BIOMECHANICS CONSTRAINS VARIABILITY IN SPATIAL STRUCTURE OF
MUSCLE COORDINATION FOR ENDPOINT FORCE GENERATION
1
M. Hongchul Sohn, 1,2J. Lucas McKay and 1,2Lena H. Ting
1
Georgia Institute of Technology, Atlanta, GA, USA
2
Emory University, Atlanta, GA, USA
email: lting@emory.edu
INTRODUCTION
Due to biomechanical redundancy, multiple muscle
coordination patterns can be used to generate the
same endpoint force. Accordingly, muscle synergy
patterns used to produce similar force vectors
during balance control varies across individuals in
both cats [1] and humans [2]. For example, in a
synergy to produce a limb loading force vector (FW1 ,
down-backward), vastus (VAST) was recruited
consistently at a high level across different animals,
whereas the activation level of gracilis (GRAC)
varied considerably across animals. On the other
hand, in producing an unloading force vector (FW2,
up-forward), sartorius (SART) was always used,
but vastus (VAST) was recruited at varying levels
across animals.
We hypothesized that the constraints of limb
biomechanics would predict the observed variability
identified experimentally. Our approach was to
identify the upper and lower bounds of individual
muscle activation levels for generating endpoint
force vectors in different directions and magnitudes.
A similar approach has been used to demonstrate
how muscle dysfunction affects the robustness of
force generation [3].
Here we were particularly interested in the relation
between the observed variability in measured EMG
signals during a behavioral task, and the spatial
structure of redundant muscle space defined by the
biomechanics, with implication to neural strategies
for selecting muscle coordination in muscle
synergies. For example, muscles with consistently
high activation would be “necessary”, and muscles
with more variability would be “optional” for a
given task. We further compared the bounds to
muscle coordination patterns predicted with
alternate neural strategies defined by scaling [4] or
minimum effort control.
METHODS
Using a detailed model of the cat hindlimb [5], we
examined how the range of permissible activation
levels of each muscle changes for the generation of
endpoint force vectors when magnitude increases
from 0 to maximal force. The posture of the static
model was matched to the kinematics of 3 cats and

used to map a 31D muscle activation vector e (0≤ em
≤1) to a 7D joint torque vector, determined by premultiplying the Jacobian transpose to a 6D
combined endpoint force and moment vector



T
F End  [ f x , f y , f z , M x , M y , M z ]T , with J F End  RFAFL e
(JT: 7×6 Jacobian transpose, R: 7×31 moment arm
matrix, FAFL: 31×31 diagonal muscle scaling factors
for active force generation).
Five experimental synergy force vectors in each cat
[1] were used as the target endpoint force vector
directions, where the maximum force magnitude in

each direction ( F WM AX , i=1~5) was found with linear
programming. We identified the maximum (upper
bound: emUB) and minimum (lower bound: emLB)
possible activations of each muscle m, in producing
 MAX
an endpoint force of   FW (α=0~1) with linear
programming at each incremental step (Δα=0.1) for
the five force directions in each cat. For each force,
we also found the muscle activation patterns
predicted by 1) scaling the pattern for the maximal
task [4], and 2) minimizing muscular effort (i.e.,
sum-squared activation).
i
i
RESULTS and DISCUSSION
The range of each muscle, which is the difference
between emUB and emLB at a given α, changed
anisotropically as the force magnitude increased
(Fig. 1A and B, shaded). When the lower bound,
emLB becomes nonzero, it corresponds to the force
magnitude (α) at which the muscle becomes
necessary (Fig. 1A and B, bottom trace), whereas
the upper bound emUBs defines the maximum
allowable activity of the muscle (Fig. 1A and B, top
trace). In many cases, these bounds converged at
α=1. However, cases existed where these bounds
did not converge (results not shown), reflecting the
redundancy within joints.
Non-negativity of muscle activation profoundly
affected the structure of the null-space associated
with zero force production at α=0. The emUBs were
limited by the relative torque-generating capability
of each muscle. For example, GRAC had emUB of 0.3
in one cat (Fig.1B) meaning that torque generated
by activating this muscle higher than 0.3 cannot be
counterbalanced by activation of other muscles. In
contrast, when negative activation was allowed, the
range spanned the full possible levels (-1≤ em ≤1) for
all muscles, showing its weak biological relevance.
In agreement with the prediction, muscles that were
consistently activated across animals were
necessary, whereas muscles that highly varied in its
recruitment level were optional. For example, emLB of
VAST became immediately nonzero for FW1 and
SART became necessary at α=0.6 for FW2 in one
animal (Fig. 1A). Similarly, GRAC and VAST
activity for FW1 and FW2, respectively, had zero emLB
at all force levels (Fig. 1B). Although the observed
variability was consistent with the model
predictions, the magnitudes of experimental force
vectors (Fig. 1A and B, vertical line) were actually
small such that the emLB s were zero for all muscles in
one cat for FW1, and in all cats for FW2.
The consistent activation of muscle at before they
are necessitated by biomechanical constraints may
conform to a neural strategy where the pattern for
the maximal task is scaled in sub-maximal tasks [4]
(Fig. 1A and B, dashed line). Accordingly, both the
scaling strategy and the minimum effort strategy
(Fig. 1A and B, dots) predicted the activation of
“necessary” muscles (VAST for FW1 and SART for
FW2) at the earliest nonzero α. However, “optional”
muscles (GRAC for FW1 and VAST for FW2) were
never selected in either strategy.
Our results show that biomechanics places bounds
on the variability of muscle activation patterns for
endpoint force generation. Recruitment of optional
muscles may reflect other goals such as stabilizing
posture.
Figure 1: Upper (top trace) and lower (bottom trace)
bounds for necessary (A) and optional (B) muscles
show the range (shaded) of allowable variability in
muscle activation for generating force in direction
of the experimental synergy force vectors FW1 (red)
and FW2 (yellow). Solutions for scaling (dashed line)
and minimum effort (dots) strategy lie within the
range, or on the lower bound.
REFERENCES
1. Torres-Oviedo G, et al. J Neurophysiol, 96, 1530-
1546, 2006.
2. Chvatal SA, et al. J Neurophysiol, 106, 999-1015,
2011.
3. Kutch JJ, Valero-Cuevas FJ. J Biomech, 44, 12641270, 2011.
4. Valero-Cuevas FJ. J Neurophysiol, 83, 1469-1479,
2000.
5. Burkholder TJ, Nichols TR. J Morph, 261, 118-129,
2004.
ACKNOWLEDGEMENTS
NIH HD 46922