J Neurophysiol 100: 1433–1454, 2008.
First published July 2, 2008; doi:10.1152/jn.01377.2007.
Modulation of Phasic and Tonic Muscle Synergies With Reaching Direction
and Speed
Andrea d’Avella,1 Laure Fernandez,1,4 Alessandro Portone,1,2 and Francesco Lacquaniti1,2,3
1
Department of Neuromotor Physiology, Santa Lucia Foundation; 2Department of Neuroscience and 3Center of Space Biomedicine,
University of Rome Tor Vergata, Rome, Italy; and 4Institute of Movement Sciences, University of Mediterranean, Marseille, France
Submitted 20 December 2007; accepted in final form 30 June 2008
INTRODUCTION
The control of limb movements is challenging because of the
kinematic, dynamic, and biomechanical complexity of the
musculoskeletal system. Much of this complexity derives from
the large number of degrees of freedom in the system, which
allows for great flexibility but makes the computations necessary for controlling arbitrary limb movements, in principle,
very complicated (Bernstein 1967). How the CNS might perform these computations has been the central topic of many
theoretical and experimental investigations and it still remains
an open question. One way to simplify the control process is to
subdivide it among task-specific control modules, each involving only a few parameters directly selected as a function of task
parameters. For example, the task of reaching a target in space,
given an initial arm configuration, is specified by a direction in
space and a distance (a vector in space). Movement duration
Address for reprint requests and other correspondence: A. d’Avella, Dipartimento di Fisiologia Neuromotoria, Istituto di Ricovero e Cura a Carattere
Scientifico Fondazione Santa Lucia, Via Ardeatina 306, 00179 Rome, Italy
(E-mail: [email protected]).
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could also be a task parameter if temporal constraints are
involved. A direct mapping of the task parameters into the
appropriate muscle activation patterns could constitute a taskspecific feedforward controller adequately solving the control
problem for reaching. However, since incalculable numbers of
different muscle patterns are potentially associated with different values of even a single task parameter, whether it is
possible to establish such a mapping without an explicit computation remains to be demonstrated.
We have recently provided support for the hypothesis that
muscle synergies constitute task-specific control modules that
generate all muscle patterns necessary for adequately performing reaching movements through simple rules involving just a
few parameters (d’Avella et al. 2006). We have shown that the
muscle patterns for fast point-to-point movements in different
directions on vertical planes, with different loads and forearm
postures, as well as for reaching movements with a reversal
and through a via-point, are well captured by the combinations
of four or five time-varying muscle synergies. By scaling in
amplitude and shifting in time these synergies, it has been
possible to account for the complex changes in the activation
waveforms of individual muscles across movement conditions,
reported in previous studies of reaching movements in vertical
planes (Buneo et al. 1994; Flanders 1991, 2002; Flanders and
Herrmann 1992; Flanders et al. 1996; Soechting and Lacquaniti 1981), as the result of simple modulation and combination
rules. However, our study focused on fast movements and on
the phasic components of the muscle activation waveforms
responsible for accelerating and decelerating the arm. In contrast, natural reaching movements occur at a range of different
speeds, depending on environmental constraints. Moreover, the
arm is maintained in stable postures before and after a reaching
movement, requiring, for an unsupported arm reaching targets
in a vertical plane, adequate muscle activation. How a controller based on muscle synergy combinations might generate
muscle patterns appropriate for movements with different
speeds and for maintaining stable arm postures is at the core of
important questions that must be addressed.
Theoretical considerations (Hollerbach and Flash 1982) suggest a straightforward strategy for generating the joint torque
profiles appropriate for movements with different speeds. The
equations of motion for an articulated arm are invariant for
changes in the timescale of the movement if the joint torques
are scaled in amplitude according to a simple rule. After
separating the total torque into an antigravity component and a
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d’Avella A, Fernandez L, Portone A, Lacquaniti F. Modulation of
phasic and tonic muscle synergies with reaching direction and speed.
J Neurophysiol 100: 1433–1454, 2008. First published July 2, 2008;
doi:10.1152/jn.01377.2007. How the CNS masters the many degrees
of freedom of the musculoskeletal system to control goal-directed
movements is a long-standing question. We have recently provided
support to the hypothesis that the CNS relies on a modular control
architecture by showing that the phasic muscle patterns for fast
reaching movements in different directions are generated by combinations of a few time-varying muscle synergies: coordinated recruitment of groups of muscles with specific activation profiles. However,
natural reaching movements occur at different speeds and require the
control of both movement and posture. Thus we have investigated
whether muscle synergies also underlie reaching at different speeds as
well as the maintenance of stable arm postures. Hand kinematics and
shoulder and elbow muscle surface EMGs were recorded in five
subjects during reaches to eight targets in the frontal plane at different
speeds. We found that the amplitude modulation of three timeinvariant synergies captured the variations in the postural muscle
patterns at the end of the movement. During movement, three phasic
and three tonic time-varying synergies could reconstruct the timenormalized muscle pattern in all conditions. Phasic synergies were
modulated in both amplitude and timing by direction and speed. Tonic
synergies were modulated only in amplitude by direction. The directional tuning of both types of synergies was well described by a single
or a double cosine function. These results suggest that muscle synergies are basic control modules that allow generating the appropriate
muscle patterns through simple modulation and combination rules.
1434
D’AVELLA, FERNANDEZ, PORTONE, AND LACQUANITI
METHODS
Experimental setup and protocol
We investigated the changes in the activity patterns in shoulder and
arm muscles across movement direction and speed during point-topoint arm reaching movements in the frontal plane. The experimental
apparatus, described in detail in our previous report on fast reaching
movements (d’Avella et al. 2006), consisted of one central sphere,
indicating a start location, eight peripheral spheres, positioned on a
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circle at 30 cm from the start on a vertical plane at approximately 45°
from each other and indicating target locations, and a handle, gripped
by the subject and provided with a reference sphere to be displaced
from start to target. Start and target spheres (4-cm diameter, made of
transparent plastic) could be illuminated from inside by a lightemitting diode (LED) and were supported by a structure that allowed
adjustment of the height of the start sphere to match the height of the
elbow of a subject standing with the upper arm along the trunk. Five
right-handed subjects (four males and one female, age range 19 –36
yr) participated in the experiments after giving informed consent. All
experimental protocols conformed to the Declaration of Helsinki on
the use of human subjects in research.
Subjects performed blocks of reaching movement trials while
standing with the start and target spheres in the frontal plane at a
distance that allowed positioning the reference sphere on the handle
close to the start sphere while keeping the upper arm vertical and the
forearm horizontal. Each trial started, when the reference sphere was
positioned at the start location, with a ready signal followed by a start
signal (computer-generated tones, 1-s delay) after which subjects were
free to choose when to start moving, although they were instructed to
reach the target within a given time interval from movement start and
to remain at the target location for ⱖ1 s (target hold period). The
target sphere LED was turned on at the time of the ready signal and
the start sphere LED was turned off at the time of the start signal. To
vary the speed of movement, the instructed movement time interval
varied across blocks and subjects were informed at the end of the trial
whether their movement was too fast or too slow ( fast and slow
signals, computer-generated tones, or high and low pitch, respectively). Five different intervals were used spanning movement times
from 240 to 820 ms. Targets in all eight directions were presented in
pseudorandom order in each block, until at least four (subject 5) or
five (subjects 1– 4) successful trials were performed in each direction.
One subject performed two blocks for each movement time interval
(subject 1, total 10 blocks), whereas all other subjects performed three
blocks for each movement time interval (total 15 blocks).
Data collection and preprocessing
We collected kinematic and electromyographical (EMG) data during the experiments. The position and orientation of a marker inserted
in the handle gripped by the subjects were recorded using an electromagnetic motion-tracking system (Fastrak, Polhemus, Colchester,
VT) at 120 Hz. The position of the reference sphere attached to the
handle, the actual arm endpoint, was computed using translation and
rotation matrices that were determined with a calibration performed
with a second marker attached to the handle in place of the reference
sphere. EMG activity from ⱕ18 muscles (see Table 1) was recorded
using active bipolar surface electrodes (DE 2.1, Delsys, Boston, MA),
band-pass filtered (20 – 450 Hz), and amplified (total gain 1,000,
Bagnoli-16, Delsys). Correct electrode placement was verified by
observing the activation of each muscle during specific movements
known to involve it (Kendall et al. 2005). Possible contamination of
the EMG recordings by electrical cross talk from adjacent muscles
was assessed by performing a cross-correlation analysis between all
pairs of channels. The peak of the normalized cross-correlation at lags
close to zero was ⬎0.2 in only seven pairs of channels (four for
subject 2 and three for subject 3). Because of the difficulty in
distinguishing cross talk from synchronous recruitment of motor units
in different muscles (Kilner et al. 2002), we did not remove these
muscles from the set used for further analyses. However, we verified
that the removal of the muscles potentially affected by cross talk did
not change any conclusion drawn from those analyses.
Data acquisition and experiment control were performed on a
workstation using custom software written in LabVIEW (National
Instruments, Austin, TX). EMG data were digitized continuously
during each block (1-kHz sampling rate, PCI-6035E, National Instruments). Kinematic data were synchronized with the EMG data by
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dynamic component and after scaling in time both components
by a factor r, if the dynamic torque component is scaled in
amplitude by r2, and the amplitude of the antigravity component is not changed, the joint motions will be scaled in time by
the same factor r. Thus if a torque profile adequate for reaching
a given spatial target at one speed is known, a simple scaling
rule allows one to generate the torque profiles for reaching that
target, along the exact same path, at different speeds, consistent
with the invariant dynamic characteristics of reaching movements observed in humans (Soechting and Lacquaniti 1981).
Such a feature of the dynamic equations for an articulated arm
suggests that the CNS might implement a simple amplitudescaling mechanism for mapping movement speed into muscle
patterns. We hypothesized that such a scaling mechanism is
realized by scaling in amplitude and in time a small number of
time-varying muscle synergies.
Maintaining the arm in a specific posture requires generating
static joint torques to balance the posture-dependent torques
arising from gravity and generating the joint stiffnesses appropriate for maintaining stability in the presence of unexpected
perturbations. Specifying the muscle activations that satisfy
these postural requirements at any arm posture along the path
of a reaching movement— because of the multiarticular nature
of the arm and the redundancy of the musculoskeletal system—
presents similar computational challenges to specifying the
phasic muscle pattern necessary for accelerating and decelerating the arm. We then also hypothesized that a combination of
a small number of muscle synergies is sufficient to generate the
muscle activations necessary for postural maintenance, thus
simplifying the postural control problem to the selection of a
few synergy combination coefficients.
In this study, we investigated the muscle patterns for reaching in the frontal plane in eight different directions over a wide
range of movement speeds. We characterized the organization
of both the postural muscle patterns, responsible for maintaining the hand stable at the end of the reaching movement, and
the entire time-course of the muscle patterns, before, during,
and after the movement, including both phasic and tonic
components. The postural muscle patterns have been decomposed as combinations of time-invariant synergies with an
iterative nonnegative matrix factorization algorithm (Lee and
Seung 1999; Tresch et al. 2006). The reaching patterns have
been decomposed as combinations of phasic and tonic timevarying synergies using a modified version of the iterative
algorithm that we have previously used for the analysis of
phasic patterns of fast movements (d’Avella et al. 2006). We
found that simple synergy amplitude and timing modulation
rules explain the variation in the muscle patterns across different directions and speeds, providing further support to the
hypothesis that muscle synergies are basic building blocks for
constructing simple, task-dependent feedforward controllers.
MUSCLE SYNERGY MODULATION WITH DIRECTION AND SPEED
TABLE
1. Summary of muscles recorded for each subject
Subject
1
2
3
4
5
Biceps brachii short head (BicShort)
Biceps brachii, long head (BicLong)
Brachialis (Brac)
Pronator Teres (PronTer)
Brachioradialis (BrRad)
Triceps brachii, lateral head (TrLat)
Triceps brachii, long head (TrLongt)
Triceps brachii, medial head (TrMed)
Deltoid, anterior (DeltA)
Deltoid, middle (DeltM)
Deltoid, posterior (DeltP)
Pectoralis major, clavicular (PectClav)
Pectoralis major, sternal (PectInf)
Trapezius, superior (TrapSup)
Trapezius, middle (TrapMed)
Trapezius, superior (TrapInf)
Latissimus dorsi (LatDors)
Teres Major (TeresMaj)
Infraspinatus (InfraSp)
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logging the time of each kinematic sample with a counter (100-kHz
clock, PCI-6602, National Instruments) synchronized with the EMG
sampling clock.
Kinematic and EMG data were then digitally low-pass filtered
(15-Hz cutoff for kinematic data, 20-Hz cutoff for EMG data after
rectification; finite-impulse response filter with zero-phase distortion)
and segmented into individual trials. The EMG waveforms were
further integrated over 10-ms intervals to reduce the size of the data
set. Data for all trials, including those with a movement time outside
the instructed interval, were visually inspected and trials with either
irregular movement or with artifacts on some EMG channel were
removed from the data set (subject 1: 7/672; subject 2: 47/766; subject
3: 56/833; subject 4: 3/748; subject 5: 21/612).
Data analysis
ENDPOINT KINEMATICS. We characterized the kinematics of the
endpoint by measuring: movement onset time, movement end time,
movement duration, maximum speed and its time of occurrence, and
movement direction in the frontal plane. Movement onset and movement end were identified as the times in which the speed profile
crossed a threshold equal to 10% of its maximum. Movement duration
(or movement time [MT]) was defined as the interval between movement onset and movement end. The movement direction was computed as the angle of rotation of the movement vector on the frontal
plane from a vector pointing medially around the axis perpendicular to
the movement plane directed forward. Thus a movement directed
upward had a 90° direction and a movement directed laterally a 180°
direction.
For each subject, all trials were ordered according
to their movement duration and the trials with a movement duration
included from the 5th to the 95th percentiles were subdivided into five
groups (see Table 2). By removing the tails of the movement duration
distribution of each subject we ensured a better degree of consistency
in the kinematic and EMG characteristics within each group. Since the
movement distance was the same in all conditions, grouping according to movement duration was equivalent to grouping according to
speed.
400-ms interval, between 900 and 500 ms before the movement onset
and between 500 and 900 ms after movement end. We then studied the
effect of the movement parameters on these postural muscle patterns
of individual trials with two-way ANOVA (averaged EMG activity of
each muscle vs. 8 directions and 5 speed groups) and multiple linear
regression (averaged EMG activity of each muscle vs. cos ␪, sin ␪, and
maximum movement speed, where ␪ is the movement direction
angle).
The variations of the postural muscle patterns after movement end
across the experimental conditions were characterized by identifying
muscle synergies from the mean postural EMG activity, averaged over
repetitions in each direction and speed group. Thus for each subject,
we computed a set of 40 vectors, each representing the average
activity of all recorded muscles in one condition (8 directions ⫻ 5
speeds). We then used a nonnegative matrix factorization algorithm
(d’Avella and Bizzi 2005; Lee and Seung 2001; Ting and Macpherson
2005; Tresch et al. 1999) to decompose each one of these muscle
activity vectors (m) as the combination of a unique set of N timeinvariant synergies (wi) multiplied by condition-specific scaling coefficients (ci)
ci wi
i⫽1
These muscle synergies capture the coordinated synchronous recruitment of groups of muscles with specific amplitude balances. Thus
they capture the spatial structure of the postural muscle patterns. We
refer to these synergies as postural. To extract a set of N synergies, the
decomposition iterative algorithm was initialized with random values
for synergies and coefficients and it stopped when the reconstruction
R2 value increased by ⬍10⫺4 over 10 consecutive iterations. At each
iteration the algorithm performed two steps: 1) it updated the synergies given the data and the coefficients; 2) it updated the coefficients
given the synergies and the data. Since both synergy and coefficients
are constrained to be nonnegative, the updates are performed efficiently using a multiplicative procedure. For each N, the extraction
was repeated 10 times and only the synergy set with the highest R2
2. Grouping of trials according to movement
direction (MT)
TABLE
MT
Subject
Group
Minimum, ms
Maximum, ms
n
1
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
333
442
517
617
717
317
400
500
600
742
225
325
425
542
675
275
342
417
517
625
275
317
392
467
608
442
517
617
717
900
400
500
600
742
905
325
425
542
675
842
342
417
517
625
844
317
392
467
608
742
119
121
118
121
119
128
130
127
133
130
140
144
142
140
146
133
135
131
135
138
109
107
106
107
107
2
3
TRIAL GROUPING.
4
5
POSTURAL MUSCLE PATTERNS AND POSTURAL SYNERGIES. To
characterize the postural activity of individual muscles before and
after the movement, we computed the mean integrated EMG over a
J Neurophysiol • VOL
冘
N
m⫽
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Muscle
1435
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D’AVELLA, FERNANDEZ, PORTONE, AND LACQUANITI
was retained. We then selected a specific number of synergies, as a
compromise between parsimony and accuracy, according to the curve
of the reconstruction R2 as a function of N. We chose the number of
synergies at which the curve showed a change in slope (a knee),
indicating that adding additional synergies did not significantly improve the accuracy of the reconstruction (d’Avella et al. 2006; Tresch
et al. 2006).
We
characterized the directional tuning of the synergy amplitude coefficients with a cosine function. For each subject, synergy, and speed, we
performed a linear regression of the synergy coefficients (c) on
movement direction (␪)
DIRECTIONAL TUNING OF POSTURAL SYNERGY COEFFICIENTS.
c ⬃ cos ␪ ⫹ sin ␪ ⫹ 1
(1)
A cos ␪ ⫹ B sin ␪ ⫹ C ⫽ 冑A2 ⫹ B2 cos 共␪ ⫺ ␪PD 兲 ⫹ C
where ␪PD ⫽ tan⫺1 (B/A) is the preferred direction. We tested the
significance of the regression with an F test and quantified the
goodness of the fit with an R2 value. We also quantified the variability
of the preferred direction across speeds by computing its angular
deviation (Batschelet 1981), defined as the square root of 2(1 ⫺ r),
where r is the length of the vector resulting from the sum of unit
vectors directed as the preferred directions divided by the number of
vectors. Finally, to test for dependence of the synergy coefficients for
both direction and speed, we also performed the linear regression
c ⬃ spmax ⫹ cos ␪ ⫹ sin ␪ ⫹ 1
where spmax is the maximum of the tangential velocity during the
movement.
EXTRACTION OF TIME-VARYING SYNERGIES FROM PHASIC MUSCLE
PATTERNS FOR FAST MOVEMENTS. To study the organization of the
phasic muscle patterns across speeds, we first normalized in time the
integrated EMG waveforms to equal movement duration. The waveform of each muscle in each trial was aligned to the time of movement
onset and resampled, using linear interpolation, with 50 samples per
movement duration, from 1 MT before movement onset to 1 MT after
movement end. Time-normalized EMG waveforms were then averaged over trials in the same direction and speed group.
We first identified time-varying synergies, as we did previously
(d’Avella et al. 2006), from the phasic muscle patterns for the fastest
movements. We found a decomposition of the time-varying muscle
activation vectors [m(t)] as a combination of N time-varying synergy
vectors, or time-varying synergy [w(t)]
m共t兲 ⫽
冘
N
i⫽1
ci wi 共t ⫺ ti 兲
each scaled by an amplitude coefficient (ci) and recruited with a
specific onset delay (ti). Thus a time-varying synergy constitutes a
collection of waveforms, each one specific for a muscle, not necessarily synchronous. We extracted time-varying synergies from the
phasic waveforms. We estimated the phasic waveforms with a subtraction procedure. We first fit a linear ramp, made by a constant EMG
level from 1 MT before movement onset, a constant level 1 MT after
movement end, and a constant slope segment from movement onset to
movement end, to the EMG waveforms to estimate their tonic comJ Neurophysiol • VOL
SIMULTANEOUS EXTRACTION OF PHASIC AND TONIC TIME-VARYING
SYNERGIES. Inspection of the EMG waveforms for slow movements
(see Fig. 2) indicated that the tonic components were in most cases
much larger than the phasic components and that their shape differed
across muscles. Thus the subtraction procedure used for the fastest
movement, based on fitting the tonic components with a linear ramp
between movement onset and end, although providing a reasonable
approximation in case of fast movements with large phasic components, was not adequate for all movement speeds. We then decided to
estimate the tonic waveforms directly from the data, with the same
approach used for the phasic components, that is, identifying a set of
phasic and tonic time-varying components whose combinations would
match the entire muscle activation waveforms. Moreover, since the
tonic components included— before and after the movement—postural EMG activations, we modeled the generation of the tonic
waveforms as linear combinations of time-varying muscle synergies,
in accordance with the model of postural activations as a combination
of time-invariant synergies. Thus we decomposed the averaged EMG
data for all directions and speed, as a combination of phasic and tonic
time-varying synergies
m共t兲 ⫽
冘
N
p
i⫽1
cPi wPi 共t ⫺ ti 兲 ⫹
冘
N
t
j⫽1
ctj wtj 共t兲
where the superscripts p and t refer, respectively, to phasic synergies
and tonic synergies. The iterative algorithm used for the extraction of
phasic synergies (d’Avella et al. 2006) was modified to allow the
simultaneous extraction of tonic synergies, defined as time-varying
synergies with a fixed timing relationship with the movement kinematics and with slowly varying activation waveforms. Thus there was
no timing coefficient associated with the tonic synergy, whose duration was equal to the duration of the time-normalized EMG waveforms, 3 MT, and their activation waveforms were low-pass filtered
after each synergy update step during the optimization. We chose a
cutoff frequency equal to the inverse of 2 MT, corresponding to the
frequency of an oscillatory sinusoidal movement from start to target
with the same normalized average speed as that of the actual movements. For each subject, we extracted the same number of phasic
synergies as the number of phasic synergies adequately capturing the
phasic muscle patterns of the fastest movements and the same number
of tonic synergies as the number of postural synergies selected to
describe the postural muscle patterns at the end of the movement. We
initialized phasic and tonic synergies with random values and iterated
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To increase statistical power, we performed the regression on individual trial data. To obtain the synergy coefficients for individual
trials, we fit the synergy extracted from averaged data on the muscle
patterns for individual trials with the same algorithm used for the
synergy extraction without updating the synergies. We then expressed
the regression coefficients as preferred direction, modulation amplitude, and offset of a cosine function
ponents (Buneo et al. 1994; Flanders and Herrmann 1992). The
constant levels before and after the movements were computed by
averaging the EMG waveforms from 1 to 0.5 MT before onset and
from 0.5 to 1 MT after offset. The phasic waveforms were then
obtained by subtracting the tonic components from the total EMG
waveforms. As for the postural synergies, for each subject, we
extracted sets with one to eight synergies using an iterative optimization algorithm and we selected the number of synergies according to
the reconstruction R2 curve. The extraction algorithm was initialized
to random values for time-varying synergies 75 samples long (1.5
MT) and it stopped when the reconstruction R2 value increased by
⬍10⫺4 over 10 consecutive iterations. At each iteration the algorithm
performed three steps: 1) it found the onset delay times, for each
synergy and movement condition, with an iterative search (matching pursuits; Mallat and Zhang 1993), given the data and the
synergies; 2) it found, for each synergy and movement condition,
a nonnegative scaling coefficient by nonnegative least squares,
given the data, the synergies, and the onset times; and 3) it updated
the synergies, given the data and the timing and amplitude coefficients for all conditions, by gradient descent of an error function
weighting the reconstruction error and the presence of large negative components in the synergy waveforms (d’Avella et al. 2006).
For each N, the extraction was repeated 10 times and only the
synergy set with the highest R2 was retained.
MUSCLE SYNERGY MODULATION WITH DIRECTION AND SPEED
1437
tangential velocity profiles bell-shaped (Fig. 1). For each subject, trials were subdivided into five groups according to
movement duration (MT, Table 2) or, equivalently, given that
movement distance was constant, to movement speed. The
rectified, low-pass filtered, and integrated EMG waveforms
recorded from 17–18 shoulder and arm muscles (Table 1) were
aligned on the time of movement onset and averaged across
trials in the same direction and speed group. These EMG
waveforms showed a tonic activity, responsible for stabilizing
arm posture against gravity during movement, and a phasic
activity associated with the generation of the forces necessary
to accelerate and decelerate the arm (Fig. 2). The postural
activity after the movement appeared modulated by movement
direction. The phasic activity during the movement changed
with movement direction and, for a given direction, increased
in amplitude with increasing speed. Our goal was to relate the
changes in the activation waveforms of individual muscles
with direction and speed to the modulation of a few muscle
synergies, the coordinated recruitment of muscle groups.
DIRECTIONAL TUNING OF TIME-VARYING SYNERGY AMPLITUDE COEFFICIENTS. We characterized the directional tuning of the time-
Postural muscle synergies
varying synergy amplitude coefficients with cosine functions. We fit
the directional dependence of the coefficients with both a single
cosine function, using the same linear model used for the postural
synergies (Eq. 1), and with the sum of two nonnegative, offset cosines
c ⬃ 关A1 cos 共␪ ⫺ ␪1PD 兲 ⫹ B1 兴⫹ ⫹ 关A2 cos 共␪ ⫺ ␪2PD 兲 ⫹ B2 兴⫹
where Ai is the amplitude modulation of the ith cosine, Bi is its offset,
and ␪PD
is its preferred direction, with the positive part function [x]⫹
i
is x for x ⬎0 and 0, otherwise. The use of a threshold nonlinearity is
necessary to describe a bimodal cosine tuning since the sum of two
cosines is still a cosine (Flanders and Soechting 1990a). To determine
the fit parameters, (A B ␪PD) for a single cosine function and
(A1 B1 ␪PD
A2 B2 ␪PD
1
2 ) for a double cosine function, we used a
nonlinear least-square iterative curve-fitting algorithm (Matlab function lsqcurvefit). We initialized the amplitude and offset parameters to
half the mean coefficient value and the preferred direction to the
values of the largest local maxima of a periodic cubic spline approximating the directional dependence of the coefficients on the movement direction. We performed this nonlinear fit, for each subject,
synergy, and speed on individual trial coefficients, determined by
fitting the synergies extracted from averaged data on the individual
trial data using a single iteration of the first two steps of the synergy
extraction algorithm. We quantified the goodness of the fit with an R2
value and the variability of the preferred direction across speeds by
computing its angular deviation. For each condition, we fit both a
single and a double nonnegative cosine function and we compared the
results with the Akaike information criterion (AIC; Akaike 1974) and
the Bayesian information criterion (BIC; Schwarz 1978). We then
selected the number of cosines according to BIC, which favors models
with smaller number of parameters than AIC (Zucchini 2000).
RESULTS
We investigated the organization of the muscle patterns for
reaching in different directions and with different speeds.
Subjects moved a reference marker attached to a handle from
one central location to eight peripheral targets in the frontal
plane with a movement duration ranging, on average across
subjects, from 285 to 846 ms. For all movement directions and
speeds the endpoint paths were approximately straight and the
J Neurophysiol • VOL
We first analyzed the postural muscle activity recorded
before and after the movement. There was no significant effect
of movement direction on the EMG activity before movement
onset in most cases (79.1%; 68/86 muscles in five subjects;
P ⬎ 0.05; two-way ANOVA; averaged EMG vs. speed group,
five levels, and direction, eight levels) but there was a significant effect of speed in half of the cases (51.2%; 44/86 cases;
P ⬍ 0.01). The dependence on speed and direction of the
postural activity at the start location is likely due to movement
preparation, since trials with the same instructed movement
duration range were performed in blocks and the time interval
analyzed followed the delivery of the target cue (see METHODS).
The averaged EMG activity after movement end depended, as
expected, on movement direction (100%; 86/86 cases; twoway ANOVA; averaged EMG vs. speed group and direction;
P ⬍ 0.01) but also depended, in most cases, on speed (82.6%;
71/86 cases). This dependence on speed is probably related to
the control of endpoint stiffness at the target location through
an increase in cocontraction for faster movements. In fact, the
EMG activity after movement end increased with the movement maximum speed in most cases (68.6%; 59/86 cases;
significant positive regression coefficient for maximum speed
in the multiple regression of averaged EMG vs. maximum
speed and movement direction; P ⬍ 0.01).
We tested whether the postural muscle patterns recorded
after movement end, i.e., with the endpoint at eight different
locations in the frontal plane, could be reconstructed by the
combination of a small number of muscle synergies. For each
subject, we identified sets of one to eight muscle synergies
from the postural muscle patterns averaged over trials in the
same direction and speed, using a nonnegative matrix factorization algorithm (see METHODS). Three postural muscle synergies were selected in all five subjects as the number of
synergies at which the curve of the reconstruction R2 had a
knee (Fig. 3). The R2 value of the reconstruction ranged from
0.90 to 0.94 across subjects, indicating that the variations in the
postural muscle patterns were well captured by the combination of three synergies.
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the extraction algorithm until the reconstruction R2 value increased by
⬍10⫺4 over 10 consecutive iterations. The phasic synergy duration
was chosen at 1.5 MT. For each subject, we repeated the extraction 10
times and selected the synergy set with the highest R2 value.
We compared the phasic time-varying synergies extracted from the
muscle patterns for all speeds with the time-varying synergies extracted from the phasic patterns for the fastest movements by computing, as a similarity measure between two synergies, the maximum
of the normalized scalar product over all possible relative time shifts
of the two synergies (d’Avella et al. 2003). We also estimated a
chance-level similarity by computing the distribution of similarities
between random time-varying synergies generated with the same
amplitude distribution as that of the data sets from which the actual
synergies were extracted and with similar smoothness (d’Avella et al.
2006). Analogously, we compared the spatial structure of the tonic
synergies after movement end with the spatial structure of the postural
synergies extracted from the postural muscle patterns. To this aim we
computed the normalized scalar product between the vector constructed with the averaged tonic synergy waveform in the interval
from 0.5 to 1 MT after movement end and the vector representing a
postural synergy. Chance similarity levels were computed generating
random synergies with the same amplitude distribution as that of the
data.
1438
D’AVELLA, FERNANDEZ, PORTONE, AND LACQUANITI
2
1
0
2
0
0.5
1
2
1
1
up
0
0
0.5
0
1
0
0.5
1
2
lateral
1
1
medial
0
0.5
0
1
speed [m s-1]
0
2
10 cm
down
1
0
0
0.5
1
2
0.5
1
2
1
0
0
0.5
1
time [s]
1
0
0
0
0.5
1
FIG.
1. Example of endpoint paths and tangential velocity profiles. Paths of the endpoint in the frontal plane for movements from one central start location
(center) to 8 peripheral targets (periphery) and corresponding speed profiles for all movements with a duration between the 5th and 95th percentiles of the
movement duration distribution for one subject (subject 1) are shown. As for all other subjects, the movement trajectories had roughly straight paths and a
bell-shaped speed profile in all experimental conditions. The speed profiles are aligned on the time of movement onset and thin vertical line is plotted, for each
trial, at the time of movement end.
Each postural synergy expressed a specific balance in the
activation of the muscles and was modulated in amplitude
across movement conditions. For example, distinctive features
of the first postural synergy identified in subject 4 (W1,
Fig. 4A) were the strong activations of the short head of biceps
brachii (BicShort), anterior deltoid (DeltA), and trapezius,
especially the inferior (TrapInf) portion. This synergy was
recruited maximally for upward targets, as indicated by the
unimodal directional tuning of the synergy amplitude coefficient (C1, Fig. 4B). The amplitude coefficient for the upward
target was also modulated by the speed of the movement. The
structure of the second synergy (W2) was characterized by a
strong activation of the long head of biceps brachii (BicLong)
and pectoralis major (PectClav and PectInf) and by null activation of superior (TrapSup) and middle (TrapMed) trapezius.
This synergy was maximally recruited for medial targets and
modulated in the preferred direction by speed as well (C2).
Finally, the third synergy (W3) showed a strong activation of
the long head of biceps brachii, middle deltoid (DeltM), and no
activation of the short head of biceps brachii and the clavicular
portion of pectoralis major (PectClav). This synergy was maximally recruited for lateral targets (C3).
The directional tuning of the recruitment amplitude of the
postural synergies was well captured by a cosine function. To
perform a statistical analysis of the dependence of the synergy
amplitude coefficients on the movement direction and speed,
we fit the data for individual trials with the synergies extracted
from averaged data (see METHODS). We performed a linear
regression of the synergy amplitude coefficients for individual
trials, separately for each subject and speed group, on the
movement direction (Fig. 5). This regression, corresponding to
fitting a cosine function (see METHODS), was significant for all
FIG. 2. Example of averaged electromyogram (EMG) patterns. The band-pass filtered (20 – 450 Hz), rectified, low-pass filtered (20 Hz), integrated (10-ms
interval), and averaged within speed groups EMG waveforms for 5 upward movements (left) and 5 downward movements (right) are shown (top) together with
the corresponding averaged speed profiles. Vertical dashed lines indicate the time of movement onset and movement end.
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MUSCLE SYNERGY MODULATION WITH DIRECTION AND SPEED
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1
2
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BicShort
BicLong
Brac
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BrRad
TrLat
TrLong
TrMed
DeltA
DeltM
DeltP
PectClav
PectInf
TrapSup
TrapMed
TrapInf
LatDors
TeresMaj
1
speed [m s-1]
InfraSp
2
1
0
1s
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D’AVELLA, FERNANDEZ, PORTONE, AND LACQUANITI
1
0.9
0.8
0.7
R2
0.6
FIG. 3. Selection of number of postural synergies. The
curves of the reconstruction R2 value as a function of the
number of postural synergies extracted, in each subject (different symbols), from the EMG patterns recorded after the movement end show a change in slope at 3 synergies (arrow),
indicating that 3 is the optimal compromise between model
accuracy and parsimony in the choice of the number of synergies.
0.5
0.4
0.3
0.2
1
2
3
4
5
6
7
8
Number of postural synergies
subjects, speeds, and synergies (75/75 cases; F test; P ⬍ 0.01).
The median of the distribution of the R2 values of all the
regressions was 0.84. In comparison, the median on the R2
values of all the regressions of the EMG activity of individual
muscles (n ⫽ 435, 5 subjects, 16 –17 muscles, 5 speeds) was
0.60, significantly lower than the median of the R2 values for
the synergy coefficients (P ⬍ 10⫺6; Wilcoxon rank-sum test).
The angular deviation of the preferred direction of the cosine
tuning across speeds, for each subject and synergy, was very
small. The median of the distribution of angular deviation
values for all subjects and synergies was 7.9° and the maximum 26.7°, indicating that the directional tuning of the synergy amplitude coefficients did not change substantially with
movement speed. Finally, we performed a linear regression of
the synergy amplitude coefficients, for each subject, on movement direction and maximum speed. All regressions were
significant (15/15; F test; P ⬍ 0.01) and in 8/15 cases there was
a significant maximum speed regression coefficient (P ⬍ 0.01;
6/15 positive; 2/15 negative). Thus some of the synergies were
also modulated in amplitude with movement speed.
In summary, the variations of the postural EMG activity in
many shoulder and arm muscles at the end of point-to-point
reaching movements with different directions and speeds were
well captured by the combinations of three postural muscle
synergies. The synergy recruitment amplitude was modulated
with direction as a cosine function with a preferred direction
that did not change with movement speed.
Phasic and tonic time-varying muscle synergies
We then considered the muscle activity recorded before,
during, and after the arm movements. Our goal was to identify
J Neurophysiol • VOL
phasic synergies (i.e., synergies responsible for accelerating
and decelerating the arm during the movement) and tonic
synergies (i.e., synergies counteracting gravity and stabilizing
posture during movement) that could explain the complex
changes observed in the muscle activation waveforms across
movement directions and speeds. To compare the muscle
patterns for movements with different durations, we normalized in time the muscle waveforms to the movement duration
of each trial.
We first determined the number of phasic muscle synergies
required to adequately reconstruct the muscle patterns for the
fastest movements. For these movements, we estimated the
tonic waveform of each muscle by fitting a linear ramp and we
subtracted it from its total waveform to obtain a phasic waveform (d’Avella et al. 2006). For each subject, we then extracted
sets of one to eight time-varying synergies from the phasic
waveforms for movements in the group of trials with the
shortest duration in all eight directions, time-normalized to unit
movement duration and averaged over trials in the same
direction (see METHODS). Each time-varying synergy spanned a
time interval equal to 1.5 MT. We selected the sets with three
synergies, in all subjects, as the number of synergies at which
the curve of reconstruction R2 had a clear change in slope
(Fig. 6). The R2 value for three synergies ranged from 0.84
to 0.90 across subjects.
We then identified, for each subject, three phasic and three
tonic time-varying synergies from the entire set of EMG
waveforms of all muscles for all movement directions and
speeds, time-normalized and averaged across repetitions. The
optimization algorithm used to extract phasic synergies was
modified to allow the identification of additional tonic syner-
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subject 1
subject 2
subject 3
subject 4
subject 5
MUSCLE SYNERGY MODULATION WITH DIRECTION AND SPEED
A
W
W
1
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W
2
3
BicShort
BicLong
Brac
BrRad
TrLat
TrLong
TrMed
DeltA
DeltM
PectInf
TrapSup
TrapMed
TrapInf
LatDors
TeresMaj
InfraSp
0
B
0.2
0.4
0
0.2
C1
0.4
C2
0
0.2
0.4
C3
lateral
1
down
speed
1
2
3
4
5
FIG. 4. Example of structure and modulation of postural synergies. A: the 3 postural synergies extracted from the postural EMG patterns recorded after the
movements in one subject (subject 4) are shown as 3 columns of horizontal bars. Each synergy, a vector in muscle activation space normalized to the Euclidean
norm, shows a specific balance in the recruitment of the different muscles. B: the amplitude coefficients that multiply each one of the 3 synergies in A to
reconstruct the postural muscle patterns for 8 target positions and 5 movements speeds (gray level) are shown in polar plots. The directional tuning is unimodal
with a small dependence on speed.
gies, defined as time-varying synergies with a fixed delay with
respect to movement onset and slowly varying activation
waveforms (see METHODS). We chose the same number of
phasic synergies as the number necessary to adequately reconstruct the phasic muscle patterns for the fastest movements and
the same number of tonic synergies as the number of postural
synergies necessary to capture the variations in the postural
muscle patterns after the movement end. The duration of the
phasic synergies was set to 1.5 MT and the duration of the
tonic synergies to 3 MT, corresponding to the entire duration
of the muscle pattern analyzed in each condition. The R2 value
J Neurophysiol • VOL
for the reconstruction of the muscle patterns with three tonic
and three phasic synergies ranged from 0.86 to 0.89 across
subjects. The phasic synergies extracted with this procedure
were in most cases similar to the phasic synergies extracted
from the phasic EMG waveforms of the fastest movements.
For each subject, we matched the three phasic synergies
extracted from all speeds to the three synergies extracted from
the fast movement only and we compared their similarity with
the similarity between random synergies (see METHODS). In 10
of 15 comparisons, the similarity was significantly higher than
that by chance (P ⬍ 0.05). The mean similarity between the
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DeltP
PectClav
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D’AVELLA, FERNANDEZ, PORTONE, AND LACQUANITI
A
B
preferred direction
synergy
1
2
amplitude
3
1
2
3
subject
1
3
4
1
5
0
up
speed
1
2
3
4
5
lateral
1 2 3 4 5
speed
offset
r2 medial
amplitude
down
FIG.
5. Cosine tuning of postural synergy amplitudes. A: polar plots of the preferred direction of the amplitude coefficients for the 3 postural synergies
extracted from all subjects (rows). In each plot, the length of the radial segment indicates the R2 value of the cosine fit of the dependence of amplitude coefficients
on movement direction for each speed (gray level). B: bar plots showing the offset (dark gray) and the modulation amplitude (light gray) of the cosine fit of the
synergy coefficients. In all cases the cosine tuning well captures the directional modulation of the postural synergies, with a preferred direction essentially
invariant with speed and, in a few cases, a dependence of the maximum of the cosine function (sum of offset and amplitude coefficients) on speed.
two sets of synergies was 0.78 ⫾ 0.13 (SD). Similarly, in most
cases the vectors obtained averaging the tonic synergy waveforms over the final 0.5 MT interval, representing the spatial
structure of the tonic synergies, were similar to the postural
synergy vectors extracted from the postural EMG activity at
the movement end. In 9 of 15 comparisons, the similarity was
significantly higher than that by chance (P ⬍ 0.05). The mean
similarity between these pairs of synergies was 0.90 ⫾ 0.16
(SD). In sum, the organization of the muscle patterns for
reaching movements in different directions and speeds was
well captured by three phasic and three tonic time-varying
J Neurophysiol • VOL
synergies extracted simultaneously from the entire set of EMG
waveforms of all conditions.
The spatiotemporal organization of the phasic and tonic
synergies showed distinctive features. In each synergy different muscles were recruited with waveforms of different
shapes and different amplitudes. The synergies identified in
subject 4 (Fig. 7) illustrate many of the features common to
all subjects. Some of the waveforms of the phasic synergies
had a single positive peak, whereas other waveforms had
more complex shapes with two or three positive and negative peaks. In some cases the peaks in the waveforms of
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1443
1
0.9
0.8
0.7
R2
0.6
0.5
0.4
0.3
1
2
3
4
5
6
Number of phasic synergies
7
different muscles were roughly aligned but different muscle
groups peaked at different times. For example, Wp1 (Fig. 7,
first column) showed an early synchronous burst in elbow
flexors (BicShort, BicLong, Brac, and BrRad), shoulder
flexors (DeltA, PectClav, and PectInf), and two portions of
the trapezius muscles (TrapSup and TrapInf) followed by a
second synchronous burst of elbow extensors (all three
heads of triceps brachii: TrLat, TrLong, and TrMed) and
latissimus dorsi (LatDors). Similarly, in Wp3 (third column),
the waveforms of middle and posterior deltoid (DeltM and
DeltP), middle and inferior trapezius (TrapMed and TrapInf), and infraspinatus (InfraSp) were recruited synchronously. In other cases the timing of the different muscle
waveforms varied more gradually. For example, in Wp2
(second column), there was an initial negative (deactivation)
peak in anterior deltoid (DeltA) and the two portions of
pectorialis, followed, in sequence, by a positive peak in
DeltP, LatDors, and teres major (TeresMaj); a negative peak
in TrapMed and TrapInf; a large burst in the three heads of
triceps; a negative peak in the two heads of biceps; a
positive peak in DeltA, PectClav, and PectInf; and, finally,
a positive peak in TrapMed, TrapInf, and InfraSp. Thus in
general, the phasic synergies showed a rather complex
spatiotemporal organization with specific waveform amplitude and timing for each synergy. Some features of this
spatiotemporal organization, such as the synchronous recruitment of shoulder and elbow flexors in the initial phase
of Wp1, are in accordance with some of the well-established
kinematic and dynamic characteristics of reaching movements, such as the tight coupling of shoulder and elbow
motions and torques (Lacquaniti et al. 1986; Soechting and
J Neurophysiol • VOL
8
Lacquaniti 1981). Finally, the different tonic synergies (Fig.
7, columns 4–6) were characterized by specific amplitude
balances among muscles, whereas the shapes of the waveforms, constrained to contain only low frequencies, were
mostly slowly varying increasing or decreasing ramps.
Muscle pattern reconstruction by synergy modulation
The example of the reconstruction of the integrated, timenormalized, and averaged EMG patterns for upward and downward movements at different speeds of subject 4 (Fig. 8)
illustrates a remarkable finding of our analysis. The EMG
patterns for movements in a given direction at different speeds,
once decomposed into phasic and tonic synergies, appear
generated by a simple rule. A specific balance of tonic synergies was combined with a specific balance of phasic synergies,
with amplitude increasing with movement speed and invariant
timing. Even for the slowest movements, with EMG waveforms made up for the most part by the tonic activity, the small
phasic components were well captured by the same balance of
phasic synergy waveform highly recruited in the fastest movements. For example, the phasic Brac and TrapMed waveforms
during the slowest upward movements (Fig. 8, column 5,
peaking close to movement onset time, first dashed vertical
line), despite their small amplitude, had a shape and a timing
remarkably similar to those for the fastest movements (Fig. 8,
column 1).
The modulation of the recruitment of the phasic and tonic
synergies across all movement directions and speeds fully
characterized the rule for the generation of the muscle patterns
as synergy combinations. Phasic synergies were modulated
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FIG. 6. Selection of the number of phasic time-varying
synergies. The curves of the reconstruction R2 value as a
function of the number of time-varying synergies extracted, in
each subject (different symbols), from the phasic EMG patterns
recorded during the fastest movements show, as for the postural
synergies, a change in slope at 3 synergies (arrow).
subject 1
subject 2
subject 3
subject 4
subject 5
1444
D’AVELLA, FERNANDEZ, PORTONE, AND LACQUANITI
phasic
p
1
W
p
2
W
tonic
p
3
W
t
1
W
t
2
W
t
3
W
BicShort
BicLong
Brac
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BrRad
TrLat
TrLong
TrMed
DeltA
DeltM
DeltP
PectClav
PectInf
TrapSup
TrapMed
TrapInf
LatDors
TeresMaj
InfraSp
mean
1.5 MT
3 MT
FIG. 7. Phasic and tonic time-varying synergies for one subject. The waveform of 3 phasic (left) and 3 tonic (right) synergies extracted from the
time-normalized muscle patterns, recorded in all conditions, in subject 4 are shown together with the mean waveform (bottom). The duration of the phasic
synergies was set to 1.5 times the movement duration (MT), whereas the tonic synergies spanned the entire duration of the analyzed muscle patterns (3 MT).
The phasic synergies have distinctive synchronous and asynchronous activation waveforms (see RESULTS). The tonic synergies are constrained to have only slowly
varying activation waveforms by the extraction algorithm (see METHODS).
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MUSCLE SYNERGY MODULATION WITH DIRECTION AND SPEED
direction
speed
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1
2
3
4
5
1
2
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4
5
BicShort
BicLong
Brac
TrLat
TrLong
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EMGs and synergy reconstruction
BrRad
TrMed
DeltA
DeltM
DeltP
PectClav
PectLow
TrapSup
TrapMed
TrapInf
LatDors
TeresMaj
synergy coefficients
InfraSp
p
p
p
p
p
p
(c 1,t 1)
(c 1,t 1)
(c 1,t 1)
t
c1
ct2
ct3
1 MT
FIG. 8. Example of reconstruction of muscle patterns by time-varying synergy combinations. The averaged time-normalized muscle patterns for 5 upward
(left) and 5 downward (right) movements at different speeds (gray area) for one subject (subject 4) are shown together with their reconstruction (black solid line)
by the combinations of the synergies of Fig. 7, with amplitude and timing coefficients indicated by the height (amplitude) and the horizontal position (timing)
of the rectangles with the colored mean synergy waveforms. Remarkably, the contribution of the phasic synergies to the muscle activation waveforms, large for
the fastest movements, can be identified consistently even in the muscle waveforms for the slowest movements, where their contribution is much smaller than
the contribution of the tonic synergies.
both in amplitude and in timing by direction and only in
amplitude by speed. Tonic synergies were modulated in amplitude by movement direction but not substantially by moveJ Neurophysiol • VOL
ment speed. This modulation pattern is illustrated by the polar
plot of the synergy amplitude and timing coefficients as a
function of movement direction for different speeds of subject
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1446
D’AVELLA, FERNANDEZ, PORTONE, AND LACQUANITI
4 (Fig. 9). The amplitude coefficients (Fig. 9A) for both phasic
(columns 1–3) and tonic (columns 4–6) synergies had a similar
directional tuning for all speeds, but the maximum of the
tuning curves depended on speed (more saturated colors corresponding to higher speeds) only for phasic synergies. Moreover, the onset time (Fig. 9B) of the phasic synergies strongly
depended on the movement direction, although it had a subtle
yet significant dependence on direction and speed in the range
of directions in which each synergy had a large amplitude
coefficient (Fig. 9B, bottom, bar plots).
Directional tuning of synergy coefficients
A
phasic
cp1
tonic
cp2
cp3
ct1
ct2
ct3
lateral
medial
amplitude
up
2
down
B
t
phasic
tp2
p
1
tp3
up
medial-up
lateral-up
timing
1
lateral
-1
3
4
5
speed
0
1
lateral-down
2
lateral-down
medial-down
down
up
medial-up
medial-down down lateral-down
lateral-down lateral
0
0
-0.2
-0.5
-0.5
-0.4
-1
-1
0
FIG.
lateral-up
9. Modulation of amplitude and timing synergy coefficients across directions and speeds for one subject. The amplitude coefficients (A) for the phasic
(left) and tonic (right) synergies and the timing coefficient (B) for the phasic synergies extracted in one subject (subject 4) across speeds (different color
saturation) are shown in polar plots as a function of movement direction. Data points are connected by a periodic spline interpolation curve. All synergies are
modulated in amplitude with movement direction but only phasic synergies are strongly modulated in amplitude by movement speed. In the range of direction
in which the phasic coefficients are highest, for each synergy, the onset time has a small but significant modulation (bar plots below the timing polar plots).
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The directional tuning of the amplitude coefficients was
captured in most cases by either a single or a double cosine
function. We performed a statistical analysis of the synergy
coefficients for individual trials. We determined the coefficients by fitting the synergies extracted from averaged data on
the individual trial data (see METHODS). We first performed a
linear regression of the synergy amplitude coefficients, separately for each subject and speed, on the movement direction.
Such linear regression amounts to fitting a single cosine with
an offset to the data. The regression was significant in all cases
for tonic synergies (75 cases: 5 subjects ⫻ 5 speeds ⫻ 3
synergies; F test; P ⬍ 0.01) and in 71 of 75 cases for phasic
synergies. The median value of the distribution of the R2 values
of all the regressions for tonic synergies (n ⫽ 75) was 0.82,
whereas the median value for phasic synergies (n ⫽ 75) was
substantially lower, 0.41. Visual inspection of the tuning
curves indicated that in many cases the tuning curve for phasic
synergies had a single large peak but also a smaller peak
roughly in the opposite direction. We then performed a nonlinear curve fit of the amplitude coefficients of the phasic
synergies with either a single cosine or the sum of two
nonnegative offset cosine functions (see METHODS). The median
of the distribution of the R2 values for single nonnegative
cosine fit was 0.46 and the median for double cosines was 0.68.
In almost all cases, a double cosine better described the
directional tuning of the phasic synergy than a single cosine,
even taking into account the larger number of fit parameters of
the double cosine fit (66/75 cases according to the BIC; 70/75
cases according to AIC; see METHODS). In comparison, the
median of the R2 values of the linear regression of the amplitude coefficients of the postural synergies extracted from the
postural EMG activity after the movement (see earlier text)
was 0.84, not significantly different from the median value for
the tonic time-varying synergy amplitude coefficients (Wilcoxon rank-sum test, P ⫽ 0.24). In contrast, the median of the
R2 values of the cosine fit of EMG activity of individual
muscles after the movements (see earlier text) was 0.60,
significantly lower than the median value for the tonic timevarying synergies (P ⬍ 10⫺6). We also compared the R2 values
for the fits with a single or a double cosine function of the
amplitude coefficients of the phasic synergies with the R2
values of the fits with single or double cosines of the largest
MUSCLE SYNERGY MODULATION WITH DIRECTION AND SPEED
Synergy amplitude modulation with speed
The preferred direction of the cosine functions describing
the directional dependence of the amplitude coefficient of tonic
and phasic synergies did not vary substantially with movement
speed. For tonic synergies (Fig. 10A), the angular deviation of
the preferred directions of the single cosine fit across speeds,
for all subjects and synergies (n ⫽ 15), ranged from 0.6 to 6.5°,
with a median value of 3.7°. In contrast, the angular deviation
of a cosine fit of the EMG activity at the end of the movement,
across speeds, for all subjects and muscle (n ⫽ 87) ranged from
1.0 to 69.5°, with a median value of 6.0°, significantly higher
than that for the tonic synergy coefficients (Wilcoxon rank-sum
test, P ⫽ 0.0071). For phasic synergies (Fig. 11A), the angular
deviation of the preferred direction of the first cosine (see
METHODS) ranged from 0.5 to 41.4°, with a median of 3.7°. In
contrast, the angular deviation of the largest integrated EMG
burst ranged from 1.3 to 71.5°, with a median of 7.7°, significantly higher (P ⫽ 0.014) than the median for the amplitude
coefficients of the phasic time-varying synergies.
The amplitude of the cosine functions depended on speed in
most cases for phasic synergies (Fig. 11B) and in a few cases
for tonic synergies (Fig. 10B). In particular, the maximum
amplitude of the cosine functions for phasic synergies appeared
J Neurophysiol • VOL
to increase more than linearly with the speed. To evaluate the
effect of speed on the synergy amplitude coefficients, we
performed a linear regression, for each subject and synergy, of
the logarithm of the amplitude coefficients for all trials, in the
direction with the highest mean coefficient, on the logarithm of
the maximum speed (Fig. 12). For phasic synergies, all regressions were highly significant (n ⫽ 15, F test, P ⬍ 10⫺6), the
median R2 value was 0.84 (range 0.66 – 0.92), and the median
regression slope was 2.03 (range 1.40 –2.73). For tonic synergies, 7 of 15 regressions were not significant (P ⬎ 0.05) and
the median slope of the 8 significant regressions was 0.32
(range 0.12– 0.69), whereas their median R2 value was 0.50
(range 0.10 – 0.58). Thus the amplitude of the phasic synergies
in the direction of maximal recruitment increased roughly as
the square of the maximum speed. In contrast, in many cases,
the amplitude of the tonic synergies did not vary significantly
with speed and, when it did, it increased with speed less than
linearly.
DISCUSSION
Analysis of the muscle patterns recorded during reaching
movements in different directions and with different speeds led
to a number of findings that support our hypothesis of a
modular control architecture based on a direct mapping of task
parameters into muscle synergy recruitment parameters. First,
the complex changes observed in the muscle activation waveforms of several shoulder and elbow muscles across movement
directions and speeds are captured by scaling in time, scaling in
amplitude, and shifting in time a few time-varying muscle
synergies. Each individual muscle waveform is reconstructed
by the sum of component waveforms belonging to different
synergies and its changes in shape and timing across movement
conditions are systematically related, through the synergy
structure, to the changes of the other muscle waveforms. Thus
simple rules allow one to specify the entire muscle pattern
by selecting a small number of synergy recruitment parameters. Second, the muscle patterns for reaching are generated by
modulating two different types of time-varying synergies:
phasic and tonic. Phasic synergies are modulated both in
amplitude and in onset timing with movement direction and
speed. Tonic synergies are modulated in amplitude with movement direction and not substantially with speed. The directional modulation of the amplitude coefficients of the tonic
synergies is well described by a cosine function, whereas the
directional modulation of the amplitude coefficients of the
phasic synergies is captured, in most cases, by the sum of two
nonnegative cosine functions. In those cases, synergies are
recruited, in the preferred directions of different cosines, with
different onset times. The dependence of the amplitude coefficients on the phasic synergies on the movement speed is more
than linear and, on average, close to quadratic. Thus the rules
mapping movement direction to muscle patterns can be expressed through simple synergy coefficient tuning functions.
Third, the muscle activity recorded after the movement, responsible for maintaining a stable posture at the target location,
is generated by the combination of a few postural, timeinvariant synergies. The spatial structure of these synergies is
similar to the spatial structure of the tonic time-varying synergies at the end of the movement, compatible with the idea
that tonic synergies are responsible for antigravity and postural
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burst of EMG activity, integrated over a 0.5 MT interval, of
individual muscles. For both synergies and muscles we chose
one or two cosines according to the BIC criterion. The median
of the R2 values for the synergies was 0.67, significantly higher
than the median value for the individual muscles, 0.48 (P ⬍
10⫺6). Finally, in most cases, when the directional tuning of a
phasic synergy was well characterized by a double cosine, the
onset time of the synergy for movements in the direction of
peak amplitude of one cosine was different from the onset in
the direction of peak amplitude of the other cosine. The mean
difference, in absolute value, over all conditions and with
amplitude tuning best described by a double cosine (according
to BIC, n ⫽ 66), between the mean onset time of all trials with
a movement direction within 30° from the preferred direction
of the first cosine and the mean onset time of all trials with a
movement direction within 30° from the preferred direction of
the second cosine was 0.35 MT, significantly different from 0
(t-test, P ⬍ 10⫺6).
Since large variations of the synergy onset time occurred in
most cases for directions with a large angular deviation from
the synergy preferred direction, whereas in directions close to
the preferred direction the timing modulation was more subtle,
we assessed the role of synergy timing modulation by evaluating the effect of fixing the onset time on the reconstruction R2
values. For each subject and synergy, we fixed the onset time
for all conditions as the mean onset time weighted by the
synergy amplitude coefficient. We found that the R2 values on
average decreased by 11.2 ⫾ 3.9% (⌬R2/R2, mean ⫾ SD),
which indicated a small but significant contribution of the
timing parameter in the muscle pattern reconstruction. In
addition, we performed a two-way ANOVA to test the effect of
direction and speed on synergy onset time, for individual trials
in directions within 60° of the synergy preferred direction of
the single or first cosine. We found a significant (P ⬍ 0.01)
main effect of speed in 13/15 cases (3 phasic synergies ⫻ 5
subjects) and direction in 14/15 cases.
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D’AVELLA, FERNANDEZ, PORTONE, AND LACQUANITI
A
synergy
B
preferred direction
1
2
amplitude
3
1
2
3
subject
1
3
4
1
5
0
up
speed
1
2
3
4
5
lateral
1 2 3 4 5
speed
offset
r2 medial
amplitude
down
FIG.
10. Cosine tuning of tonic time-varying synergies. As for the directional tuning of the postural synergies (Fig. 5), the polar plots of the cosine tuning
preferred direction (A) and offset and amplitudes (B), for different speeds, indicate that the directional dependence of the tonic time-varying synergies is well
captured by a cosine function and does not change substantially with speed.
control and phasic synergies are responsible for overcoming
inertia and accelerate and decelerate the arm.
Synergy model formulation and identification
We have used two types of synergy models to characterize
the coordinated recruitment of groups of muscles underlying
the control of reaching. Postural muscle activity after the
movement end was modeled as the combination of timeinvariant synergies, whereas muscle activity before, during,
and after the movement was modeled as the combination of
time-varying synergies. A time-invariant synergy expresses a
relationship in the activations of different muscles that does not
J Neurophysiol • VOL
depend on time. This was a natural choice for capturing the
muscle patterns during the maintenance of a static arm posture.
In contrast, a time-varying synergy expresses a spatiotemporal
(across muscles and time) relationship in the muscle activations. Such a relationship is described by a collection of
activation waveforms that can be different across muscles and
can capture both synchronous and asynchronous muscle recruitment. Thus a time-varying muscle pattern can be described by just two recruitment parameters—amplitude and
onset time—for each synergy. However, a time-varying pattern
can also be described by time-varying combinations of timeinvariant synergies. We chose to characterize the regularities in
the time-varying patterns before, during, and after the move-
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MUSCLE SYNERGY MODULATION WITH DIRECTION AND SPEED
A
B
preferred direction
synergy
1
2
1449
amplitude
3
1
2
3
subject
1
3
4
1
5
0
1 2 3 4 5
speed
up
speed
1
2
3
4
5
lateral
single cosine
double cosine: first
double cosine: second
r2 medial
offset
amplitude
down
FIG. 11. Cosine tuning of phasic time-varying synergies. The directional tuning of the phasic synergies was captured better by the sum of 2 nonnegative offset
cosine functions (blue: first cosine function according to peak amplitude; red: second cosine) than by a single cosine function (gray) across movement speeds
(different color saturation or gray level) in most cases (see RESULTS). A: the preferred directions on the single (solid gray line) or double (blue solid line and red
dashed line) cosine functions did not, in most cases, depend on speed. The second cosine function often had a preferred direction approximately opposite to that
of the first cosine. B: the modulation amplitude (more saturated bar) and offset (less saturated bar) of the cosine functions showed a clear dependence of speed.
ment using a time-varying model because of the parsimony
of this model with respect to the time-invariant model. In
fact, the latter requires specifying the entire activation
waveform for each synergy to describe a time-varying muscle pattern (d’Avella et al. 2006). Moreover, when we extracted time-invariant synergies from the same timenormalized muscle patterns used to extract phasic and tonic
time-varying synergies (data not shown), we found that it was
difficult to distinguish phasic and tonic time-invariant synergies because most synergies showed both phasic and tonic
components in their activation waveforms. This is not surprisJ Neurophysiol • VOL
ing since, with a time-invariant model, any temporal regularity
in the muscle patterns can only be captured by the synergy
activation waveforms, and time-invariant synergies can be
separated into phasic and tonic only if the phasic and tonic
muscle activities are generated by orthogonal synergies, a
condition that does not appear to occur in practice. In contrast,
tonic time-varying synergies can be separated from phasic
synergies according to their spatiotemporal structure by requiring that their waveforms have frequencies not higher than the
movement frequency and that their onset is fixed with respect
to movement onset. In summary, we believe that parsimony
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2
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D’AVELLA, FERNANDEZ, PORTONE, AND LACQUANITI
2
Cp1, up
Cp2, down
Cp3, lateral
r2 = 0.87
b = 2.00
r2 = 0.77
b = 2.67
r2 = 0.91
b = 1.73
Ct1, up
r2 = 0.10
b = 0.12
Ct2, down
Ct3, up-medial
r2 = 0.25
b = 0.32
r2 = 0.00
b = 0.04 (n.s.)
p = 0.65
1
log C
0
-1
-3
-4
-1
0
1
log max speed
-1
0
1
-1
0
1
-1
0
1
-1
0
1
-1
0
1
FIG. 12. Example of dependence of synergy amplitude coefficients on movement speed. The scatterplots of the logarithm of the amplitude coefficient (C) for
individual trials of the 3 phasic and the 3 tonic synergies extracted in one subject (subject 4) as a function of the logarithm of the maximum speed (in m/s), for
all trials in the direction of maximal recruitment of each synergy, show that the dependence on speed for the phasic synergies (b: slope of linear regression, dashed
line; r2: coefficient of determination of the linear regression; all 3 regressions significant) is more than linear, whereas the dependence of the tonic synergies,
when significant (Ct1 and Ct2), is less than linear.
and the possibility of separating phasic and tonic synergies
according to their spatiotemporal structure support our choice
of a time-varying synergy model. However, our data and our
approach based on characterizing the regularities of the muscle
patterns during unperturbed reaching movements do not allow
us to distinguish experimentally between a time-varying and a
time-invariant model. Such a distinction might be possible by
testing the predictions of the two models on the effect of
experimental manipulations. Although the specific conclusions
may not be relevant for the control of reaching movements in
primates, such an approach has recently been pursued on
spinalized frogs by perturbing proprioceptive feedback (Kargo
and Giszter 2008).
Time-invariant postural synergies were identified with a
nonnegative matrix factorization algorithm (Lee and Seung
1999, 2001; Tresch et al. 1999, 2006). This algorithm iteratively finds a set of nonnegative synergy vectors, common to
all conditions, and a set of nonnegative combination coefficients, specific to each condition, that minimize the reconstruction error. This decomposition of muscle patterns into timeinvariant synergy vectors and time-varying combinations of
coefficients is related, and in many cases similar (Tresch et al.
2006), to that obtained with other algorithms such as principal
component analysis (PCA) (Mardia et al. 1979), factor analysis
(Basilevsky 1994), and independent component analysis (Bell
and Sejnowski 1995; Makeig et al. 1997). These algorithms
have been used to analyze the muscle patterns recorded in
different species and in different behaviors in many studies
(Cheung et al. 2005; Hart and Giszter 2004; Ivanenko et al.
2003, 2004; Krishnamoorthy et al. 2003; Olree and Vaughan
1995; Patla 1985; Saltiel et al. 2001; Ting and Macpherson
2005; Torres-Oviedo et al. 2006; Tresch et al. 1999; Weiss and
Flanders 2004). In contrast, time-varying synergies were identified with an iterative optimization algorithm developed specifically for identifying spatiotemporal components in the musJ Neurophysiol • VOL
cle patterns (d’Avella and Tresch 2002) and, to date, used only
in a few studies (d’Avella and Bizzi 2005; d’Avella et al. 2003,
2006). A time-varying decomposition approach based on PCA
has been recently used (Klein Breteler et al. 2007), although
this approach does not allow for independent time shift of the
muscle synergies.
Synergy models are useful only if the regularities in the
motor output expressed by the synergies provide information
on the functional architecture of the controller. Although the
assumption that regularities in the output of a system reflect its
internal organization is common in science, there are some
concerns about the application of these approaches to the
analysis of biological motor systems. First, it must be ensured
that the regularities are not due to poor sampling of the motor
output. For example, in examining movements in only one
condition one might erroneously take a stereotypical muscle
pattern as a centrally organized synergy simply because there
is not enough variability in the task specification. By sampling
eight reaching directions and a broad range of movement
speeds, we believe we have found an adequate compromise
between task variability and duration of the experimental
sessions. Muscle patterns change significantly across directions
and speeds (see Fig. 2), thus indicating that the structure
captured by the synergies is not simply due to a lack of
variability in the data. Second, if the synergies are organized by
the CNS and not mere statistical description of the regularities
in a specific data set, they should be able to capture the
regularities in the muscle patterns recorded in conditions not
included in the data used for the synergy identification. For
reaching movements, we have previously tested the robustness
of the time-varying synergies extracted from point-to-point
movements by showing that they can adequately reconstruct
the muscle patterns in novel dynamic and postural conditions
as well as patterns underlying more complex via-point and
reversal movements (d’Avella et al. 2006). For postural con-
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MUSCLE SYNERGY MODULATION WITH DIRECTION AND SPEED
trol, Torres-Oviedo and collaborators (2006) have shown that
the time-invariant synergies extracted from the hindlimb muscle patterns of cats during support surface translations in
different directions at a normal fore– hind paw distance capture
the muscle patterns recorded at different fore– hind paw distances as well as during support surface multidirectional rotations. Finally, a direct validation of the synergy models, as for
any model based on the parsimonious description of the regularities of the motor output, can come only from testing causal
predictions of the model, such as the effects of manipulations
of the controller. One exciting possibility is to perform such a
test in the context of motor learning and skill acquisition.
For all subjects, three phasic time-varying synergies were
selected to characterize the organization of the phasic muscle
patterns across movement directions and speed according to
the curve of the reconstruction R2 for time-varying synergies
extracted from the phasic muscle patterns of the fastest movements (Fig. 6), as in our previous study of fast reaching
movements (d’Avella et al. 2006). The point on the curve at
which the slope markedly decreased was taken as an indication
of the correct number of synergies. The rationale for this
selection is that, until all synergies capturing structural variation in the data are included, the reconstruction R2 increases
rapidly and that, after that point, only a small amount of
variation due to noise is captured by additional synergies.
Similarly, three time-varying tonic synergies were selected
because the curve of the reconstruction R2 for postural timeinvariant synergies showed a clear change in slope (Fig. 3).
We found that the structure of the time-varying synergies
extracted from the phasic muscle patterns of the fastest movements was in most cases similar to the structure of the phasic
time-varying synergies extracted together with the tonic synergies from the entire set of muscle patterns. This result
suggests that the structure of the phasic synergies is robust for
changes in speed, supporting their role as basic modules for the
generation of dynamic control signals. Moreover, the extraction of time-varying synergies from the subset of the data
collected in our previous study (d’Avella et al. 2006) containing the same movement conditions of the fast movements of
the present study (center-out point-to-point movements to eight
targets in the frontal plane with an unloaded handle and neutral
forearm posture) resulted in the selection of the same number
of three synergies for all nine subjects of that study. In 125 of
135 (92.6%) comparisons between the structure of the three
best-matching pairs of synergies in the sets of synergies extracted from the phasic muscle patterns of the nine subjects of
that study and from the five subjects of the present study, the
synergies were significantly more similar than by chance (P ⬍
0.05, with the procedure described in METHODS for comparing
time-varying synergies). Thus it appears that the difference
between the number of synergies selected in our previous study
(four to five) and that of the present study (three), depends on
the larger number of experimental conditions of our previous
study, which was nonetheless restricted to fast movements
only, including movements in both frontal and sagittal planes
and in both center-out and out-center orientations. In fact, the
structure of each one of the three phasic synergies identified in
this study (Fig. 7) appears related to the structure of five
J Neurophysiol • VOL
synergies extracted from fast movements in our previous study
(see Figs. 6 and 10 in d’Avella et al. 2006) through simple
combinations. For example, the first synergy in this study (Wp1)
is roughly matched by a combination of the first and the third
synergy of our previous study (W1 and W3). Consistently, all
three synergies have a similar directional tuning on the frontal
plane (Fig. 9 in both papers), whereas the two synergies
extracted in our previous paper have different tuning on the
sagittal plane. In sum, extracting synergies from a variable but
not exhaustive movement repertoire might result in a partial
resolution of the fundamental modular unit when these units
are recruited in a correlated fashion across the conditions
tested. Stronger evidence for a synergistic organization in
human arm movements will therefore come from an investigation of a broad range of natural movements, as done previously for human locomotion (Ivanenko et al. 2005) and in
lower vertebrates (d’Avella and Bizzi 2005).
Synergy modulation with direction and speed
The directional modulation of the synergy amplitude coefficients is characterized by a cosine tuning for postural (Fig. 5)
and tonic (Fig. 10) synergies and by a single or double cosine
tuning for phasic synergies (Fig. 11). Cosine tuning is characteristic of motor cortical neurons (Caminiti et al. 1991; Georgopoulos et al. 1982) and might represent an optimal strategy
for force generation with redundant actuators in the presence of
signal-dependent motor noise (Todorov 2002) or a general
tuning property of motor variables encoding time derivatives of
a position-dependent function (Zhang and Sejnowski 1999),
such as muscle shortening rates (Mussa-Ivaldi 1988). At the
muscle level, cosine tuning might also derive from a minimumeffort criterion (Fagg et al. 2002). Cosine tuning has been
observed in the muscle activations during step-tracking movements of the wrist in different directions (Hoffman and Strick
1999). However, the tuning observed in shoulder and elbow
muscles during reaching movements (Flanders et al. 1996) or
isometric force production (Flanders and Soechting 1990b;
Pellegrini and Flanders 1996) is described by more complex
functions than a single cosine function or even a double cosine
function. A key issue here is the time interval considered for
estimating the directional tuning. When it is possible to clearly
separate an agonist and an antagonist activation time interval,
as in most cases for the wrist muscles (Hoffman and Strick
1999), the rectified and integrated EMGs show a single cosine
tuning. However, when the timing of muscle activation gradually shifts with movement direction and the directional tuning
is computed integrating the EMGs over the time interval with
the largest activity (Flanders et al. 1996), the tuning functions
have multiple peaks. Thus a possible explanation of these
differences in directional tuning is that independently modulated components of the EMG activation waveforms may
be associated with a single cosine tuning but, because these
components can be shifted in time and scaled in amplitude differently across direction, the combined tuning can be
quite complex and can vary over the time course of the
movement. This explanation is in accordance with the existence of fixed relationships among these individual muscle
components expressed by muscle synergies and with our
observation of single and double cosine tuning of the synergy amplitude coefficients. In particular, when the phasic
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Number and structure of synergies
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J Neurophysiol • VOL
Postural and tonic synergies
Our results suggest that muscle synergies underlie both the
control of movement and the control of posture. Both control
tasks must deal with the complexity of a multiarticular arm and
both tasks might benefit from a low-dimensional representation
of the musculoskeletal characteristics provided by muscle synergies. Maintaining a stable static arm posture requires coordinating the activity of many muscles spanning shoulder and
elbow joints to generate the appropriate net torques and endpoint stiffness (Perreault et al. 2004). Postural time-invariant
synergies (Fig. 4A) might allow one to find an approximated
solution to the problem of finding an arm posture with the hand
in a desired position in space and the appropriate tonic muscle
activations for maintaining that posture against gravity and
against unexpected postural perturbations by selecting a few
synergy amplitude coefficients. Thus postural synergies might
allow one to implement a direct mapping of high-level postural
goals into muscle patterns.
Movement involves transitions between static postures and
requires appropriate muscle torques and arm impedance (Burdet et al. 2001) for balancing gravity and maintaining stability
in each intermediate posture as well as muscle torques for
accelerating and decelerating the joints. Tonic and phasic
time-varying muscle synergies appear naturally related to the
generation of antigravity and dynamic torques, respectively.
Such a decomposition of the torque generation process might
be a consequence of the different scaling rules for the two
torque components necessary for achieving speed-invariant
endpoint paths that the CNS might have exploited to simplify
its implementation of a controller.
Neural implementation of a synergy-based controller
The neural control of arm movements is implemented in a
distributed cortical and subcortical network involving many
different structures of the CNS. However, the anatomical
organization and the physiological characteristics of the motor
cortex suggest that it might have a specific role in the neural
implementation of a controller relying on muscle synergy
combinations. The divergent connectivity patterns of the axons
of corticospinal neurons in the motor cortex (Schieber 2001),
making synaptic contacts onto spinal interneurons (Shinoda
et al. 1979, 1981) and motoneurons (Cheney and Fetz 1985;
Fetz and Cheney 1980; McKiernan et al. 1998) across several
spinal segments, is an adequate anatomical substrate for the
organization of muscle synergies. Furthermore, the extensive
horizontal, intrinsic axon collaterals in the motor cortex (Huntley and Jones 1991) together with the widespread and overlapping representation of individual muscles on the cortical
surface (Rathelot and Strick 2006) might underlie the spatial
and temporal patterning of the activity across the population of
corticospinal neurons. In humans, focal transcranial magnetic
stimulation of the motor cortex shows increased excitability of
elbow and wrist muscles during a reaching movement involving coactivation of shoulder muscles, suggesting the involvement of common cortical circuits in the synergistic recruitment
of these muscles (Devanne et al. 2002). Finally, recent experimental results in anesthetized cats indicate that the EMG
outputs of two cortical points simultaneously stimulated are
additive (Ethier et al. 2006, 2007). These results suggest that
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synergies are recruited according to a double cosine, the
preferred directions of the two cosine functions are associated with different onset times, suggesting that each cosine
tuning function is associated with a different biomechanical
role in the control of the movement.
Phasic synergies are strongly modulated in amplitude
with movement speed (Figs. 11 and 12), whereas tonic
synergies, in most cases, are not (Fig. 10). As mentioned in
the INTRODUCTION, scaling in amplitude and in time the
dynamic torques, by an amplitude scaling factor equal to
the square of the time-scaling factor, and in time only the
antigravity torques generates invariant endpoint paths (Atkeson and Hollerbach 1985; Hollerbach and Flash 1982).
Since our EMG data have been time-normalized to equal
movement time, amplitude scaling of the phasic synergies
with speed suggests that these synergies are involved in the
generation of the appropriate dynamic torques for controlling movement trajectories with invariant paths. Similarly,
the lack of significant amplitude modulation of the tonic
synergies with speed suggests that they are mainly involved
with the generation of the appropriate antigravity torques
along the movement path.
Our results on the amplitude scaling of phasic and tonic
time-varying synergies with speed are compatible and extend previous results on the scaling of individual muscle
waveforms based on the normalization of the EMG time
base (Flanders and Herrmann 1992). After scaling the time
base of the EMG waveforms and endpoint kinematics so that
the movement times at different speeds are equal, the
waveforms associated with phasic synergies scale in amplitude with speed but the waveforms associated with tonic
synergies do not. However, when individual muscle activation waveforms are not time-normalized, the phasic components of individual muscle waveforms appear to correlate
best with templates whose time base is scaled in time
differently for different muscles (Buneo et al. 1994;
Flanders 2002). For example, during reaching movements in
one direction on the sagittal plane, Buneo and collaborators
reported that the time base of the phasic component of
anterior deltoid scaled with movement time more than the
time base of the phasic component of biceps brachii. We
found that the waveforms of most muscles, including anterior deltoid and biceps brachii (see Fig. 8), are well captured
by the combination of the synergy components after time
normalizing to equal movement time, i.e., with the same
time base factor for all muscles. These observations are not
incompatible. Muscle waveforms generated by combining
different components, each scaled in time by the same factor
but each shifted in time and scaled in amplitude independently, might correlate best with a template whose time base
is scaled by a factor different from the common scaling
factor of the components. However, the synergy model used
in this analysis, which assumes equal scaling for all synergies and captures a time-scaling mechanism common to all
synergies, would miss any additional synergy-specific scaling. Thus it will be interesting to refine our model to allow
for independent time-scaling of the synergies. To this aim,
we are developing a novel synergy extraction algorithm
capable of identifying a synergy timescale in addition to the
amplitude scale and time shift identified by the current
algorithm.
MUSCLE SYNERGY MODULATION WITH DIRECTION AND SPEED
the corticospinal circuitry is capable of implementing a physiological mechanism allowing for linear summation of the
muscle activations associated with the neural representation of
different synergies. Thus we speculate that a time-varying
muscle synergy might be driven by the dynamic activation of
a specific population of neurons in the motor cortex. The
spatial structure of the synergy might be determined by the
selective recruitment of spinal interneuronal and motoneuronal
populations by the set of corticospinal neurons active at each
specific time in the time course of the synergy. As time flows,
the activity distribution in the cortical population will evolve
along an attractor trajectory in neural space, thus determining
the spatiotemporal structure of the synergy.
Our results suggest that the combination of muscle synergies
is a general strategy that the CNS uses for implementing the
sensorimotor transformations necessary for reaching. A direct
mapping of goals and initial states into synergy modulation
coefficients might allow for constructing an inverse internal
model without an explicit representation of the dynamic behavior of the musculoskeletal system. Such a mapping might
exploit the structure of the equation of motion of the arm to
generate the torques appropriate for controlling reaching movements with invariant kinematic features through simple scaling
rules expressed as the amplitude and timing modulation of a
small number of time-varying synergies. Thus the knowledge
of the dynamics of the system would be implicitly incorporated
in the structure of the synergies. Such a control architecture—
based on a low-dimensional representation of motor output
provided by the synergies and low-dimensional mapping of
task-relevant sensory information into synergy recruitment
parameters—might also simplify motor learning and motor
adaptation.
ACKNOWLEDGMENTS
We thank Y. Ivanenko and V. C. K. Cheung for comments on the manuscript.
GRANTS
This work was supported by the Italian Ministry of Health, Fondo per gli
Investimenti della Ricerca di Base and Progetti di Ricerca di Interesse Nazionale from the Italian Ministry of University and Research and a Disturbi del
Controllo Motorio e Cardiorespiratorio grant from the Italian Space Agency.
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