a virtual isometric task, such as in this and other studies (Berger and d'Avella
2014; Berger et al. 2013; de Rugy et al. 2013). Virtual tasks overcome the difficulty of obtaining

37

complex biomechanical models, as the model can be defined by the experimenter. This way, the
physics of the system are linear and known, and can be used in simulations in a straightforward
way. A previous study using this approach showed that the reconstruction of isometric forces in
an EMG-controlled task using muscle synergy decomposition is acceptable only when the number
of synergies is equal to the number of considered muscles (de Rugy et al. 2013). Otherwise, the
reconstruction quality quickly degrades even when the number of synergies is derived from widely
used heuristic rules (de Rugy et al. 2013). These results have been criticized because only five
wrist muscles were analyzed, which are considered few for reliable muscle synergy analysis, and
the only performance metric used was the final error, which is dependent on feedback mechanisms,
introducing a bias that may misrepresent offline reconstructions of the forces using muscle
synergies (Berger and d'Avella 2014). However, decreasing the number of synergies is associated
with larger residual components of the EMG, which we showed to play an important role in task
performance. Thus, our results further expand on the view posited by (de Rugy et al. 2013), with
the additional contribution of not being limited by a theoretical reconstruction of forces, but by
directly manipulating the contribution of the residual component of the EMG to isometric force to
highlight its importance in the execution of the task. This parallels results in hand gesture
recognition, where the features obtained through dimensionality reduction techniques that allow
classification of the gestures account for only small portions of the total variance in the data
(Marjaninejad and Valero-Cuevas 2019). These results emphasize the need of a shift within the
community in the criteria used to evaluate the goodness of muscle synergies extracted through
dimensionality reduction methods such as NMF.
Our results suggest that humans produce muscle activations that cannot be fully accounted for by
linear combinations of low-dimensional sets of muscle synergies, as extracted via NMF. This is

38

not in conflict with the notion that the CNS can use muscle synergies as building blocks of
movement embedded in neural circuits, as shown by numerous animal studies (Bizzi et al. 1991;
Caggiano et al. 2016; Mussa-Ivaldi et al. 1994; Tresch et al. 1999). Additionally, synergy control
in a myoelectric task (isolating msyn from r) has been shown to produce cursor trajectories similar
to control through individual muscles, showing that synergy control may be useful for myoelectric
interface applications (Berger and d'Avella 2014). However, this does not necessarily imply that
the CNS is limited to a small number of muscles synergies structurally defined in neural circuits.
It is entirely possible that the CNS readily learns and exploits a large number of task-dependent
muscle synergies, which may vary from individual to individual (Nazarpour et al. 2012). In this
sense, the role of muscle synergies could be viewed as a source of flexibility in the repertoire of
possible motor commands and stability in movement execution, as opposed to a way to simplify
the control of movement by limiting the number of control inputs (Latash et al. 2007).
Alternatively, muscle synergies could be the result of exploiting redundancy to produce optimal
movements (Todorov and Jordan 2002).
The main limitation of our study is that our experimental design was originally conceived to test
motor learning of virtual surgeries (to be presented in a follow-up manuscript). Therefore, the
virtual surgery trials were presented in blocks after transitions from baseline trials. This could have
induced a small amount of learning in the trials immediately following the perturbation or engaged
exploratory behaviors due to the saliency of the perturbation. These factors could explain why our
initial error estimates do not match the experimental data more closely, in addition to the variability
in force not accounted for by the EMG-force mapping. Nonetheless, because the observed
differential effect between both virtual surgeries was large and robust, the results of a study that
addresses these limitations would probably not be very different from our current results.

39

Another limitation is that we used a highly specific isometric task, but the results may be
generalizable to other isometric tasks involving leg or finger forces. However, this is also a strong
point, as our task manipulates the residuals to study their effects on the performance of the task,
which would be difficult to implement in tasks involving movement due to the need of an elaborate
model of the relationship between EMG and movement.
Because our results suggest that muscle activation residuals contain task-relevant activity even in
high-dimensional synergy sets, it is possible that a linear model of muscle synergies may be
insufficient to explain motor behaviors. Certainly, movement kinematics and dynamics are highly
non-linear, which implies that a linear controller would only be effective locally, and that synergies
themselves should vary accordingly with body kinematics and dynamics. Therefore, we think that
it would be beneficial to develop non-linear models of muscle synergies and tools for muscle
synergy extraction based on these models.
Overall, our results indicate that current muscle synergy identification techniques wrongly
attribute the fraction of unexplained variability in the EMG signals to noise. Our study is not able
to discern whether the organization of the residual component of the EMG is due to the inadequacy
of an additive linear model of muscle synergies, additional muscle synergies left out of the analysis
by the 90% variance or other heuristic rules, or other possible sources like individual muscle
control. Studies that achieve a high quality reconstruction of the measured EMG (R
2
> 0.98) while
preserving a low-dimensional set of synergies compared to the number of analyzed muscles may
not suffer from this problem. However, our results indicate the need for cautious interpretations of
results derived from synergy extraction techniques based on heuristics with lenient accuracy levels.
In particular, studies that aim to infer neural structures through EMG recordings should carefully
consider the role of the residual components of the EMG signals.

40

Chapter 3. Differences in learning rates after virtual
surgeries are determined by the statistics of post-
surgery movements

Authors: Victor R. Barradas, Yasuharu Koike and Nicolas Schweighofer

Abstract

A virtual surgery task is a myoelectric control task that simulates isometric force production after
a tendon transfer surgery by altering a baseline mapping between muscle activations and force.
The virtual surgery also interacts with muscle synergies, represented as modules whose linear
combinations produce the observed muscle activations, resulting in compatible or incompatible
surgeries. Successful learning of a compatible surgery task requires retuning of only the synergy
activation coefficients, while learning an incompatible surgery requires a complete change in the
structure of the synergies. Experimental evidence shows that learning in a compatible surgery task
is faster than in an incompatible surgery. This difference in learning rates has been attributed to a
hypothesized stability of muscle synergy representations in the Central Nervous System. Here we
use a computational model of the task to offer an alternative explanation based on the statistics of
post-surgery movements. Our model is based on learning an inverse model of the task through
direct inverse modeling. Importantly, our model does not include explicitly defined muscle
synergies in its structure. We show that each kind of surgery differentially affects the variance of
the forces produced by the motor system, which defines the shape of the cost function that direct
inverse modeling seeks to minimize through gradient descent. Incompatible surgeries are
associated with highly elliptical cost functions, in which learning proceeds slowly, whereas
compatible surgeries are associated with rounder cost functions, allowing faster learning.

41

Therefore, our findings indicate that differences in the speed of learning in the virtual surgery
experiment can be accounted for by differences in the post-surgery force statistics. This suggests
that muscle synergy stability may not be solely responsible for differences in learning rates in
compatible and incompatible virtual surgeries.

Introduction

Throughout their lives, humans must learn a variety of motor skills to interact successfully with
their surroundings. Growing evidence indicates that part of this learning process corresponds to
the adaptation and formation of internal models of the body and the environment (Golub et al.
2015; Kawato 1999; Wolpert et al. 1998). Internal models are neural circuits that can “simulate”
the physical relationships between motor commands and body states (forward internal models), or
issue suitable motor commands to generate desired body states (inverse internal models) (Wolpert
et al. 1998). Internal models are essential for the execution of skilled movements in which sensory
feedback is too slow to be useful. Therefore, internal models must adapt in the face of changes in
the environment or the body by integrating information about the changes so as to cancel their
effects.
In addition, it has been hypothesized that the central nervous system (CNS) groups the control of
functionally similar muscles into modules called muscle synergies (Lee 1984). This would reduce
the number of variables that the CNS needs to control to produce a movement, mitigating the
complexity of controlling the redundant muscle system (Bizzi et al. 1991), which could
theoretically activate in an infinitely large number of ways to produce the same joint torques.

42

Therefore, in the muscle synergy hypothesis, internal models of the body must capture the structure
of the muscle synergies to issue appropriate motor commands to reach desired body states.
A widely used paradigm to study motor adaptation is modifying the mapping between movements
and sensory feedback, as in an arm reaching task where movements are rotated or mirrored in the
visual scene. According to the muscle synergy hypothesis, adaptation to these type of tasks could
occur by modulating the spatial activation of existing muscle synergies, bringing about the
necessary changes in the sensorimotor mapping, and thus in the internal model of the task (Berger
et al. 2013). Therefore, it is reasonable to assume that learning a task that requires the acquisition
of new muscle synergies or the modification of existing ones will be a slower process than learning
a task that requires only a new sensorimotor mapping.
Indeed, (Berger et al. 2013) found this to be the case, in support of the muscle synergy hypothesis.
This was accomplished using the virtual surgery paradigm, which simulates tendon transfer
surgeries. The virtual surgery alters the pulling forces of arm muscles in a virtual mapping from
EMG to two-dimensional isometric force at the wrist, which affects performance during a reaching
task. Compatible virtual surgeries modify the synergy-force mapping such that the existing muscle
synergies still span the force space, requiring only a retuning of the sensorimotor mapping for
successful task completion. In contrast, incompatible surgeries are a class of virtual surgeries that
specifically alter the synergy-force mapping such that forces associated with synergy activity are
constrained along only one direction on the plane, impeding force production in the orthogonal
direction. Therefore, they require a re-structuring of the existing muscle synergies or the
acquisition of new synergies for successful task completion. As expected, directional errors in
reaching decreased at a slower rate during the incompatible surgery than during the compatible
surgery, reflecting slower learning.

43

Here, we present a computational model that reproduces the difference in learning rates in the
incompatible and compatible virtual surgery tasks. In our model an inverse internal model of the
virtual surgery tasks is learned using direct inverse modeling (Kuperstein 1988; Sanger 2004).
Learning takes place by mapping motor commands that the model generates to the actions that the
motor command produces after the surgery is introduced. This is implemented by minimizing the
error between the generated motor command and the motor command that the model would have
produced if the generated action had been its target (Kuperstein 1988; Sanger 2004).
The fundamental feature of this model is that it reproduces the difference in learning rates between
incompatible and compatible surgeries without explicitly defining muscle synergies in the
structure of the model. Instead the difference in learning rates is a consequence of differences in
the statistics of the forces produced after the onset of both kinds of virtual surgeries. Specifically,
the incompatible surgery produces forces with a highly predominant component in the direction to
which the forces associated with synergies were constrained, and with only a small component in
the orthogonal direction. In contrast, the compatible surgery produces forces that are more evenly
distributed in all directions, similar to the unperturbed task. In consequence, the variability of
produced forces during the incompatible surgery is greatly limited when compared to the
compatible surgery. Therefore, there is a difference in the quality of the samples from which the
internal model can learn, which can explain the differences in the speed of learning the virtual
surgery task.
Our results suggest that the observed differences in the speed of learning incompatible and
compatible virtual surgeries may not be exclusively due to the stability of muscle synergies and
the difficulty of acquiring new synergies. Differences in the variability of produced forces between
the tasks may also contribute to the learning rate difference. Therefore, it is possible that the size

44

of the originally observed differences due to muscle synergy stability has been overestimated, and
that the effect of a difference in produced force variability must be taken into account in subsequent
studies.

Methods

Overview. We adapted a previously developed model (Sanger 2004) to study learning during
compatible and incompatible virtual surgeries. The model simulates isometric production of planar
forces by an arm given different target directions in a task analogous to a study by (Berger et al.
2013). Importantly, we reduced the output force to its direction to focus exclusively on
feedforward control. The model allows the introduction of virtual surgeries to simulate the learning
process of correcting the error produced by the virtual surgery. These computations are performed
by an inverse model of the arm, representing the CNS, which generates motor commands for the
arm given a desired force direction. During learning after the onset of the virtual surgery, the
inverse model attempts to reduce errors in the produced force direction by minimizing errors in
the motor command. This has the effect of remapping motor commands to the actual force
directions they produce, instead of the force directions expected before the virtual surgery. This
process corresponds to a direct inverse modeling approach (Jordan and Rumelhart 1992).
Model structure and implementation. The model learns an inverse model f
-1
of f, the motor
system after applying a virtual surgery (Figure 3.1a). A target with direction θd is taken from the
space of all possible targets Θd. The inverse model f
-1
receives θd

to generate a motor command m.
The motor command m is transformed by the virtual surgery T (if the virtual surgery is applied –
T is the identity otherwise), and then propagates through the forward model f. This produces a

45

realized direction θr in the space of realizable directions Θr. Notice that in general, Θr is a subspace
of Θd. Therefore, θr may not be the same as θd, which produces a sensory error. Then, f
-1
receives
θr, so that f
-1
(θr) is the motor command mθ that the inverse model would have produced if the target
direction had been θr. This allows the computation of the motor error m - mθ, which is used for
learning the inverse model, where m is used as the motor command that should be associated to
θr.

Figure 3.1. Structure of the computational model.
a. Model diagram. b. Structure of the internal model. The input is an RBF representation of a direction θ.
The internal model is a single-layer network and is fully defined by W, the weights of the connections
between the input and the output.

Specifically, following application of a virtual surgery, the motor system f is represented as:
𝐟 (𝐦 ) = 𝑎𝑡𝑎 𝑛 2(𝐌𝐓 σ(𝐦 )) (3.1)
where M is a muscle activation to force mapping, T is the virtual surgery, and m is a motor
command vector produced by the inverse model (see next section). These variables have the same
dimensions as their experimental counterparts described in the previous section. A sigmoidal
transformation σ(m) converts each element of m into a value between 0 and 1. This ensures that
the muscles in the model produce only pulling forces as defined by M.
The inverse model f
-1
is given by a weighted sum of radial basis functions:

46

𝐟 −𝟏 (θ) = 𝐖𝛟 (θ) (3.2)
ϕ
𝒊 (θ) = exp(−ln(2)(
|θ
𝒄𝒊
−θ|
𝝎 /𝟐 )
𝟐 ) (3.3)
where θ is the direction of the isometric force produced by the arm, W is a 10 × N matrix of
weights, Φ(θ) is a column vector of N radial basis functions evaluated at θ, ϕi(θ) is each one of N
radial basis functions with centers equally spaced around the workspace evaluated at θ, θci is the
center of each radial basis function, and ω is the full width at half maximum of the radial basis
function. We set N = 50 in the simulations.
Learning. The learning process aims to minimize the motor error given by the quadratic cost
function
J =
1
2
‖𝐦 − 𝐖𝛟 (θ
r
) ‖
2
(3.4)
by finding an appropriate set of weights W. A local minimum of J can be found by using a gradient-
descent learning rule, given by
𝐖 𝒏 +𝟏 = 𝐖 𝒏 + η(𝐦 − 𝐖 𝒏 𝛟 (θ
r
))𝛟 (θ
r
)
T
(3.5)
where Wn is the weight matrix at the current trial, Wn+1 is the weight matrix at the next trial, and
η is the learning rate.
Estimation of maximum achievable learning rates. In the general case, a given cost function
can be approximated as a second order Taylor expansion around the minimum of the cost function.
J(𝐰 ) = J(𝐰 ∗
) +
1
2
(𝐰 − 𝐰 ∗
)
T
𝐇 (𝐰 − 𝐰 ∗
) (3.6)
𝐇 ij
=
∂J
∂𝐰 i
∂𝐰 𝐣 (3.7)

47

where w is a vector of learned weights, w* is the vector of weights at the minimum of the cost
function, and H is the Hessian of the cost function, a matrix of second derivatives of the cost
function with respect to w. To align the coordinate system given by the weights w with the
principal axes of the cost function, a coordinate transformation of the cost function can be achieved
by the eigen-decomposition of H
J(𝐯 ) = J(𝐯 ∗
) +
1
2
𝐯 T
𝚲𝐯 (3.8)
𝐯 = 𝐔 (𝐰 − 𝐰 ∗
) (3.9)
where Λ is a matrix containing the eigenvalues of H in its diagonal and U is a matrix containing
the eigenvectors of H in its columns. This transformation aligns the original axes of w with the
eigenvectors of H under the new coordinates v. Therefore, the update equation of v using gradient
descent is given by
𝐯 𝐧 +𝟏 = 𝐯 𝐧 + η𝚲𝐯 (3.10)
Where η is a learning rate parameter. Therefore, the convergence rate of the gradient descent
algorithm to a minimum in J is determined by the dynamics of equation 3.10. Because Λ is a
diagonal matrix, the dynamics of learning are decoupled along each axis of v. In consequence, the
convergence rate to a minimum around the vicinity of the minimum is limited by the slowest time
constant in equation 3.10, which corresponds to λmin, the smallest eigenvalue of Λ and H (Bishop
1995). Under these circumstances, the fastest possible convergence rate is a function of λmin/λmax,
the ratio of the largest and smallest eigenvalues of H (Bishop 1995). If both eigenvalues have
similar magnitudes, the shape of the cost function J is relatively spherical, and convergence to the
minimum can be relatively fast. On the other hand, a large disparity in the magnitudes of these

48

eigenvalues corresponds to a highly eccentric ellipsoidal J, and convergence to the minimum will
be comparatively slower (Figure 3.2).

Figure 3.2. The shape of the cost function influences the convergence rate to the minimum in gradient
descent.
This illustration shows a cost function with only two adaptive parameters w 1 and w 2. We used gradient
descent to find the minimum in two different cost functions with different shapes, where one of the cost
functions is more spherical than the other, indicated by the level curves. This can be described by the ratio
λ min/λ max, where a larger value corresponds to a more spherical shape. We set the learning rates to a value
that is close to the maximum possible learning rate. Convergence to the minimum is reached in fewer
iterations in the more spherical cost function.

We aimed to calculate λmin and λmax of H in order to determine the fastest possible learning rate
during the virtual surgery tasks in our model. Let us assume that our network has only one output,
the activation of muscle n. Then
J =
1
2
‖𝐦 − 𝐰 𝒏 𝛟 (θ
r
) ‖
2
(3.11)
where wn is the row in the original matrix of weights W corresponding to output n. The gradient
for J is then
𝛁 J = (𝐦 − 𝐰 𝒏 𝛟 (θ
r
))𝛟 (θ
r
)
T
(3.12)
Taking the derivative of the gradient ∇J with respect to wn results in the Hessian H.
𝐇 = 𝛟 (θ
r
)
T
𝛟 (θ
r
) (3.13)

49

Because in general there are many realized directions θr used as training inputs, the Hessian
corresponds to the average of the Hessian matrices obtained for each training input (Lecun et al.
1991)
𝐑 =
1
p
∑ 𝛟 (θ
r𝑖 )
T
𝛟 (θ
r𝑖 ) ≅ 𝐇 p
i=1
(3.14)
where p is the number of inputs presented to the network and R is an N × N matrix, which
correspond to the sample covariance matrix of the inputs Φ(θr). Therefore, for a single layer
network with one output, H is solely a function of the inputs to the network, and can be
approximated by R, the sample covariance matrix of the inputs (Lecun et al. 1991).
Our model is a single layer network, but it has N outputs corresponding to the muscle activation
of each muscle. However, because in a single layer network the weights wn that produce the output
n do not affect the rest of the other outputs, the cost function J can be reformulated as the sum of
the cost functions for each individual output as represented in equation 3.11.
𝐽 =
1
2
[‖𝐦 − 𝐰 𝟏 𝛟 (θ
r
) ‖
2
+ ‖𝐦 − 𝐰 𝟐 𝛟 (θ
r
) ‖
2
+ ⋯ + ‖𝐦 − 𝐰 𝒏 𝛟 (θ
r
) ‖
2
] (3.15)
Therefore, it is possible to optimize each subordinate cost function separately. Furthermore, the
Hessian matrix is identical for every subordinate cost function because the Hessian matrix is
independent of wn. In consequence, the learning dynamics are the same for all outputs of the
network. Therefore, we can investigate the dynamics of learning by limiting our analysis to the
Hessian matrix of the one input network, for which we aimed to compute λmin and λmax of R
(equation 3.14).
Because the virtual surgeries transform the output forces of the network, the distribution of θr used
as inputs for learning changes accordingly. Therefore, we analyzed the resulting covariance

50

matrices after applying Ti and Tc to estimate the fastest possible learning rates under each
condition (see next section).
Because we found that some of the obtained R matrices were ill-conditioned, which results in
unreliable computations of the smallest eigenvalues. Therefore, we used a different metric based
on a more compact representation of the network inputs Φ(θr). Φ(θr) is a high-dimensional
representation of θr, which is merely a one-dimensional variable derived from two-dimensional
force vectors g. Since both Φ(θr) and g representations contain the same information about θ, we
used these g forces as a representation of the network input, and computed a low-dimensional
covariance matrix R’
𝐑 ′ =
1
p
∑ 𝐠 𝑇 𝐠 p
i=1
(3.16)
We then obtained λmin/λmax of R’ for each simulated subject under all surgery conditions, which
provided us with estimates of the maximum learning speeds for each condition..
Muscle activation-force mapping. Forces produced at the hand with the arm in a static posture
can be approximated as a linear function of the activations of muscles that actuate the shoulder
and elbow (Berger et al. 2013):
𝐠 = 𝐌𝐦 (3.17)
where g is a two-dimensional force vector, m is a ten-dimensional vector of muscle activations,
and M is a 2 × 10 matrix, in which each column indicates the force produced by each muscle,
which is scaled by m.

51

We created 15 simulated subjects based on 15 different M mappings. These mappings were
obtained experimentally from human subjects during an isometric reaching task, as described in
(Barradas et al. 2019).
Baseline model initialization. In order to perform the simulations, initial values of weights W of
the network are required. With these initial values, the model must be able to produce forces
corresponding to the desired target, and therefore produce plausible muscle activations.
We obtained the baseline state for the network by assuming that humans approximate an optimal
controller that minimizes effort to generate isometric forces in different directions (Fagg et al.
2002). Effort is defined as the sum of squared muscle activations when generating a force:
u = ∑ m
i
2 10
i
(3.18)
where u is the effort term, and mi are the muscle activations for each muscle. Therefore, we found
the set of optimal muscle activations that produced target forces in the horizontal plane ranging in
direction from 0 to 360° and in magnitude from 1 to 50 N. The optimization procedure was
implemented by solving a quadratic program with the effort term as the minimization objective
(equation 3.18), equation 17 as an equality constraint, and the non-negativity of muscle activations
m
i
≥ 0, 𝑎𝑠 as inequality constraints. We confirmed that the optimal muscle tuning curves correlate
reasonably well to the muscle activation curves computed in previous experiments (Barradas et al.
2019) (see Appendix).
Subsequently, we used the derived muscle activations as well as the radial basis representation of
their associated force direction (equation 3.3) as training pairs to train the RBF network in equation
2 to derive the network weights. For this step we trained the network using a standard gradient-
descent based procedure to find a set of network weights W for the minimization of the squared

52

error between the optimal muscle activations and the muscle activations produced by the network.
Figure 3.3 shows the steps to obtain the initial state of the model m opt

Figure 3.3. Process for initializing the inverse model f
-1
.
First, EMG-force mappings are taken from experimental data (Barradas et al. 2019) and used as the M
mapping in f. Next, optimal muscle activations m opt to produce forces in all directions θ of the two-
dimensional space are computed assuming that effort is minimized. Then, f
-1
is trained using the force
directions θ and desired muscle activation from the optimal computation m opt in the previous step, providing
the baseline network weights W. The network is trained using gradient descent to minimize the error
between the output of the network m and the optimal muscle activation m opt. Next, muscle synergies S are
extracted from the muscle activations m that f
-1
produces in response to desired directions that span the
two-dimensional space using NMF. Finally, incompatible T i and compatible T c virtual surgeries are built
based on M and S using the methods detailed in (Berger et al. 2013)

Muscle synergy extraction. We used non-negative matrix factorization (NMF) (Lee and Seung
1999) to extract muscle synergies from the model by using the set of muscle activations it produced
after a forward pass through the network when setting the desired reaching direction to different
targets from 0 to 360°. The synergy extraction can be represented as
𝐦 = 𝐒𝐜 + 𝐫 (3.19)
where S is a 10 × N matrix that contains the identified synergies in its columns with N being the
number of synergies, c is an N-dimensional vector of synergy activations and r is a vector of
residuals to account for portions of m not accounted by Sc. We selected N using a series of
heuristic rules as specified in previous studies (Barradas et al. 2019; Berger et al. 2013).
Virtual surgeries. Following the methods described in (Berger et al. 2013), we designed virtual
surgeries that were either incompatible or compatible with the muscle synergies extracted by

53

nonnegative matrix factorization (NMF) . A virtual surgery modifies the EMG-force mapping (M)
by applying a linear transformation in muscle space:
𝐌 ′ = 𝐌𝐓 (3.20)
where T is a 10 × 10 matrix that constitutes the transformation or virtual surgery.
Incompatible virtual surgeries are designed such that muscle activations m produced by synergy
combinations Sc are restricted to generate forces along only one dimension of the force space,
while the resulting EMG-force mapping M’ spans the whole force space. Therefore, theoretically,
any force can still be produced by a new combination of muscle activations m’, but in practice,
produced forces are biased towards one dimension of the plane. In contrast, compatible virtual
surgeries are designed such that muscle activations m produced by synergy combinations Sc span
the whole force space.
For each simulated subject, we designed an incompatible and a compatible surgery. We designed
both surgeries according to the methods described in (Berger et al. 2013). The incompatible
surgeries were built by finding Ti that maps muscle activations m in the column space of S into
the null space of M. On the other hand, the compatible virtual surgeries were built by finding Tc
that maps m into muscle activations contained in a space outside of the null space of M. More
details can be found in (Berger et al. 2013). We randomly generated both incompatible and
compatible surgeries and applied them to the output of the inverse models to calculate the initial
average error under that particular surgery. We selected surgeries that produced an initial average
error close to 60°. This was decided based on previously reported results for the mean initial error
during a compatible virtual surgery (Berger et al. 2013).

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Simulation procedure. We ran two sets of simulations for each simulated subject using the Ti and
Tc virtual surgeries. Each simulation trial was defined as one cycle of target presentation, force
production, propagation of the generated direction through the inverse model, and a learning step.
Targets were the same as in the experimental procedure. We used the M and T matrices computed
from 15 experiment participants to define 15 virtual subjects.
The simulations consisted of a baseline phase with 72 trials and a virtual surgery phase with 288
trials, following the protocol defined in (Berger et al. 2013), except for the washout phase. Trials
consisted in reaching to one of eight equally spaced directions in the plane. The target order in
each simulations was randomly generated within sets of eight targets. The simulation protocol for
the compatible and incompatible surgeries was the same.
We set the number of RBFs to N = 50. The model takes two parameters: ω and η, which are the
full width at half maximum of the RBFs and the learning rate, respectively. For each parameter,
we defined a set of possible values, and ran simulations for all virtual subjects using all possible
combinations from the defined set of parameter values. The defined parameters values were ω v =
[5° 10° 15° 20° 30° 40° 50° 60° 80° 100°] and ηv = [0.01 0.015 0.02 0.05]. For each simulated
subject and parameter combination, we ran 10 different simulations with a different target order,
and averaged the obtained learning curves.
Learning rates in simulations. We averaged the error in direction for every set of eight targets in
order to obtain smoothed learning curves. For each simulated subject and parameter combination,
we then averaged the smoothed learning curves for all 10 simulation runs. We fitted the simulated
initial direction error curves for each condition with first order exponentials and used the learning
rate parameter to compare it to the maximum learning rate estimates computed beforehand.

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Results

Simulated subjects adapt to a higher degree to compatible surgeries than to the
incompatible surgeries
We created 15 simulated subjects based on experimental data that produced muscle activations
similar to humans (Appendix), and we ran simulations for each simulated subject under each
surgery condition based on the learning rule in equation 3.5.
Figure 3.4 shows an example of force production for one simulated subject before, at the onset and
at the end of training for each kind of virtual surgery. At the surgery onset the incompatible surgery
produces force patterns with a component that is predominantly aligned with the line of action of
the virtual surgery. In contrast, the easy surgery produces a force pattern that is similar to a rotation
of the original force axes, and spans the force space more uniformly than the incompatible
surgeries. By the end of training the initial force direction error is reduced with respect to the
surgery onset in both cases. However, the extent to which errors were reduced during the
compatible surgery is much larger than during the incompatible surgery. This suggests a faster
learning rate during the compatible surgery.

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Figure 3.4. Forces at each target before, at the onset and at the end of training during the incompatible and
compatible virtual surgeries for a representative simulated subject.
Each colored line represents the force produced by the model for the eight different targets.

Learning compatible virtual surgeries is faster than learning incompatible
surgeries
We found the combination of the RBF width (ωv = 30°) and the learning rate parameter (ηv = 0.02)
that provided the minimum error fit to the direction error experimental data in (Berger et al. 2013)
across all simulated subjects (R
2
= 0.91). Figure 3.5b shows the average error in force direction
during each surgery across all simulated subjects for this parameter combination. The compatible
surgery shows faster learning than the incompatible surgeries. This indicates that our model
adequately captures the differences in learning rate between the incompatible and compatible
surgeries. Other parameter combinations produce qualitatively similar results.

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Figure 3.5. Motor and performance error during the virtual surgery tasks.
The learning rate parameter η v was fixed at 0.02 and the width of the RBF was fixed at 30° for all simulated
subjects, as this was the best fit to the experimental data (Berger et al. 2013). a. Motor error during learning
compatible and incompatible virtual surgeries. The motor error corresponds to the value of the cost function
J (equation 3.4) and is the quantity minimized by direct inverse modeling. As predicted, the motor error
decreases at a lower rate during the incompatible surgery. b. Simulated and real initial direction error during
learning compatible and incompatible virtual surgeries. However, many other parameter combinations
show qualitatively similar results. Error bars indicate the standard deviation. Dotted lines show data
published in (Berger et al. 2013) for the incompatible and compatible surgeries.

Larger learning rate in learning compatible than incompatible virtual surgeries
We analyzed the covariance of the low dimensional representations of the directions of the forces
that the simulated motor system produced after the application of both the compatible and the
incompatible surgeries (equation 3.1). As detailed in the Methods section, the ratio of the smallest
and largest eigenvalues λmin/λmax of the covariance matrix R’ provide an estimate of the largest
possible learning rate during each virtual surgery, as these eigenvalues describe the shape of the
cost function (equation 3.4).
Thus, we calculated the λmin/λmax ratio of the covariance matrix R’ of forces g for each simulated
subject before and after applying the compatible or the incompatible surgeries. Figure 3.6 shows

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an estimation of the probability density associated with the λmin/λmax ratio under each surgery
condition. The λmin/λmax ratio for the incompatible surgery condition concentrates at zero,
predicting that learning is slow in this condition, whereas the compatible surgery produces
eigenvalue ratios that are considerably larger than the incompatible surgery, predicting that
learning is faster in this condition.

Figure 3.6. Probability density distributions of the eigenvalue ratio before the surgeries and at the onset
of both surgeries.
The eigenvalue ratio is used to predict differences in the speed of learning between compatible and
incompatible virtual surgeries. It is based on the λ min/λ max ratio of the covariance matrix R’ of the two-
dimensional forces g produced by the motor system. While both kinds of surgeries produce smaller
eigenvalue ratios than the baseline condition, the incompatible surgery distributions have a sharp peak close
to zero, which is associated with slow learning. Therefore, in general, faster learning is expected for the
compatible surgeries.

We then compared the estimated maximum learning speed calculated via the eigenvalues of the
low dimensional R’ covariance matrix with the actual speed of learning in the simulations for each
surgery condition. For each simulated subject we estimated the learning rate under each surgery

59

condition via an exponential fit to the error in initial direction curve. Figure 3.7 shows the
relationship between λmin/λmax and the estimated learning rate for each simulated subject (R
2
=
0.75). As for Figure 3.5, we set the RBF width ωv = 30° with a fixed learning rate parameter ηv =
0.02.

Figure 3.7. Predicted maximum learning rates vs estimated learning rates.
The black line indicates the results of a robust linear fit. The estimated learning rate is proportional to the
maximum predicted learning rate with the λ min/λ max eigenvalue ratio across all simulated subjects.

As predicted, learning was faster during the compatible surgeries than during the incompatible
surgeries, as compatible surgeries were associated with larger λmin/λmax ratios. There is a
reasonably good linear correspondence between the ratio of the eigenvalues and the speed of
learning in the simulation. This indicates that the variability of the force directions θ r used as
learning inputs to the inverse model produced by the incompatible surgery is less uniform in all
dimensions than the compatible surgery, which produces a highly ellipsoidal cost function

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associated with slow learning. This supports our hypothesis that the differences in learning rates
observed in incompatible and compatible surgeries could arise from differences in the variability
of the inputs to the internal model.

Discussion

Experiments based on the virtual surgery paradigm have shown that subjects performing an
isometric reaching task after undergoing an incompatible virtual surgery learn to perform the task
at a slower rate than when subject to a compatible virtual surgery (Berger et al. 2013). Incompatible
surgeries are built such that they constrain synergetic muscle activations to produce forces along
only one dimension. Therefore, breaking out of the existing muscle synergies is necessary to
achieve good performance in the task. On the contrary, compatible surgeries do not impose such
constraints in force production, requiring only a retuning of the existing muscle synergies for
completion of the task. Based on these observations, the differences in learning rates between
incompatible and compatible surgeries has been interpreted as evidence for the muscle synergy
hypothesis. Here we show that both kinds of surgeries affect the statistics of the post-surgery
movements in a differential way. Incompatible surgeries allow very little variance in the forces
that are perpendicular to the direction in which they constrain force production, whereas
compatible surgeries allow similar variances in all dimensions of force. We use a computational
model based on direct inverse modeling to simulate the virtual surgery tasks, and show that these
unbalanced variances in the dimensions of force production can also account for the differences in
learning rates in the two tasks. Importantly, our model is able to account for the observed difference
in learning rates without explicitly defining muscle synergies in its structure. This weakens the

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evidence that the incompatible and compatible surgery experiment provides in favor of the muscle
synergy hypothesis.
The muscle synergy hypothesis posits that the CNS groups functionally similar muscles into
modular control units to simplify the control of movement, since the body has exceedingly more
degrees of freedom than required to perform a given action. Numerous evidence in animal models
indicates that the stimulation of different groups of motor neurons in the spinal cord leads to a
superposition of the force fields that each group of neurons would produce when stimulated in
isolation (Bizzi et al. 1991; Caggiano et al. 2016; Mussa-Ivaldi et al. 1994; Tresch et al. 1999).
This indicates that muscle synergies may be implemented in spinal circuits and triggered by the
motor cortex and other supraspinal structures (Overduin et al. 2012).
Due to the highly invasive nature of the methods used in animal models, it has proven difficult to
obtain direct evidence of muscle synergies in humans. Most methods to study muscle synergies in
humans involve the use of statistical learning techniques that aim to identify low-dimensional
patterns of muscle activity recorded by means of EMG during a motor task. However, it is not
clear whether the extracted low-dimensional patterns arise from modular neural circuits or whether
they arise from biomechanical constraints in the performed tasks (Burkholder and van Antwerp
2013; Kutch and Valero-Cuevas 2012; Steele et al. 2015).
Recently, a clever experiment using an EMG-computer interface showed that a task that requires
learning new synergies to be performed (the incompatible virtual surgery) is learned at a slower
rate than a task that simply requires the retuning of existing synergies (the compatible virtual
surgery). Assuming that muscle synergies arise from considerably stable neural circuits, this
experiment supports the idea that the identified low dimensional patterns of activity using
statistical learning techniques are a consequence of neural circuits representing muscle synergies.

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However, we show through simulations with a computational model of the virtual surgery tasks
that the observed differences in learning rates between the compatible and incompatible virtual
surgeries could arise because of differences in the statistics of the virtual movements produced
after the introduction of the surgeries. Concretely, the incompatible virtual surgeries produce
movements with directions that are highly biased towards the line of action of the surgery. Thus,
subjects do not experience enough movements along the perpendicular axis, which results in an
overall lower quality of the sample movements available for learning. This ultimately results in
slow learning of the task. These issues are not present during the compatible surgery, where the
samples available for learning are more varied, allowing faster learning.
The computational model that we used is based on a direct inverse learning approach to learn
inverse models. Direct inverse modeling is a good learning strategy in the absence of an
approximate internal model of the task that can be used to extrapolate a more accurate model. This
is especially relevant in the virtual surgery task, which produces a change in the virtual physics of
the arm, for which the existing model is generally inadequate. Furthermore, mathematically, a
direct inverse modeling approach minimizes errors in the action space (motor error), and
corresponds to learning a left inverse model of the task. On the other hand, learning a task by
minimizing errors in the observation space (sensory error) corresponds to learning a right inverse
of the task. A previous study showed that, in the linear case, the gradients in the action and
observation spaces are guaranteed to deviate from each other by less than 90 degrees, and therefore
it is possible to concurrently minimize the sensory error and the motor error, obtaining a right
inverse, by using direct inverse modeling (Rolf and Steil 2014). Therefore, for a linear system, a
direct inverse modeling approach is indistinguishable from a sensory error minimization learning

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scheme, for instance, in a distal learning scheme, in which a forward model is used to provide the
gradient for learning an inverse model.
Other learning strategies including reinforcement learning and random exploration may be
possible. In fact, in our direct inverse modeling approach it is possible to include exploration noise
in the desired direction of force, which increases the speed of learning in both the incompatible
and compatible virtual surgeries, as this leads to a richer variability in the input to the model,
facilitating learning. However, we observed that the difference in learning rates between surgeries
remains qualitatively similar. Another possible random exploration scheme could introduce motor
noise in the output. In this case, a muscle synergy based model could reproduce the difference in
learning rates. This is because random exploration in the space of synergy inputs would only
produce forces that lie on the line of action of the incompatible surgery, making it difficult to learn
the task.
Even though our model does not contain explicitly defined muscle synergies, low-dimensional
patterns can be discerned by applying dimensionality reduction techniques to the output of the
obtained optimal controller. As in an experimental setting, we based the selection of the number
of extracted synergies on the heuristic rule that the synergies must reconstruct at least 90% of the
variance in the muscle activation signals. This usually leaves a residual portion of the variance in
the muscle activations which muscle synergies cannot account for. Therefore, the effects of the
virtual surgeries on the residual component of the signal are different from those on the synergy
component, which were specified when designing the surgeries (Barradas et al. 2019). We
therefore presume that the residual component of the signal is responsible for the small amount of
variability in the perpendicular direction of the incompatible surgery. Increasing the number of
synergies in the muscle synergy extraction would decrease the size of the residuals, which would

64

be associated with even a smaller variability along the perpendicular direction. In consequence,
we posit that the residuals are responsible for the small amount of learning during the incompatible
surgery. Therefore, a controller that can only produce synergetic activity would be unable to
improve performance even a modest amount, as the covariance matrix of the produced forces
would be singular, making the ratio of its eigenvalues zero.
Our computational approach may be useful to model tendon transfer surgeries. Tendon transfers
are a common surgical intervention to treat paresis in disorders such as spinal cord injury (Kozin
et al. 2010). These surgeries are generally greatly beneficial to the patients, but often times they
are not carried out due to concerns about whether the patient will be able to adapt to the new tendon
configuration. Therefore, our modeling framework could be used to make predictions on the speed
of adaptation to the tendon transfer surgery and to suggest exercises to promote faster adaptation.
Other inverse modeling paradigms such as feedback-error learning (Kawato and Gomi 1992),
distal learning (Jordan and Rumelhart 1992) and reinforcement learning exist. Feedback-error
learning is based on building inverse internal models by using corrective feedback signals as
training inputs to the inverse model. However, this assumes that there is some knowledge about
the motor system that allows for effective feedback corrections. In the case of the virtual surgery
task, this may not be possible because the virtual physics of the task change, which could make
the feedback corrections ineffective. On the other hand, reinforcement learning paradigms explore
a range of motor command combinations to find solutions to the task. However, for a large number
of muscles, the exploration space may be too large to explore. If the exploration strategy relies
instead on exploring combinations of synergy activations, then it might be feasible to explore the
whole space. However, there may be no learning during the incompatible task because this strategy
would not have a mechanism to form new muscle synergies without making ad-hoc adjustments

65

to the model. Therefore, in the face of a change in the virtual physics of the motor system, and a
high dimensional motor command space where exploration might be unfeasible, direct inverse
modeling is a good strategy for motor learning.

Appendix

Muscle tuning curves: experimental vs minimum effort simulation

In building the inverse models f
-1
we assumed that isometric forces produced by the arm result
from a minimum effort control strategy in a similar way in which it has been proposed for forces
produced by the wrist joint (Fagg et al. 2002). In Figure 3.8 we show normalized tuning curves of
ten muscles (trapezius, posterior deltoid, middle deltoid, anterior deltoid, pectoralis major, triceps
long head, biceps, triceps lateral head, brachioradialis and pronator teres) averaged across 15
subjects obtained in a previous experiment (Barradas et al. 2019) and the minimum effort
normalized tuning curves obtained from our optimal control simulation, as described in Methods.
The simulated tuning curves closely match the experimental tuning curves in most cases (R
2

=0.72), although for two of the muscles (trapezius and anterior deltoid) the fit is poor. This could
be due to a high variability in the tuning curves of these muscles in the experimental subjects.
Additionally, the optimal control simulation did not include any free parameters that could be
adjusted to better fit the data. Based on the similarity of the muscle activations produced by the
simulation and the observed data, we determined that the simulated inverse models produced
adequate motor outputs for the simulation of the incompatible and compatible virtual surgeries.

66

Figure 3.8. Experimental and simulated muscle tuning curves for ten muscles.
The simulated optimal muscle activations used to initialize our model closely resemble the experimental
data, indicating that the model produces adequate muscle activations for further learning simulations.

Maximum learning rate predictions and their relationship with failure of
learning

Let us assume a system in which the motor system f, defined in equation 3.1, is a modular
controller, such that σ(m) can be expressed completely in terms of linear combinations of the
modules, or synergies, as in equation 3.19 minus the residuals. Then the motor system can be
represented as
𝐠 = 𝐌𝐓𝐒𝐜 (3.21)
where c is the output of the inverse model f
-1
in response to a desired force g*, and is an N-
dimensional vector with N equal to the number of muscle synergies in the system.

67

We can estimate the fastest possible rate of convergence by investigating the covariance matrix of
a set of g, which defines the statistics of the inputs to the learning system based on equations 3.16
and 3.21:
𝐑 ′
=
𝟏 𝒑 ∑ (𝐌𝐓𝐒 𝐜 𝒊 )
T
𝐌𝐓𝐒 𝐜 𝒊 𝒑 𝒊 =𝟏 (3.22)
where p is the number of input samples, and R’ is the covariance matrix of g forces.
In the case of an incompatible surgery the product TS is rank-deficient by design. Consequently,
R’ is also rank-deficient and the ratio λmin/λmax is zero. This predicts that learning the incompatible
surgery will be infinitely slow in the dimension of the eigenvector corresponding to λmin, which is
equivalent to failure of learning. This is in agreement with the failure of learning framework
(Sanger 2004), which states that failure of learning occurs when the motor system and the internal
model form a projector operator, as is the case with TS. This allows the motor error (equation 3.4)
to be minimized, but does not translate to a reduction in the sensory error. Under this view, failure
of learning occurs because there is no variability in one of the dimensions of the input, making it
impossible to learn.

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Chapter 4. Gradual and abrupt virtual surgeries elicit
different muscle activation strategies

Authors: Victor R. Barradas, Toshihiro Kawase, Woorim Cho, Yasuharu Koike and Nicolas
Schweighofer

Abstract

Large errors in task performance when attempting to learn a new motor task may result in failure
to learn. This can be mitigated by gradually adjusting the difficulty of the task such that initial
errors are small, which facilitates error correction. An incompatible virtual surgery is a task in
which the mapping from EMG signals from a group of muscles to virtual forces used to control a
virtual object is arbitrarily altered from an intuitive configuration to a configuration in which new
muscle synergies must be learned to achieve good performance in the task. Learning in this task is
usually slow and incomplete because the abrupt introduction of the surgery induces large errors in
performance. Incomplete virtual surgeries also promote increased joint co-contraction and
stiffness, as the surgery tends to restrict forces along one dimension, and subjects attempt to break
this restriction. Here we compare the effects of introducing the incompatible surgery abruptly and
gradually. Because the initial errors are small when introducing the surgery gradually, we
hypothesized that the gradual surgery would allow to reach better performance levels than the
abrupt surgery. We also expected that better performance in the task would be associated with
reductions in joint stiffness, as has been reported for a number of motor tasks. However, we
unexpectedly found no differences in final performance between the abrupt and gradual groups,
but better performance was indeed associated with lower joint stiffness. Furthermore, the abrupt
group was more efficient than the abrupt gradual in the sense that better task performance was

69

achieved with less joint stiffness, indicating different strategies for muscle activation between the
two groups.

Introduction

Humans are remarkably adept at acquiring new motor skills and adapting existing motor skills to
new environments. One of the mechanisms responsible for the attainment of motor skills is error-
based learning, which uses information about the error in previous task attempts to produce actions
that cancel out the error in subsequent attempts. However, errors that are too large can result in
failure to learn the task because they do not provide relevant information for error correction
(Sanger 2004). Such large errors could be avoided by a coach guiding the learner’s movements
toward the desired outcome (Sanger 2004), or by gradually increasing the difficulty of the task
such that initial errors are small (Sanger 1994).
Furthermore, gradually introducing a visuomotor rotation results in larger after-effects, which
suggests a more elaborate internal model of the task (Kagerer et al. 1997). This may be because
the central nervous system (CNS) attributes errors caused by gradual perturbations to changes in
the properties of the body, while it attributes the effects of abrupt perturbations to the environment
(Kluzik et al. 2008). Gradually introduced perturbations also facilitate the retention of the learned
motor skills in more general contexts (Kluzik et al. 2008), which could make it a valuable resource
in the area of rehabilitation. Additionally, the effectiveness of gradual protocols for learning has
been demonstrated in sensorimotor transformation tasks in adult owls (Linkenhoker and Knudsen
2002), phoneme discrimination in children with language learning deficits (Tallal et al. 1996), and
learning of Morse code (Clawson et al. 2001).

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Here we examined the effects of applying a perturbation gradually in the context of an EMG-
computer interface task, in which the position of an object in a virtual environment is controlled
by applying virtual forces on it derived from an EMG-force mapping. The EMG-force mapping
produces force estimates based on the activation of muscles, as measured by EMG. The
perturbation, also called an incompatible virtual surgery (Berger et al. 2013), consists of altering
the EMG-force mapping in such a way that new patterns of muscle activations, or muscle
synergies, are needed to overcome the perturbation.
Incompatible virtual surgeries are hard to learn, which manifests in slow learning rates (Berger et
al. 2013) and in incomplete learning even after long practice sessions. Therefore, we hypothesized
that the gradual introduction of the virtual surgery would facilitate learning the task with respect
to an abrupt introduction of the surgery, as small errors are easier to correct than large errors.
Additionally, it has been proposed that along with the acquisition of an internal model of the task,
learning a motor skill also involves an impedance controller that modulates the stiffness of limbs
to reduce the effects of perturbations (Franklin et al. 2003). At the beginning of motor learning,
the CNS has not yet developed an adequate internal model of the task, so it relies on increasing
the stiffness of the limbs to minimize the effect of perturbations (Darainy and Ostry 2008; Osu et
al. 2002). As learning progresses and task performance improves, the internal model of the task
becomes more accurate, and the CNS decreases limb stiffness and muscle co-contraction, which
is no longer necessary to counteract the perturbation (Osu et al. 2002; Thoroughman and Shadmehr
1999).
Our initial observations indicated the possibility of acquiring more efficient control strategies, that
is, strategies involving lower limb stiffness, when subject to the gradual virtual surgery than when
subject to the abrupt gradual surgery. Therefore, we aimed to find differences in the relationship

71

between joint stiffness and task performance when learning the incompatible virtual surgery task
abruptly and gradually.
Surprisingly, we found that the gradual and the abrupt introduction of the virtual surgeries did not
result in significant differences in the degree to which the surgery was learned, that is, both groups
showed equivalent task performances by the end of the experimental procedure. However, in
accordance with previous studies, we found that lower joint stiffness was associated with better
performance in the task. Furthermore, we found differences in the relationship between joint
stiffness and task performance when learning the task abruptly versus gradually. Participants in
the abrupt group were able to achieve better performance with lower joint stiffness than
participants in the gradual group. We speculate that this is an indication of the acquisition of a
slightly more elaborate internal model when learning the virtual surgery abruptly.

Materials and Methods

Subjects. Twenty two right-handed subjects (mean age, 26.2 ± 3.1 years, SD; 18 males)
participated in the experimental task. All procedures were approved by the Institutional Review
Board of the Tokyo Institute of Technology.
Experimental setup. Each participant sat on a racecar seat while gripping a handle located at the
height of the base of their sternum with their right hand. The arm posture corresponded to an elbow
flexion of around 90° and the elbow was supported on a stand at approximately the same height as
the hand. A splint was used to immobilize the hand, wrist and forearm. Participants were instructed
to lean on the back of the seat for the duration of the experiment. The base of the handle was
attached to a 6 axis force transducer (Dyn Pick; Wacoh-Tech Inc.) used to measure isometric

72

forces. The force transducer was mounted on a 2-D sliding rail to allow for an adjustable
configuration for each participant. The virtual environment was displayed on a computer screen
placed at the height of the participants’ eyes at a distance of around 1 m. The virtual environment
consisted of a circular red cursor (1 cm diameter), and several ring-shaped white targets (2 cm
diameter) on a black background. We recorded surface EMG activity from 10 muscles crossing
the shoulder and elbow joints: pronator teres, brachioradialis, biceps brachii long head, triceps
brachii lateral head, triceps brachii long head, anterior deltoid, middle deltoid, posterior deltoid,
pectoralis major, and middle trapezius. Active bipolar electrodes (DE 2.1; Delsys) were used to
record EMG activity. EMG signals were bandpass filtered (20-450 Hz) and amplified (gain 1000,
Bagnoli-16; Delsys). Force and EMG recordings were digitized at 2 kHz using a USB analog-to-
digital converter (USB-6259; National Instruments). The movement of the cursor was simulated
using mass-spring-damper dynamics according to:
𝒑 ̈ = −
𝑏 𝑚 𝒑 ̇ −
𝑘 𝑚 𝒑 + 𝐹 (𝑡 ) (1)
where p is a vector containing the x and y positions of the cursor on the screen and its derivatives
are indicated in dot notation, m is the system’s mass, k is the stiffness, b is the damping coefficient
and F(t) is the force recorded by the force transducer (during force control) or the estimated force
by the EMG-force mapping (during EMG control) (m = 0.05 kg, b = 100 kg/s). k was calculated
as a function of the maximum voluntary force (MVF) (described in the next section), so that targets
with equal percentages of MVF required the same cursor displacement across participants. The
mass-spring-damper dynamics filter the EMG signals further to make them suitable for controlling
the movement of the cursor.
Experimental protocol. In all phases of the experiment, participants used their right arm to
perform a number of isometric force tasks. These tasks required the displacement of a cursor on a

73

visual display from a center position to one of eight targets uniformly distributed around the center.
Participants first performed a force control task and then an EMG control task (Figure 4.1a). In the
force control task, the cursor was controlled via forces applied by the arm on a load cell (force
control). In the EMG control task, the cursor was controlled by a linear approximation of the force
derived from EMG measurements of 10 arm muscles (EMG control).

Figure 4.1. Experimental schedule and virtual surgery construction.
a. Experimental schedule. Following a maximum voluntary force (MVF) block, participants performed the
force control task. Simultaneous recording of EMG and force data in this task were analyzed to extract
muscle synergies, to produce the baseline EMG-force mapping, and to construct abrupt and gradual
incompatible virtual surgeries. Participants then performed the EMG control tasks, starting with a
familiarization block, followed by baseline, and then either the abrupt or the gradual virtual surgery.
Washout blocks followed the virtual surgery. b – d. Virtual surgery construction: example from one
participant. c. EMG-force mapping extracted after the force control task. Top: Each arrow represents the
estimated force on the horizontal plane that a single muscle would produce when fully activated in isolation
from the rest of the muscles (columns of the 2 x 10 M matrix). Bottom: Forces produced by each of the
muscle synergies extracted after the force control task (columns of 2 x N MS matrix, where N is the number
of synergies). Before applying any virtual surgery, these forces span the 2-dimensional plane. b. Abrupt

74

incompatible virtual surgery. We designed the incompatible surgery by defining a matrix T 12 that rotates
the synergy force vectors so that they became collinear at an angle of 135° degrees, while minimizing the
angles between the original and transformed EMG to force mapping (M and MT 12 respectively). The
incompatible virtual surgery T 12 was introduced abruptly in the first virtual surgery block and was held for
the remainder of the virtual surgery blocks (until block 15). d. Gradual virtual surgery. We incrementally
rotated the vectors in MS in 12 uniform steps from the initial MS to MT 12S. Each rotated virtual surgeries
was introduced at the beginning of each block in the virtual surgery blocks. For visualization purposes, only
steps 1, 6 and 12 of the gradual virtual surgery are shown.

The force control task started with a maximum voluntary force (MVF) block, in which participants
were instructed to produce a maximum voluntary force in each of eight directions spanning the
horizontal plane, with two trials for each direction. The mean MVF was calculated as the mean of
the maximum forces recorded across all trials. For each muscle, the value at the 95 percentile of
the recorded EMG signal across all trials was used to normalize the values of EMG from the
corresponding muscle in all subsequent tasks.
Participants then performed an isometric reaching task by applying force to reach targets in the
virtual environment. The recorded force and EMG signals during this task were processed to
compute the EMG-force mapping, extract muscle synergies, and construct the virtual surgeries.
Targets were arranged radially in eight directions and required 5, 10, 15 or 20% of MVF to be
reached. Each trial started by displaying the target at the central position. The central position
corresponded to the position of the cursor when no forces were applied. After placing the cursor
inside the central target for two seconds, the central target disappeared and one of the radial targets
appeared. After reaching each target, both the cursor and the target disappeared from the screen
and participants were asked to hold the applied force as steadily as possible for two seconds. Next,
the cursor and the central target reappeared and participants were asked to move the cursor back
to the center. After this, another trial began. Each target was presented three times, with a total of

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96 trials. Targets were presented in a randomized order. Trials were repeated if participants failed
to reach a target.
Next, cursor control was switched to EMG control without the participants’ knowledge. The first
EMG control block was a familiarization block, and was followed by an incompatible surgery
introduced either abruptly or gradually (Figure 4.1a). The virtual surgery task consisted of three
phases: baseline, virtual surgery, and washout, which consisted of 2, 15, and 3 blocks, respectively.
Each block consisted of 48 trials: three trials for each of the eight targets at a magnitude of 5%
MVF randomized within target sets containing each one of the eight targets. The level of baseline
noise in each EMG signal was measured at the start of every block while the participant was
relaxed. This baseline noise was subtracted from the EMG signals measured during the
corresponding block.
We randomly assigned participants to one of two groups. In the abrupt group (11 subjects), the
virtual surgery was introduced abruptly, that is, the EMG-force mapping suddenly changed from
the baseline mapping to the transformed mapping at the onset of the first virtual surgery block, and
remained constant until the last virtual surgery block. In the gradual group (12 subjects), the virtual
surgery was introduced incrementally by rotating the individual muscle forces in the EMG-force
mapping a small amount at the beginning of each virtual surgery block until the designed
transformed mapping was reached at the onset of virtual surgery block 12. The complete virtual
surgery continued until the end of block 15, so that subjects performed the task under the complete
virtual surgery for four blocks (see Virtual surgery construction section for details). After this, the
baseline EMG-mapping was restored and subjects performed the washout blocks.

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EMG-force mapping. Force produced at the hand with the arm in a static posture can be
approximated as a linear function of the activations of muscles that actuate the shoulder and
elbow (Berger et al. 2013):
𝐟 = 𝐌𝐦 (2)
where f is a two-dimensional force vector produced on the horizontal plane, m is a ten-
dimensional vector of muscle activations, composed by normalized EMG signals recorded from
ten muscles simultaneously, and M is a 2 × 10 matrix that maps muscle activations to forces, as
represented in the upper panel of Figure 4.1c. We determined M using the same procedure
described in the Materials and Methods section of Chapter 1.
Synergy extraction and number of synergies. We used non-negative matrix factorization (NMF)
(Lee and Seung 1999) to extract muscle synergies from the EMG signals collected during the main
force control subtask:
𝐦 = 𝐒 𝐜 (3)
where S is a 10 × N matrix that contains the identified synergies in its columns with N being the
number of synergies, and c is an N-dimensional vector of synergy activations. Equation 3 assumes
perfect matrix factorization (no residual EMG activity).
We extracted muscle synergies and determined the number of synergies using the same procedure
described in the Materials and Methods section in Chapter 1.
Construction of virtual surgeries. For each participant, we designed an incompatible virtual
surgery. Incompatible virtual surgeries are designed such that muscle activations m produced by
synergy combinations Sc are restricted to generate forces along only one dimension of the force
space, while the resulting EMG-force mapping M’ spans the whole force space (Figure 4.1b). We

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designed the incompatible virtual surgeries according to the procedure to build an easy
incompatible surgery described in the Materials and Methods section in Chapter 1.
Subjects in the abrupt group experienced the resulting easy incompatible virtual surgery for all 15
blocks in the virtual surgery phase, whereas subjects in the gradual group experienced a series of
intermediate EMG-force mappings between the baseline mapping and the fully transformed
mapping. We generated ten intermediate mappings by incrementally and simultaneously rotating
each force vector from its initial configuration in the baseline mapping to its final configuration in
the fully transformed mapping in 12 equally spaced steps (Figure 4.1d).

Data analysis
Task performance metric. We used the initial angular error as a metric to quantify task
performance during the experiment. The initial angular error was calculated for each trial as |θtarget
- θcursor| where θtarget is the direction of the target, and θcursor is defined as the direction of the line
segment that joins the point at which the cursor exits a 2 cm diameter circumference at the center
of the screen and the position of the cursor 100 ms after exiting the circumference. We averaged
the initial angular error for the targets within sets of eight trials.
Index of muscle co-contraction around the joint (IMCJ). We used the index of muscle co-
contraction around the joint (IMCJ) to evaluate the level of co-contraction around the elbow and
shoulder joints during the virtual surgery (Osu et al. 2002). This metric has been shown to correlate
well with joint stiffness (Osu et al. 2002). Based on EMG recordings of a group of muscles, it
estimates the torque generated by each muscle around each joint, and sums the absolute value of
all torques around each joint. Therefore, antagonistic torques contribute to higher IMCJ values.

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To obtain estimates of the shoulder and elbow joint torques based on EMG measurements, we used
a description of the biomechanics of the arm in an isometric state (Valero-Cuevas 2009):
𝐟 = 𝐉 −T
(𝛉 )𝛕 (4)
where f is a two-dimensional force generated at the wrist, θ is a two-dimensional vector containing
the shoulder and elbow joint angles, which defines the posture of the arm, J(θ) is the 2 × 2 Jacobian
of the geometric model of the arm with respect to joint velocities at the θ posture, and τ is a two-
dimensional vector of shoulder and elbow joint torques. Combining equations 2 and 4 we can
estimate the net joint torques based on the measured muscle activations,
𝛕 = 𝐉 T
(𝛉 )𝐌𝐦 = 𝐁𝐦 (5)
where B is the product of the Jacobian and EMG-force mapping matrices, which results in a 2 ×
10 matrix. To find the IMCJ, we take the absolute value of each torque element in B and multiply
it by its corresponding muscle activation:
𝐁 = [b
ij
] (6)
𝐂 = [|b
ij
|] (7)
IMCJ = 𝐂𝐦 (8)
where bij is the element in the i
th
row and j
th
column of B, C is the 2 × 10 matrix of non-signed
torques produced by each muscle around each joint, and IMCJ is a vector containing the IMCJ
index for each joint.
We calculated the value of the IMCJ during the last 0.2 s of every trial during the EMG control
phase of the experiment.

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Additionally, we computed estimates of the IMCJ in the transformed EMG-force mapping spaces,
as the function of muscles may change after the surgery, and co-contraction may be defined
differently in the new space. Therefore, we considered
𝛕 𝐓 = 𝐉 T
(𝛉 )𝐌𝐓𝐦 = 𝐁 𝐓 𝐦 (6)

Results

Abrupt and gradual virtual surgeries produced equivalent task performance

The initial error for the abrupt group showed a sudden increase after the onset of the virtual surgery,
and then decreased as learning progressed until seemingly reaching an asymptote. On the other
hand, the initial error for the gradual group showed no difference with respect to the performance
at baseline for the first five blocks of the surgery, after which it started to increase steadily. By the
end of the first virtual surgery, performance was equivalent for participants in both the abrupt and
gradual groups (Figure 4.2). Both groups also showed similar aftereffects during the washout phase
of the experiment.

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Figure 4.2. Mean initial angular error for abrupt and gradual virtual surgeries.
Each block was subdivided into six target sets, each comprising eight trials (eight different targets). The
targets aligned with the incompatible surgery’s line of action were removed. The initial angular errors at
the remaining six targets were averaged for each target set. The blue trace shows the mean initial angular
error of all twelve participants in the gradual group. The red trace shows the mean initial angular error for
all eleven participants in the abrupt group. Error bars show the s.e. at 95% C.I.

Gradual surgery induced greater joint stiffness than abrupt surgery
We calculated the IMCJ based on the baseline EMG-force mapping M at the end of each trial to
assess the joint stiffness during the virtual surgery. The abrupt surgery produced a sudden increase
in stiffness at the onset of the virtual surgery in both the shoulder and elbow joints that decreased
modestly by the end of the surgery. On the other hand, the gradual surgery produced no apparent
increase in stiffness during the initial blocks of the surgery, but steadily increased as the
intermediate surgeries approached the incompatible virtual surgery. The IMCJ trends produced by
both the abrupt and the gradual surgeries have a similar shape to their respective initial error

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curves. However, the IMCJ level for both the shoulder and elbow joints by the end of the virtual
surgery was significantly higher during the gradual surgery than during the abrupt surgery (Figure
4.3a), indicating increased co-contraction of the shoulder and elbow joints when learning the
virtual surgery gradually.
We also calculated the IMCJ based on the transformed EMG-force mapping MT (Figure 4.3b).
Whereas the IMCJ at the shoulder joint is qualitatively similar to the IMCJ obtained for the
baseline mapping for both abrupt and gradual conditions, there are no significant differences in the
IMCJ based on the transformed mapping at the elbow joint between the abrupt and the gradual
surgeries.

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Figure 4.3. Mean IMCJ in shoulder and elbow joints during the abrupt and gradual virtual surgeries.
a. IMCJ based on baseline EMG-force mapping M. b. IMCJ based on the transformed mapping after the
surgery MT. Each block was subdivided into six target sets, each comprising eight trials (eight different
targets). The blue trace shows the mean IMCJ of all twelve participants in the gradual group. The red trace
shows the mean initial angular error for all eleven participants in the abrupt group. Error bars show the s.e.
at 95% C.I.

Task performance is proportional to co-contraction index
We found that in the last block of the virtual surgery, participants who generated low co-
contraction levels as measured by the IMCJ based on the baseline EMG-force mapping also
produced smaller errors in initial direction (Figure 4.4). This is true for both the abrupt and the
gradual surgeries. We fitted the relationship between IMCJ and the initial direction error in the last

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block of the virtual surgery across the participants in each group with a robust linear model. The
slope in the model for the gradual group (R
2
= 0.73) is smaller than the slope for the gradual group
(R
2
= 0.76), indicating that participants in the abrupt group achieved worse performance than
participants in the gradual group for equivalent levels of co-contraction. The relationship between
IMCJ based on the transformed mapping and the average initial direction error is qualitatively
similar.

Figure 4.4. Good performance in the virtual surgery task is associated with low levels of joint co-
contraction.
For both the abrupt and gradual groups, there is a directly proportional relationship between the norm of
the IMCJ at the shoulder and elbow and the mean initial direction error in the last block of the virtual
surgery. Black lines represent the output of a robust linear regression model obtained for each model
separately.

Discussion

Gradually increasing the difficulty of a task facilitates the learning of motor skills (Sanger 1994),
as gradual increments in difficulty result in smaller performance errors compared to increasing the

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difficulty abruptly, which is associated with larger errors, and consequently, with a larger
propensity to failed learning in the task (Sanger 2004). Furthermore, gradual perturbations in motor
tasks such as visuomotor rotations (Kagerer et al. 1997; Saijo and Gomi 2010) and reaching in
viscous force fields (Kluzik et al. 2008) may induce the CNS to attribute errors in performance to
changes in the body, as opposed to changes in the environment when the perturbation is abrupt
(Kluzik et al. 2008). This suggests that gradual perturbations produce longer lasting effects in
learning (Criscimagna-Hemminger et al. 2010; Kluzik et al. 2008), which could be a significant
finding for the rehabilitation of motor deficits.
Here, we tested whether gradually introducing an incompatible virtual surgery could promote
better final performance than the same task introduced abruptly. Learning an incompatible virtual
surgery requires the acquisition of new patterns of muscle activation (Berger et al. 2013), and
introducing it abruptly generally results in a failure to adequately perform the task, as the motor
commands required before and after introducing the surgery differ greatly. Therefore, gradually
introducing the virtual surgery could facilitate the necessary transitions in suitable muscle
activations to avoid failure of learning.
However, our results indicate that gradually introducing the virtual surgery may not be sufficient
to facilitate enhanced task performance. We did not find significant differences in performance
between the abrupt and gradual surgeries during the final blocks of the surgery. During the early
blocks in the gradual virtual surgery, performance stayed at levels close to baseline, and as the
surgery progressed, performance deteriorated. This could be an indication that the gradual surgery
steps may not be equivalent in difficulty, as late surgery steps are more difficult to adapt to than
earlier steps. Therefore, training in the more difficult surgery steps may have been insufficient to
acquire an appropriate motor strategy for the next surgery step, resulting in rapidly deteriorating

85

performance. This could be addressed in further experiments where surgery steps are introduced
adaptively based on task performance, rather than following a fixed number of trials.
Better training schedules could also be designed by taking into account the covariance of the
generated forces after every surgery step. In Study 2, we showed that learning in virtual surgeries
that result in highly biased force production along the axis of the surgery is slower than in surgeries
where force variance is similar along all direction. It is likely the case that for our current method
of introducing the gradual virtual surgeries, the ratio of the smallest and largest eigenvalue of the
force covariance matrices decreases rapidly as the surgery progresses. Therefore, this metric could
be a good indicator to prescribe the length of training of each surgery step in an informed way and
to track motor performance during the task.
Despite the fact that the gradual protocol did not enhance learning of the virtual surgery with
respect to the abrupt protocol, we observed that different strategies for muscle activation emerge
when experiencing the virtual surgery abruptly and gradually. In both groups better task
performance was associated with lower joint co-contraction in agreement with previous studies
(Darainy and Ostry 2008; Osu et al. 2002). However, unexpectedly, subjects in the gradual group
produced an overall higher co-contraction of muscles around each joint. Our findings indicate that
in the gradual surgery there was a tendency to show higher co-contraction of antagonistic muscles
around each joint for the same level of task performance than in the abrupt surgery. This could be
an indication that the gradual surgery promotes a resistance to change the baseline motor strategy
as changes in the environment are not salient enough. As such using the baseline motor strategy at
the end of surgery requires higher muscle co-contractions. In contrast, the abrupt surgery induced
lower co-contraction for equivalent levels of performance, which could indicate a slight change in

86

the internal model of the task, as an impedance controller could relax its output as the internal
model becomes more accurate (Franklin et al. 2003).
It is possible that the measured co-contractions do not correspond to co-contraction in the task
space, as muscles in the transformed EMG-force mappings may become functionally different to
their role in the baseline mapping. Therefore, we also calculated the IMCJ based on the
transformed EMG-force mappings. Under this assumption, we found that subjects in the gradual
group produced lower co-contraction around the elbow joint, when compared to the co-contraction
estimates based on the baseline EMG-force mapping. This may indicate that the activations of
elbow muscles are adjusting to a strategy that does not correspond to co-contraction in the space
of the transformed EMG-force mapping. This, however, did not occur for the shoulder joint,
indicating that even in the transformed space, different muscle activation strategies were adopted
when experiencing a perturbation gradually or abruptly.

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Chapter 5. Concluding remarks and future work

In this dissertation we have explored various aspects of motor learning during different virtual
surgery tasks. We consider virtual surgeries a promising tool for rehabilitation of the hemiparetic
upper-limb after stroke because of their potential to promote the learning of new muscle co-
activation patterns, which could promote the decoupling of the abnormal synergies seen in elbow
and shoulder muscles.
In particular, we consider incompatible virtual surgeries an insightful testbed for studying motor
learning in non-disabled subjects. The reason for this is that a virtual surgery can highly restrict
force production of the limb to patterns reminiscent of those produced by the affected arms of
stroke survivors. Therefore, the process of learning in an incompatible virtual surgery could
produce patterns similar to those of the abnormal synergies in stroke survivors, suggesting that the
abnormal synergies could be reshaped into less restrictive ones.
Our results suggest that the residual activity from matrix factorization methods for muscle synergy
extraction plays an important role in task performance. The nature of the virtual surgery can have
a great impact on the residuals and consequently, in task performance. Even though two
incompatible virtual surgeries can have identical effects on the muscle synergy to force mapping,
their effects on the EMG-force mapping can be very different. In fact, if the matrix factorization
algorithm produced no residuals, both EMG-force mappings would be identical. Therefore, an
incompatible virtual surgery would be impossible to learn, as there would be no variability in the
virtual forces perpendicular to the line of action of the surgery, which we found is essential for our
model to learn, as seen in Chapter 3.

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This suggests that muscle activity that is not captured by muscle synergy activity could be
exploited to enhance learning during a virtual surgery task. New types of muscle synergies could
be designed that aim to maximize the force variability produced by the residual muscle activity.
This could potentially facilitate faster and more effective learning of the designed surgery in both
the non-disabled and stroke populations.
One of our long-term goals is attempting to reshape the affected arm muscle activation patterns of
stroke survivors into patterns that are coupled to a lesser degree. We consider that an intervention
similar, but opposite in effect to the gradual introduction of an incompatible surgery, could
promote the reshaping of the affected muscle activation patterns. This intervention assumes that
the synergy-force mapping after a stroke is close to that of a non-disabled subject’s mapping after
applying an incompatible virtual surgery. Therefore, a virtual surgery that makes an almost
incompatible synergy-force mapping more similar to a compatible mapping that spans the force
space more broadly could immediately contribute to better performance. Removing this virtual
surgery abruptly would probably result in a sudden drop in performance, and the extent of learning
would be low due to the induced large errors. Thus, removing the virtual surgery gradually could
mitigate this effect.
However, our current results in non-disabled subjects indicate that gradually introducing the virtual
surgery may not be sufficient to facilitate enhanced task performance. Despite this fact, we
observed that different strategies in muscle activation emerge when experiencing the virtual
surgery abruptly and gradually. Unexpectedly, subjects in the gradual group produced an overall
higher co-contraction of muscles around each joint. The emergence of two different motor
strategies, suggests that it may be possible that improvements in performance could be observed

89

if the schedule of the gradual introduction of the surgery was tailored to be adaptive as a function
of task performance, or more time to learn in each gradual step was provided.
More studies are obviously necessary to research the viability of using the virtual surgery paradigm
as a tool for rehabilitation post-stroke, but we hope that this set of studies provides a basis to further
expand on the viability and effectiveness of this approach.

90

References

Alessandro C, Delis I, Nori F, Panzeri S, and Berret B. Muscle synergies in neuroscience and
robotics: from input-space to task-space perspectives. Frontiers in Computational Neuroscience
7: 16, 2013.
Barradas VR, Kutch JJ, Kawase T, Koike Y, and Schweighofer N. When 90% of the variance
is not enough: residual EMG from muscle synergy extraction influences task performance. Biorxiv
2019.
Beer RF, Ellis MD, Holubar BG, and Dewald JPA. Impact of gravity loading on post-stroke
reaching and its relationship to weakness. Muscle & Nerve 36: 242-250, 2007.
Berger DJ, and d'Avella A. Effective force control by muscle synergies. Frontiers in
Computational Neuroscience 8: 13, 2014.
Berger DJ, and d'Avella A. Towards a myoelectrically controlled virtual reality interface for
synergy-based stroke rehabilitation. Converging Clinical and Engineering Research on
Neurorehabilitation Ii, Vols 1 and 2 15: 965-969, 2017.
Berger DJ, Gentner R, Edmunds T, Pai DK, and d'Avella A. Differences in adaptation rates
after virtual surgeries provide direct evidence for modularity. Journal of Neuroscience 33: 12384-
12394, 2013.
Bernstein N. The co-ordination and regulation of movement. Oxford, UK: Pergamon, 1967.
Berret B, Delis I, Gaveau J, and Jean F. Optimality and modularity in human movement: from
optimal control to muscle synergies. In: Biomechanics of anthropomorphic systems. Cham:
Springer, 2019, p. 105-133.
Bishop CM. Neural networks for pattern recognition. Oxford university press, 1995.
Bizzi E, and Cheung VCK. The neural origin of muscle synergies. Frontiers in Computational
Neuroscience 7: 6, 2013.
Bizzi E, Mussa-Ivaldi FA, and Giszter S. Computations underlying the execution of movement-
a biological perspective. Science 253: 287-291, 1991.
Brunnstrom S. Movement therapy in hemiplegia: a neurophysiological approach. New York:
Harper and Row, 1970.
Burkholder TJ, and van Antwerp KW. Practical limits on muscle synergy identification by non-
negative matrix factorization in systems with mechanical constraints. Medical & Biological
Engineering & Computing 51: 187-196, 2013.
Caggiano V, Cheung VCK, and Bizzi E. An optogenetic demonstration of motor modularity in
the mammalian spinal cord. Scientific Reports 6: 15, 2016.
Cheung VCK, d'Avella A, Tresch MC, and Bizzi E. Central and sensory contributions to the
activation and organization of muscle synergies during natural motor behaviors. Journal of
Neuroscience 25: 6419-6434, 2005.

91

Cheung VCK, Turolla A, Agostini M, Silvoni S, Bennis C, Kasi P, Paganoni S, Bonato P, and
Bizzi E. Muscle synergy patterns as physiological markers of motor cortical damage. Proceedings
of the National Academy of Sciences of the United States of America 109: 14652-14656, 2012.
Clark DJ, Ting LH, Zajac FE, Neptune RR, and Kautz SA. Merging of healthy motor modules
predicts reduced locomotor performance and muscle coordination complexity post-stroke. Journal
of Neurophysiology 103: 844-857, 2010.
Clawson DM, Healy AF, Ericsson KA, and Bourne LE. Retention and transfer of morse code
reception skill by novices: Part-whole training. Journal of Experimental Psychology-Applied 7:
129-142, 2001.
Criscimagna-Hemminger SE, Bastian AJ, and Shadmehr R. Size of error affects cerebellar
contributions to motor learning. Journal of Neurophysiology 103: 2275-2284, 2010.
d'Avella A, and Lacquaniti F. Control of reaching movements by muscle synergy combinations.
Frontiers in Computational Neuroscience 7: 7, 2013.
d'Avella A, Portone A, Fernandez L, and Lacquaniti F. Control of fast-reaching movements
by muscle synergy combinations. The Journal of Neuroscience 26: 7791, 2006.
d'Avella A, Saltiel P, and Bizzi E. Combinations of muscle synergies in the construction of a
natural motor behavior. Nature Neuroscience 6: 300-308, 2003.
Darainy M, and Ostry DJ. Muscle cocontraction following dynamics learning. Experimental
Brain Research 190: 153-163, 2008.
de Rugy A, Loeb GE, and Carroll TJ. Are muscle synergies useful for neural control? Frontiers
in Computational Neuroscience 7: 13, 2013.
de Rugy A, Loeb GE, and Carroll TJ. Muscle coordination is habitual rather than optimal.
Journal of Neuroscience 32: 7384-7391, 2012.
Delis I, Berret B, Pozzo T, and Panzeri S. Quantitative evaluation of muscle synergy models: a
single-trial task decoding approach. Frontiers in Computational Neuroscience 7: 21, 2013.
Dewald JP, Pope PS, Given JD, Buchanan TS, and Rymer WZ. Abnormal muscle coactivation
patterns during isometric torque generation at the elbow and shoulder in hemiparetic subjects.
Brain 118 ( Pt 2): 495-510, 1995.
Fagg AH, Shah A, and Barto AG. A computational model of muscle recruitment for wrist
movements. Journal of Neurophysiology 88: 3348-3358, 2002.
Franklin DW, Osu R, Burdet E, Kawato M, and Milner TE. Adaptation to stable and unstable
dynamics achieved by combined impedance control and inverse dynamics model. Journal of
Neurophysiology 90: 3270-3282, 2003.
Gill PE, and Wong E. Methods for convex and general quadratic programming. Mathematical
Programming Computation 7: 71-112, 2015.
Golub MD, Yu BM, and Chase SM. Internal models for interpreting neural population activity
during sensorimotor control. Elife 4: 28, 2015.
Harris CM, and Wolpert DM. Signal-dependent noise determines motor planning. Nature 394:
780-784, 1998.

92

Hug F. Can muscle coordination be precisely studied by surface electromyography? Journal of
Electromyography and Kinesiology 21: 1-12, 2011.
Ivanenko YP, Cappellini G, Dominici N, Poppele RE, and Lacquaniti F. Coordination of
locomotion with voluntary movements in humans. Journal of Neuroscience 25: 7238-7253, 2005.
James G, Witten D, Hastie T, and Tibshirani R. An introduction to statistical learning with
applications in R Introduction. In: Introduction to statistical learning: with applications in R. New
York: Springer, 2013, p. 1-14.
Jordan MI, and Rumelhart DE. Forward models: Supervised learning with a distal teacher.
Cognitive Science 16: 307-354, 1992.
Kagerer FA, ContrerasVidal JL, and Stelmach GE. Adaptation to gradual as compared with
sudden visuo-motor distortions. Experimental Brain Research 115: 557-561, 1997.
Kawato M. Internal models for motor control and trajectory planning. Current Opinion in
Neurobiology 9: 718-727, 1999.
Kawato M, and Gomi H. A computational model of four regions of the cerebellum based on
feedback-error learning. Biological cybernetics 68: 95-103, 1992.
Kluzik J, Diedrichsen J, Shadmehr R, and Bastian AJ. 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.
Kozin SH, D'Addesi L, Chafetz RS, Ashworth S, and Mulcahey MJ. Biceps-to-Triceps
Transfer for Elbow Extension in Persons With Tetraplegia. Journal of Hand Surgery-American
Volume 35A: 968-975, 2010.
Krakauer JW, and Carmichael ST. Broken movement: The neurobiology of motor recovery
after stroke. MIT Press 2017.
Kuperstein M. Neural model of adaptive hand-eye coordination for single postures. Science 239:
1308-1311, 1988.
Kutch JJ, and Valero-Cuevas FJ. Challenges and new approaches to proving the existence of
muscle synergies of neural origin. Plos Computational Biology 8: 11, 2012.
Latash ML, Scholz JP, and Schoner G. Toward a new theory of motor synergies. Motor Control
11: 276-308, 2007.
Lecun Y, Kanter I, and Solla SA. Eigenvalues of covariance matrices: application to neural-
network learning. Physical Review Letters 66: 2396-2399, 1991.
Lee DD, and Seung HS. Learning the parts of objects by non-negative matrix factorization.
Nature 401: 788-791, 1999.
Lee WA. Neuromotor synergies as a basis for coordinated intentional action. Journal of Motor
Behavior 16: 135-170, 1984.
Linkenhoker BA, and Knudsen EI. Incremental training increases the plasticity of the auditory
space map in adult barn owls. Nature 419: 293-296, 2002.
Marjaninejad A, and Valero-Cuevas FJ. Should anthropomorphic systems be “redundant”? In:
Biomechanics of Anthropomorphic SystemsSpringer, 2019, p. 7-34.

93

Merletti R, Aventaggiato M, Botter A, Holobar A, Marateb H, and Vieira TM. Advances in
surface EMG: recent progress in detection and processing techniques. Critical Reviews™ in
Biomedical Engineering 38: 2010.
Mussa-Ivaldi FA, Giszter SF, and Bizzi E. Linear combinations of primitives in vertebrate motor
control. Proceedings of the National Academy of Sciences of the United States of America 91:
7534-7538, 1994.
Nazarpour K, Barnard A, and Jackson A. Flexible cortical control of task-specific muscle
synergies. Journal of Neuroscience 32: 12349-12360, 2012.
Neptune RR, Clark DJ, and Kautz SA. Modular control of human walking: A simulation study.
Journal of Biomechanics 42: 1282-1287, 2009.
Osu R, Franklin DW, Kato H, Gomi H, Domen K, Yoshioka T, and Kawato M. Short- and
long-term changes in joint co-contraction associated with motor learning as revealed from surface
EMG. Journal of Neurophysiology 88: 991-1004, 2002.
Overduin SA, d'Avella A, Carmena JM, and Bizzi E. Microstimulation activates a handful of
muscle synergies. Neuron 76: 1071-1077, 2012.
Roh J, Rymer WZ, and Beer RF. Robustness of muscle synergies underlying three-dimensional
force generation at the hand in healthy humans. Journal of Neurophysiology 107: 2123-2142,
2012.
Roh J, Rymer WZ, Perreault EJ, Yoo SB, and Beer RF. Alterations in upper limb muscle
synergy structure in chronic stroke survivors. J Neurophysiol 109: 768-781, 2013.
Rolf M, and Steil JJ. Explorative learning of inverse models: A theoretical perspective.
Neurocomputing 131: 2-14, 2014.
Saijo N, and Gomi H. Multiple motor learning strategies in visuomotor rotation. Plos One 5: 11,
2010.
Sanger TD. Failure of motor learning for large initial errors. Neural Computation 16: 1873-1886,
2004.
Sanger TD. Neural-network learning control of robot manipulators using gradually increasing
task-difficulty. IEEE Transactions on Robotics and Automation 10: 323-333, 1994.
Santuz A, Ekizos A, Janshen L, Baltzopoulos V, and Arampatzis A. On the methodological
implications of extracting muscle synergies from human locomotion. International Journal of
Neural Systems 27: 15, 2017.
Schmidt RA, Zelaznik H, Hawkins B, Frank JS, and Quinn Jr JT. Motor-output variability: a
theory for the accuracy of rapid motor acts. Psychological review 86: 415, 1979.
Shuman BR, Schwartz MH, and Steele KM. Electromyography data processing impacts muscle
synergies during gait for unimpaired children and children with cerebral palsy. Frontiers in
Computational Neuroscience 11: 9, 2017.
Steele KM, Tresch MC, and Perreault EJ. Consequences of biomechanically constrained tasks
in the design and interpretation of synergy analyses. Journal of Neurophysiology 113: 2102-2113,
2015.

94

Steele KM, Tresch MC, and Perreault EJ. The number and choice of muscles impact the results
of muscle synergy analyses. Frontiers in Computational Neuroscience 7: 9, 2013.
Tallal P, Miller SL, Bedi G, Byma G, Wang XQ, Nagarajan SS, Schreiner C, Jenkins WM,
and Merzenich MM. Language comprehension in language-learning impaired children improved
with acoustically modified speech. Science 271: 81-84, 1996.
Thoroughman KA, and Shadmehr R. Electromyographic correlates of learning an internal
model of reaching movements. Journal of Neuroscience 19: 8573-8588, 1999.
Ting LH, and Macpherson JM. A limited set of muscle synergies for force control during a
postural task. Journal of Neurophysiology 93: 609-613, 2005.
Todorov E, and Jordan MI. Optimal feedback control as a theory of motor coordination. Nature
Neuroscience 5: 1226-1235, 2002.
Torres-Oviedo G, Macpherson JM, and Ting LH. Muscle synergy organization is robust across
a variety of postural perturbations. Journal of Neurophysiology 96: 1530-1546, 2006.
Torres-Oviedo G, and Ting LH. Subject-apecific muscle synergies in human balance control are
consistent across different biomechanical contexts. Journal of Neurophysiology 103: 3084-3098,
2010.
Tresch MC, Cheung VCK, and d'Avella A. Matrix factorization algorithms for the identification
of muscle synergies: Evaluation on simulated and experimental data sets. Journal of
Neurophysiology 95: 2199-2212, 2006.
Tresch MC, and Jarc A. The case for and against muscle synergies. Current Opinion in
Neurobiology 19: 601-607, 2009.
Tresch MC, Saltiel P, and Bizzi E. The construction of movement by the spinal cord. Nature
Neuroscience 2: 162-167, 1999.
Valero-Cuevas FJ. A mathematical approach to the mechanical capabilities of limbs and fingers.
Progress in Motor Control: a Multidisciplinary Perspective 629: 619-633, 2009.
van Beers RJ, Haggard P, and Wolpert DM. The role of execution noise in movement
variability. Journal of Neurophysiology 91: 1050-1063, 2004.
Wolpert DM, Miall RC, and Kawato M. Internal models in the cerebellum. Trends in Cognitive
Sciences 2: 338-347, 1998.
Wright ZA, Rymer WZ, and Slutzky MW. Reducing abnormal muscle coactivation after stroke
using a myoelectric-computer interface: A Pilot Study. Neurorehabilitation and Neural Repair 28:
443-451, 2014.
Zhou P, and Rymer WZ. Factors governing the form of the relation between muscle force and
the EMG: A simulation study. Journal of Neurophysiology 92: 2878-2886, 2004.

Abstract (if available)

Abstract Stroke is one of the leading causes of long-term disability in the world. One of its possible consequences is upper limb hemiparesis, which is characterized by an abnormal coupling of the movements of the elbow and shoulder joints, resulting in a significant loss of motor coordination and a high impact on the quality of life of stroke survivors. This abnormal coupling is associated with the abnormal co-activation of elbow flexor and shoulder abductor and adductor muscles. Currently, most rehabilitation treatments do not directly target the recovery of motor coordination, but rather, the recovery of function, which promotes the use of compensatory strategies to circumvent the impaired coordination. Therefore, there is a need to develop new rehabilitation treatments that directly promote the improvement of motor coordination. A promising approach is the use of electromyography (EMG) -computer interfaces, which allow interaction with a virtual environment through a mapping from muscle activations, as measured by EMG, to variables in the virtual environment. Virtual surgeries are tasks in an EMG-computer interface that map EMG onto forces in the virtual space, and mimic tendon transfer surgeries by assigning each muscle a pulling force in the virtual environment that is different from its actual pulling force. This kind of task has been used to induce learning of new patterns of muscle activation in non-impaired subjects, suggesting a potential use in stroke rehabilitation to abolish the abnormal coupling of elbow and shoulder muscles. Our long-term goal is to assess the feasibility of using virtual surgeries as a component of rehabilitation treatments for hemiparesis after stroke. However, it is first necessary to study the effects of virtual surgeries on motor learning in non-impaired subjects thoroughly. We present three studies to this effect. The first study shows the importance of residual muscle activation, which is often ignored, in the execution of the virtual surgery task. The second study proposes that differences in learning rates observed when learning compatible and incompatible virtual surgeries are due to the statistics of the produced virtual forces. Finally, the third study characterizes motor learning during the gradual and abrupt introduction of virtual surgeries. We hope that this set of studies can be used as a starting point for building a framework to use virtual surgeries in the rehabilitation of hemiparesis after stroke.

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Asset Metadata

Creator Barradas Patino, Victor Ramon (author)

Core Title Virtual surgeries as a tool for studying motor learning

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Biomedical Engineering

Publication Date 10/30/2019

Defense Date 08/28/2019

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

Tag direct inverse modeling,motor learning,muscle synergies,myoelectric computer interfaces,OAI-PMH Harvest,residuals,stroke rehabilitation,virtual surgeries

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Schweighofer, Nicolas (committee chair), Kutch, Jason (committee member), Sanger, Terence (committee member)

Creator Email vbarrada@usc.edu,vrbarradasp@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-230388

Unique identifier UC11674989

Identifier etd-BarradasPa-7891.pdf (filename),usctheses-c89-230388 (legacy record id)

Legacy Identifier etd-BarradasPa-7891.pdf

Dmrecord 230388

Document Type Dissertation

Rights Barradas Patino, Victor Ramon

Type texts

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

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

Repository Name University of Southern California Digital Library

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