Muscle synergies obtained from comprehensive mapping of the primary motor cortex forelimb representation using high-frequency, long-duration ICMS

While muscle synergies have been investigated in various muscle activity sets, it is unclear whether and how synergies may be organized in the cortex. We have investigated muscle synergies resulting from high-frequency, long-duration intracortical microstimulation (HFLD-ICMS) applied throughout M1. We compared HFLD-ICMS synergies to synergies from voluntary movement. While synergies can be identified from M1 stimulation, they are not clearly related to voluntary movement synergies and do not show an orderly topographic organization across M1.

Abstract

Simplifying neuromuscular control for movement has previously been explored by extracting muscle synergies from voluntary movement electromyography (EMG) patterns. The purpose of this study was to investigate muscle synergies represented in EMG recordings associated with direct electrical stimulation of single sites in primary motor cortex (M1). We applied single-electrode high-frequency, long-duration intracortical microstimulation (HFLD-ICMS) to the forelimb region of M1 in two rhesus macaques using parameters previously found to produce forelimb movements to stable spatial end points (90–150 Hz, 90–150 μA, 1,000-ms stimulus train lengths). To develop a comprehensive representation of cortical output, stimulation was applied systematically across the full extent of M1. We recorded EMG activity from 24 forelimb muscles together with movement kinematics. Nonnegative matrix factorization (NMF) was applied to the mean stimulus-evoked EMG, and the weighting coefficients associated with each synergy were mapped to the cortical location of the stimulating electrode. Synergies were found for three data sets including 1) all stimulated sites in the cortex, 2) a subset of sites that produced stable movement end points, and 3) EMG activity associated with voluntary reaching. Two or three synergies accounted for 90% of the overall variation in voluntary movement EMG whereas four or five synergies were needed for HFLD-ICMS-evoked EMG data sets. Maps of the weighting coefficients from the full HFLD-ICMS data set show limited regional areas of higher activation for particular synergies. Our results demonstrate fundamental NMF-based muscle synergies in the collective M1 output, but whether and how the central nervous system might coordinate movements using these synergies remains unclear.

NEW & NOTEWORTHY While muscle synergies have been investigated in various muscle activity sets, it is unclear whether and how synergies may be organized in the cortex. We have investigated muscle synergies resulting from high-frequency, long-duration intracortical microstimulation (HFLD-ICMS) applied throughout M1. We compared HFLD-ICMS synergies to synergies from voluntary movement. While synergies can be identified from M1 stimulation, they are not clearly related to voluntary movement synergies and do not show an orderly topographic organization across M1.

cortical coordination of muscle activity and movement is a fundamental question in neuroscience. In particular, reach-and-prehension movements of the arm and hand offer a major motor control challenge by requiring the coordinated activity of dozens of muscles across multiple joints (Bernstein 1967; Flash and Hogan 1985; Hogan 1985; 1988). Evidence from studies of cortical organization, spinal efferents, and voluntary electromyography (EMG) support control simplification by storing and recalling template-like synergy profiles that evoke EMG patterns in a predetermined fashion. In this system, instead of controlling each muscle individually, the brain may reduce the degrees of freedom in the neuromuscular system by utilizing fewer synergies than the total number of individual muscles available (Ajiboye and Weir 2009; Capaday et al. 2009; d’Avella and Bizzi 2005; d’Avella et al. 2006, 2003; Ethier et al. 2006; Flanders et al. 1994, 1996; Hayes et al. 2014; Macpherson 1991; Overduin et al. 2008, 2015; Safavynia et al. 2011; Saltiel et al. 2001; Ting and Macpherson 2005; Ting et al. 2015; Torres-Oviedo and Ting 2007; Weiss and Flanders 2004). The variety of movements needed to capture all, or most, of the synergies potentially needed to control a limb has only been addressed recently by acknowledging the fact that synergy results depend on data set completeness and constraints in the task or biomechanics of the movements (Steele et al. 2013, 2015). In this study we have taken an atypical approach to identifying synergies, in a manner similar to Overduin et al. (2012). In addition to analyzing synergies based on recording EMG activity during a range of different voluntary movements, we asked, what are the synergies represented in the neural output linkages from motor cortex to motoneurons and muscles? We evoked synergistic activation of muscles and associated movements by systematically applying trains of high-frequency, long-duration intracortical microstimuli (HFLD-ICMS) to primary motor cortex (M1).

We extracted synergies using nonnegative matrix factorization (NMF) from EMG activity resulting from HFLD-ICMS applied systematically throughout the forelimb representation of M1 (Lee and Seung 2001; Ting and Macpherson 2005; Torres-Oviedo et al. 2006) (appendix 1). We also extracted synergies from naturally occurring EMG activity associated with voluntary reaching movements to target locations throughout the monkey’s workspace. This is comparable to a recent study in which microstimulation was applied to 46 M1 precentral gyrus locations in two macaques (Overduin et al. 2012, 2014). The authors found that the EMG patterns could be reduced to linear sums of synergistic muscle coactivation patterns. Overduin and colleagues concluded that synergies may reasonably be stored in corticomotoneuronal circuits (Overduin et al. 2012, 2014). Here we have increased the specificity of the mapping by recording EMG from stimulation at 197 cortical locations in two macaques systematically covering the full extent of the M1 forelimb representation in the precentral gyrus. We theorized that electrical stimulation of cortical output may reflect the underlying organization of neural circuits for cortical control of movement, and so any synergies represented in stimulus-evoked EMG would provide a representation of potential muscle synergies available in the output of primary motor cortex.

The primary aim of this study was to identify the number and properties of muscle synergies obtained from NMF analysis of EMG activity associated with activation of cortical output by HFLD-ICMS applied systematically throughout the full extent of M1 cortex. Our secondary aims were 1) to compare synergy profiles and weighting coefficients for sets of HFLD-ICMS-evoked EMG data and 2) to compare HFLD-ICMS synergies with synergies derived from a voluntary movement-related EMG data set for each monkey. Our hypothesis was that the presence of synergies in EMG evoked by HFLD-ICMS applied to M1 would represent a synergy-based organization of M1. Our results showed only a few similarities between voluntary and HFLD-ICMS-evoked data and across monkeys. These synergies showed high probabilities of coactivating muscles with opposing joint actions and did not show an orderly topographic organization across M1. Our findings contribute to understanding the mechanisms and organization of muscle coordination in cortical networks from focal HFLD-ICMS applied to M1 and potentially to natural movement.

METHODS

Behavioral Task

The left M1 region representing the right forelimb was systematically mapped using HFLD-ICMS in two awake male rhesus macaques performing a reach-and-prehension task with their right forelimb. Each monkey was trained, as described previously (Park et al. 2004), in a cyclical reach-and-prehension task using its right forelimb. The task involved three distinct movement segments shown in Fig. 1: 1) depression of a lever near the waist for 1 s to activate a food reward, 2) reach and grasp of a banana pellet at shoulder level, and 3) transportation of the food pellet to the mouth. During training and data collection the monkey was seated in a custom primate chair, designed for comfort, with its left forelimb restrained, a restraining plate at the waist, and a Plexiglas shield in front of its face to prevent access to the cranial implant, microelectrode positioner, electrode, and cables. The shield contained a small hole that allowed the monkey to put the food pellets in its mouth.

Fig. 1.

Diagram of monkey engaged in three-location reach-to-grasp task: 1, home plate lever; 2, food-well; and 3, mouth. Approximate marker cluster setup also pictured (not to scale).

Surgical Procedures

Following training, each monkey was surgically implanted with a titanium recording chamber positioned over the left forelimb M1 cortex and subcutaneously tunneled EMG electrodes as described previously (McKiernan et al. 1998, 2000; Park et al. 2001). Head movements were limited with a flexible head restraint system that restricted turning of the head (McKiernan et al. 1998, 2000). An X–Y microelectrode positioner was attached to the chamber and used to position a single electrode over targets in M1 cortex. Glass insulated platinum-iridium electrodes with impedances of 0.7–1.5 MΩ were slowly lowered through the dura and into the cortex to record neural activity and perform stimulation.

EMG electrodes were surgically implanted into 24 muscles (Table 1) by subcutaneously tunneling wires to the muscles from a cortical connector (Amphenol, Wallingford, CT) secured with dental acrylic near the cortical recording chamber. Pairs of multistranded stainless steel wires (Cooner Wire) were inserted into each muscle to a length of ~2–3 cm with ~5 mm separation between the ends of the wires. Placement accuracy was tested by observing muscle twitches induced from short stimulus trains (Grass SD9 Stimulator, West Warwick, RI), and cross talk was tested before recording (Hudson et al. 2010; Park et al. 2000; Van Acker et al. 2013). All surgeries were performed as directed by the standards of the Guide for the Care and Use of Laboratory Animals, published by the United States Department of Health and Human Services and the National Institutes of Health. The protocol was also reviewed and approved by the Institutional Animal Care and Use Committee.

Table 1.

Muscles recorded with EMG, listed by their primary joint of action

Shoulder	pectoralis major (PEC), teres major (TMAJ), latissimus dorsi (LAT), anterior deltoid (ADE), posterior deltoid (PDE)
Elbow	triceps long head (TLON), triceps lateral head (TLAT), dorsoepitrochlearis (DE), brachioradialis (BR), brachialis (BRA), biceps long head (BIL), biceps short head (BIS)
Wrist	extensor carpi radialis (ECR), extensor carpi ulnaris (ECU), flexor carpi radialis (FCR), flexor carpi ulnaris (FCU), palmaris longus (PL)
Digit	extensor digitorum communis (EDC), extensor digitorum 2 and 3 (ED23), extensor digitorum 4 and 5 (ED45), flexor digitorum superficialis (FDS), flexor digitorum profundus (FDP)
Intrinsic hand	abductor pollicis brevis (APB), first dorsal interosseus (FDI)

Data Acquisition

Motion capture.

Motion capture data were recorded simultaneously with EMG activity using the Vicon Nexus system (Los Angeles, CA). Two clusters, each containing 10 markers, were secured with soft Velcro fabric on semirigid plastic sheets curved around the upper and lower forelimb (Fig. 1). For each data session, four static trials of 1 s each were collected. For these, the forelimb was held in a consistent orientation and one of four bony landmarks was manually probed, including the acromion process of the shoulder, the lateral epicondyle of the elbow, and the radial and ulnar styloid processes of the wrist. The locations of these landmarks were later calculated over the time of the trials for that session. Data were buffered so collection began 500 ms before stimulation onset for each trial and were collected for a total of 5 s. Kinematic data were sampled at a frequency of 100 Hz.

HFLD-ICMS protocol.

HFLD-ICMS was applied to a series of single M1 locations while the monkey performed the behavioral task. Stimulation typically interrupted the voluntary movement and caused the monkey’s forelimb to move involuntarily (Griffin et al. 2014). At the start of each data collection section, a manual hydraulic microdrive (FHC) coupled to a custom X–Y positioner was used to position a microelectrode at or near putative lamina V sites in M1 cortex. Electrode track spacing along the surface of the precentral gyrus was 1 mm. Down the anterior wall of the precentral gyrus stimulation sites were spaced at 0.5-mm intervals (Cheney and Fetz 1985; Park et al. 2001). The location of layer V was based on electrode depth, the presence of large spike waveforms, and the presence of poststimulus effects (PStEs) in stimulus-triggered averages (StTAs) at 15 μA (Park et al. 2004). Each set of trials at a single site was collected in a block and the site locations were selected randomly throughout the days of data collection.

Individual stimuli were symmetrical biphasic pulses, with an initial cathodic pulse followed immediately by an anodic pulse each of which were 0.2 ms in duration. EMG data were collected from 24 muscles of the forelimb using intramuscular electrodes with hardware set gains, filtered from 30 Hz to 1 kHz (Grass P511) and digitized at 4,000 Hz. Cross talk between EMG electrodes was examined by compiling and comparing EMG-triggered averages to poststimulus effects (Buys et al. 1986; Van Acker et al. 2013). Based on this analysis, first dorsal interosseus in monkey X was deleted from the database.

HFLD-ICMS was triggered manually when the monkey was at one of the three discrete locations (Fig. 1) within the task. Stimulation parameters were between 90 and 150 Hz and between 90 and 150 μA as previously determined to be effective for evoking limb movement and stabilization at a spatial end point (Van Acker et al. 2014). A stable spatial end point was defined as translocation with subsequent stabilization of the wrist in space for the remaining duration of stimulation (Van Acker et al. 2014). EMG and motion capture data were imported into MATLAB (MathWorks, Natick, MA) for postprocessing and analysis. The HFLD-ICMS data was broken into two sets for analysis: a subset with five trials from each of the site locations in M1 (ALL), and a subset of the ALL sites that included up to five trials that reached stable spatial end points (END) (described in detail below, in Stable spatial end point determination).

Voluntary movement protocols.

A data set for voluntary movement-related EMG activity coupled with motion analysis data was recorded from each monkey performing a reach-and-prehension task. For monkey A, motion and EMG were captured while the monkey performed the same fixed-target reach-and-prehension task used throughout the HFLD-ICMS recording sessions. Blocks of approximately five trials of the cyclical reaching task were collected throughout multiple recording days. For monkey X, a variable-target reach-and-prehension task was used, in which a food reward was presented throughout the available workspace to elicit a wide range of voluntary reaching movements and to recreate some of the range of movements evoked with HFLD-ICMS. Voluntary movements were primarily in front of monkey X, limited laterally and superiorly by the limits of eye movements and peripheral vision (the head was restrained), medially by forelimb range of motion, and inferiorly by the chair waist plate. The majority of these trials were collected in a single session, with blocks of approximately five trials at each reaching target location. The data set was pooled with a few blocks of trials of monkey X performing the standard fixed-target task on multiple days.

Data Analysis

Stable spatial end point determination.

Motion capture was used to determine when the radial tuberosity of the forelimb (wrist) reached a stable position in 3D space, as described previously (Van Acker et al. 2014). Briefly, the end point was calculated by using a modified point cluster technique to calculate the location of the wrist in relation to a cluster of markers on the distal forelimb segment. The overall speed of the wrist was calculated from the derivative of the virtual wrist positional coordinates. A stable spatial end point was defined as the time when the speed of the virtual wrist marker dropped below a speed threshold. The speed threshold of 372.1 mm/s was defined as 25% of the maximum speed for both monkeys (1,488.1 mm/s) determined during initial data collection, which allowed for a best match of algorithm to real-time visual assessment of stable spatial end points (Van Acker et al. 2014).

Processing EMG data.

Using MATLAB software (MathWorks), EMG data were high-pass Butterworth filtered with a cutoff frequency of 35 Hz and were full-wave rectified about the mean. Stimulus artifacts were removed using a weighted averaging method of ±2 ms surrounding each stimulus pulse. EMGs were then low-pass filtered at 40 Hz and for each day, the maximum observed EMG value was used to normalize each muscle’s recordings (Park et al. 2000; Van Acker et al. 2014). For HFLD-ICMS trials, the EMG activity from each muscle was averaged across the stimulus window of 1,000 ms. For the voluntary movement trials, the EMG from each muscle was averaged in 100-ms windows over the 1,500-ms trial duration, resulting in 15 data points for each trial (Ting and Macpherson 2005; Torres-Oviedo et al. 2006). This processed EMG data formed the data matrix for NMF analysis.

Datasets included for analysis.

Three primary EMG data sets were included in the synergy analysis for each monkey: two HFLD-ICMS-evoked EMG data sets and one data set of voluntary movement-evoked EMG. Stimulus-evoked synergies were calculated from a set of data obtained from all cortical stimulation sites (ALL) in M1 and a subset of data from only sites with stable spatial end points (END). Both sets included approximately five trials at each cortical location with stimulus parameter ranges from 90 to 150 μA and from 90 to 150 Hz, with preference given to trials with parameters that were more likely to be effective (near 110 μA and 110 Hz). The set of five trials nearest the preferred parameters (110 μA and 110 Hz) was selected for the ALL data set, while the five trials that achieved a stable spatial end point nearest the preferred parameters was selected for the END data set (Van Acker et al. 2014). Thus all of the END cortical locations were included in the ALL cortical locations, but the individual trials varied. Voluntary data (VOL) included EMG from a reach-and-prehension task: fixed-target for monkey A and variable-target for monkey X, as described above under Voluntary movement protocols. For monkey X in particular, every attempt was made to introduce as much variability in the voluntary movement as possible, but movements were still constrained by the monkey chair. Monkey A did not adapt well to the free-ranging, multitarget reaching task so only the standard fixed target reaching task was used.

In addition to the two HFLD-ICMS-evoked EMG data sets, the effect of using a smaller percentage of sites was investigated (10–100% in 10% intervals) by progressively adding back data from 10% of the ALL data set sites in random order. These were termed random site data sets (RS). For example, the 20% RS data set included all the sites in the 10% RS data set, plus another randomly selected 10% of sites from the ALL data set. Synergies for this set were fully recalculated 20 times, with 20 different random permutations of sites. Synergies and R values for each data set from 10% of sites through 100% of sites were then sequentially averaged and compared. The purpose of this exercise was to investigate how much data from our ALL data set was needed to find reliable and consistent synergies. Accurate synergies depend on a complete sampling of the control space (Burkholder and van Antwerp 2012), but the amount of variation in actual movement of the forelimb was uncontrollable during ICMS. Thus we instead investigated how consistent synergy results were by iteratively and consecutively adding 10% of the stimulated cortical locations from the ALL data set of each monkey.

Nonnegative matrix factorization.

NMF was used to determine the muscle synergy patterns present in forelimb EMG activity from each of the data sets and subsets (Lee and Seung 2001). NMF factors a large data set into two components: synergies and weighting coefficients (Fig. 2). The user must specify the number of synergies, or rank, as all the synergies and weighting coefficients are completely recalculated and can be completely different at each specified rank. Synergies are nonorthogonal vectors, normalized to unit length, which describe muscle activation groups (Paatero and Tapper 1994; Torres-Oviedo and Ting 2010). Muscles are listed in each synergy with nonnegative values activated relative to each other and are renormalized to have the most significant muscle in each synergy represented with a value of 1. Weighting coefficients are vectors containing values by which each synergy is multiplied for each trial in the data set. The EMG pattern can then be summarized as follows: EMG = V = W * H (Lee and Seung 2001), where V is a (m × t) matrix of the original EMG data with t time bins or trials and m muscles; W is a (m × k) matrix with k basis column vectors, or synergies, of unit length, with representations for each muscle in each synergy; and H is a (k × t) matrix containing the weighting coefficients for each synergy for each time bin or trial in the original data set (Berry et al. 2007; Donoho and Stodden 2003; Lee and Seung 1999, 2001; Ting and Macpherson 2005; Torres-Oviedo et al. 2006). See appendix 1.

Fig. 2.

Synergy concept. Depiction of how NMF synergies would theoretically combine linearly to generate a variety of muscle activation patterns. C is the weighting coefficient vector, W is a synergy vector (with elements of w), and mn is a motoneuron. Figure modified from Ting and Macpherson (2005).

Rank selection: variation accounted for.

The NMF process was repeated with increasing specified rank, or number of synergies. The rank needed to describe the data was decided post hoc based on a threshold of 90% of the overall variability accounted for (VAF) for an increased rank, where VAF = 1 – SSE/TSS (d’Avella et al. 2006; Torres-Oviedo et al. 2006; Torres-Oviedo and Ting 2010). VAF is the uncentered Pearson correlation coefficient, SSE is the sum of the squared errors, and TSS is the total sum of the squares, taken with respect to zero (d’Avella et al. 2006; Torres-Oviedo et al. 2006; Torres-Oviedo and Ting 2010).

Synergy profiles and weighting coefficients used.

Synergies were identified and normalized to the maximum value in each synergy. The scaled synergies represent both the amount of variation in each muscle’s data explained by that synergy (Paatero and Tapper 1994) and how the muscle activation levels correlate relative to one another (Lee and Seung 2001; Torres-Oviedo et al. 2006; Torres-Oviedo and Ting 2010). For example, when a synergy is activated, a muscle with a value of 1 in the synergy vector designates that the muscle would be activated at a level higher (proportional to its maximum activation) than a muscle at a level of 0.5. The actual amount of muscle activity generated would depend both on the relative scaling of the muscles in the synergy vector, and how high the total synergy is activated by the weighting coefficients. Synergy vectors were plotted as bar graphs and arranged around unit circles to show muscle relationships for the different data sets. To compare synergies from two data sets, two synergy vectors were first matched by selecting the pairs with a minimum dot product between the two vectors. Significant similarity of two synergies was determined by finding the correlation coefficient between the two vectors with a significance level of P < 0.05 (Overduin et al. 2012). In a manner that is distinct from principal component analysis, synergies in the figures are arranged based on the best match between two data sets. This, coupled with the fact that any recalculation of a different number of synergies from the same data set will yield different results, means that it is impossible to draw conclusions about what the synergies from other ranks would represent.

In addition to looking at the number of synergies needed to explain an EMG data set, and the profile of those synergies, the weighting coefficients from the HFLD-ICMS-evoked EMG data set from all M1 stimulus sites (ALL data set) were mapped to the forelimb region of M1. Weighting coefficients from the NMF results identify the level of activation of each synergy in each trial. For these maps, the weighting coefficients for each synergy from the three to five trials recorded from each stimulus site were averaged. This average was plotted on M1 cortical maps at the location of the stimulation.

RESULTS

Data Set Characteristics

An example data set of EMG recordings of a single cortical site’s response to HFLD-ICMS and voluntary movement is shown in Fig. 3. The characteristics of the two HFLD-ICMS-evoked EMG data sets for synergy analysis, including the mean frequency and intensity of the trials used in each stimulus-evoked data set, are summarized in Table 2. The random data sets (RS) were omitted for simplicity’s sake and used later.

Fig. 3.

EMG activity at a single cortical site derived from HFLD-ICMS that interrupted voluntary activity to move the monkey's hand from the pellet feeder to a stable end point near the mouth (A) and voluntary movement over the same feeder-to-mouth trajectory (B). HFLD-ICMS was collected at 110 Hz and 110 µA for 1 s. starting when the hand was at the feeder. The gray shading denotes the HFLD-ICMS duration (A) and voluntary movement (B) from the feeder to the mouth approximating the same time frame and movement trajectory of the left column (HFLD-ICMS). HFLD-ICMS and voluntary movement records are averaged across five trials. All HFLD-ICMS EMG data were processed to remove stimulus artifacts. Note that our previous work has shown that HFLD-ICMS blocks natural activation of cortical output neurons and produces a pattern of muscle activation that is entirely stimulus driven (Griffin et al. 2011; Cheney et al. 2013). Therefore, it is not surprising that the detailed muscle activation patterns observed for voluntary movement (right column) and HFLD-ICMS (left column) are dissimilar, although many basic features generally match, for example, activity in elbow and shoulder flexors. For muscle abbreviations, see Table 1.

Table 2.

Characteristics of the stimulus-evoked EMG data sets used for synergy analysis

	Monkey A		Monkey X
	All HFLD-ICMS sites (ALL)	Sites from HFLD-ICMS with stable spatial end points (END)	All HFLD-ICMS Sites (ALL)	Sites from HFLD-ICMS with stable spatial end points (END)
Sites included, no.	122	94	75	74
Total trials included, no.	603	319	374	280
Range of trials per site, no.	3–5	1–5	4–5	1–5
Mean trials per site, no.	4.9 ± 0.3	3.4 ± 1.6	5.0 ± 0.1	3.8 ± 1.5
Sites at the preferred 110 µA, 110 Hz, %	67.4%		72.0%
Mean frequency, Hz	108.7 ± 7.8	112.2 ± 7.8	110.5 ± 7.8	109.8 ± 7.8
Mean intensity, µA	108.5 ± 7.6	111.0 ± 7.7	110.5 ± 7.8	109.8 ± 7.8
Overlap in trials, no.	178		147
Number of trials in voluntary movement data sets (VOL)	71 (fixed-target)		104 (variable-target)

Values are means ± 1 SD.

As can be seen in Fig. 4, fewer synergies were needed to explain voluntary movement-evoked EMG than stimulus-evoked EMG. Two or three synergies were needed to account for 90% of the variation in the VOL EMG, while four or five synergies were required for the ALL and END data sets. These sets of synergies were used in further analysis, while other sets were discarded.

Fig. 4.

Overall variation accounted for (VAF) for the six data sets for increasing number of synergies. Monkey A’s data (“A”) is shown with circles, monkey X’s data (“X”) is shown with x’s. Data from the ALL sites data set is orange, green shows data from the sites with stable end points (END), and purple denotes data from voluntary movement (VOL). Red boxes mark the first rank to cross the 90% VAF threshold, which were designated for further analysis. Note that A VOL is derived from the fixed-target task, while X VOL is derived from the variable-target task. Note that the first crossing of the 90% threshold for the ALL and END data points for monkey X are nearly superimposed and contained in one box.

Synergy Profiles Compared Within Each Monkey

When the profiles of each synergy set were examined within each monkey, three of the four or five synergies from the monkey’s two stimulus-evoked data sets were similar. For monkey A, the four END synergies data set were matched to the best four of the five ALL synergies and three of the four pairs were significantly correlated (P < 0.05; range R = 0.2653–0.7648) (Fig. 5, top and middle rows, dark gray plots). Monkey X had four synergies for each stimulus-evoked data set and all 4 pairs were significantly correlated (P < 0.05; range R = 0.4600–0.9923) (Fig. 5, top and middle rows, light gray plots). As a reference point, when each ALL synergy set was compared across synergies within one monkey, no pairs were significantly correlated. The largest correlation for monkey A was between A2 and A4 (labeled in Fig. 8) at 0.29, P = 0.17 (min R = −0.39, mean R = −0.09 ± 0.24) and for monkey X was between X2 and X3 at R = 0.16, P = 0.57 (min R = −0.42, mean R = −0.18 ± 0.18).

Fig. 5.

The synergies profiles shown are the 2–5 synergies needed to reach the 90% VAF threshold for each data set. Left columns, monkey A; right columns: monkey X. Synergies are from top row, ALL sites stimulated near 110 μA, 110 Hz (ALL); middle row, all sites with stable spatial end points (END); bottom row, voluntary data (VOL, including fixed-target task data for A and variable-target task for X). Synergies in the middle and bottom panels are aligned with the closest match to the ALL synergies. Pearson correlation coefficients (R) to the ALL sites data set are printed under the end point subset and the voluntary data set, synergies (*P < 0.05). The muscle order is listed on the left (*FDI was only collected for monkey A).

Fig. 8.

Radial plots for stimulus-evoked synergies from ALL cortical sites (A) and VOL data compared across monkeys (B) (monkey A in the red dashed lines and monkey X in blue). ALL data synergies are from the top row of the bar plots in Fig. 5, while VOL data synergies correspond to the bottom row in Fig. 5. Synergies are arranged in each set in order of best to worst correlation. VOL data is from the fixed-target task for monkey A and the variable-target task for monkey X. C: the radial plot legend shows shades of purple designating different regions of the arm from most proximal to most distal, while muscles with higher activations are labeled on each plot. The muscles are arranged in 15° increments around a unit circle where the outermost circle has value of 1, and the dashed circles show levels of muscle activation at 0.25, 0.5, and 0.75.

Synergies from the voluntary data sets were less similar to the synergies from stimulus-evoked EMG. The three VOL synergies from monkey A (Fig. 5, bottom row, dark gray plots) were matched with the three most similar synergies from the five synergies in the ALL data set, resulting in two significant correlations (P < 0.05) and an average correlation of 0.5157 (mean P = 0.106, range R = 0.2137–0.7587). For monkey X, the two VOL synergies (Fig. 5, bottom row, light gray plots) and the four ALL synergies had only one significant match and the average of the two best correlations was 0.3485 (mean P = 0.2721, range R = 0.1350–0.5620). Although the voluntary synergies have some similar characteristics to the stimulus-evoked synergies, they were less similar than the two sets of stimulus-evoked synergies. However, this is not unexpected since the END data set has overlapping trials with the ALL data set.

Synergy profiles and the variation of the original data accounted for by those synergies were calculated for data subsets (RS synergies, 10–90% of ALL stimulated M1 sites) in 10% intervals and recalculated 20 times (Figs. 6 and 7). When 50% of the sites were included, all the synergies were represented and NMF converged on the same rank as the 100% data set in some of the repetitions, but it did not do so consistently. R values were consistently greater than 0.5 when 60% of the sites were present. Most of the averaged P values reached significance at P < 0.05 (and thus are not shown). However, 80% of the sites were required for monkey X to have consistent rank convergence and for monkey A representation of all the synergies and 100% of the sites were needed for consistent rank convergence.

Fig. 6.

Monkey A RS (random data sets) synergy R values and profiles. A: the correlation coefficients of each percentage subset were averaged over the 20 repetitions and the mean and standard deviations are plotted vs. increasing percentage of data for each synergy. B: the synergy profiles, averaged over 20 full recalculations, shown for the minimum rank needed to achieve 90% VAF, for different percentages of M1 stimulation sites included for synergy analysis. Synergies are aligned with the closest matching synergy from the 100% ALL synergies data set. Pearson correlation coefficients (R) are given for each synergy matched to a synergy calculated from 100% of the M1 sites. The muscle order for each bar plot is given in the legend at the right. The brackets below each synergy designate that the synergy was not always present over the 20 repetitions. The titles above each synergy set show the number of synergies that were required for 90% VAF.

Fig. 7.

Monkey X RS synergy R values and profiles. Description is the same as Fig. 6.

Comparison of Synergies Across Monkeys

When both sets of stimulus-evoked synergies were compared across monkeys (Fig. 8), three synergies in each set were significantly correlated (P < 0.05). The other synergy profiles were not significantly correlated and therefore specific to each individual monkey. The first synergy (Fig. 8, top left plot) shows the synergy that is best matched between the monkeys in the ALL data set, with a correlation coefficient of 0.7832 (P < 0.001).

What movements would be produced by the synergies observed? Despite statistically significant similarity in synergies across monkeys, a muscle-by-muscle analysis shows differences in the synergy profiles that would be expected to produce some variability in the movement that would be evoked if the synergy were activated in isolation. The first synergy in Fig. 8A (ALL data, synergy 1) would produce elbow flexion, digit/wrist extension, and flexion of the digits, and the movement would be expected to be very similar for both monkeys. By the same logic, activation of synergy 2, which was also significantly correlated across monkeys, would tend to adduct and medially rotate the shoulder and extend the elbow, with monkey A additionally showing flexion of the digits. However, looking closely, the high levels of representation of muscles with opposing functions, such as the elbow flexor brachialis coupled with triceps, and flexor digitorum profundus with extensor digitorum 2 and 3, emphasizes the presence of cocontraction in this synergy. Synergy 3 was significantly similar between monkeys and would produce shoulder extension and lateral rotation, wrist extension, and hand flexion. In monkey X, it would also produce cocontraction of shoulder flexors and medial rotation, strong wrist extension and adduction, and an assortment of wrist/hand muscles acting in flexion and extension. Activation of synergy 4 in monkey A would produce cocontraction of antagonistic muscles resulting in shoulder stabilization with elbow extension. Synergy 4 in monkey X was dominated by simple elbow flexion. Synergy 5 in monkey A would yield finger and wrist flexion.

None of the voluntary synergies (Fig. 8B) of monkey A’s performance significantly correlate with the voluntary synergies of monkey X, although there were some similarities in wrist and digit action for synergy 1. Monkey A had a pattern of coactivation of antagonistic muscles at one or more joints for all three of the VOL synergies. Monkey X notably had very little shoulder representation and no elbow representation in its two VOL synergies, despite the task requiring first elbow extension and then flexion, suggesting that the elbow movement of the VOL task was represented in the 10% of the variation that remained unaccounted for by the synergy set. One or two VOL synergies in each monkey would have high levels of cocontraction in different directions if all the muscles in the synergy profile were activated simultaneously. In the analysis above, it is important to remember that during the actual movements, whether voluntary or HFLD-ICMS evoked, the synergies are not generally activated in isolation but as combinations used to produce the movement observed.

Weighting Coefficient Characteristics and Cortical Representation

Figure 9 shows the weighting coefficients averaged over available trials and mapped with a “heat” scale to the electrode position in M1 for the ALL data set. Gray regions are sites where StTA did not yield any forelimb representation, due to a border location, or in the case of monkey X, due to difficulty locating layer V in the depth of the precentral gyrus. The cortical maps of weighting coefficients appear to have small areas of higher activation for a particular synergy. Map A1 (monkey A, synergy 1 in Fig. 9), in particular, has high activations along tracks down the bank of the central sulcus. Map X1 also has a high activation track down the bank of the central sulcus but also has a large area of high activation on the surface near the convexity of the gyrus. The high activations in the anterior bank of the central sulcus for A1 and X1 correspond nicely to synergy 1 (Fig. 5), which is dominated by the wrist/hand muscles known to be represented heavily in the bank of the gyrus (Park et al. 2001). The maps of weighting coefficients for other synergies show less distinct regions of higher activation in the sulcus. High activation regions are smaller and appear more random, or, in the case of A3, a large part of the anterior bank of the sulcus is activated to a nearly uniform medium-low level. In most maps, there appear to be single or very small groups of sites with high activations (red/orange/yellow sites) surrounded by very low activations (bluish sites). Synergies that were more similar in profile also had maps of weighting coefficients that visually appeared more similar.

Fig. 9.

Heat maps showing the average weighting coefficients for each synergy at each cortical stimulation location for the ALL data set synergies. Coefficients were averaged over the 3–5 trials available. Synergies are ordered to match the radial plots with the best matches across monkeys being first. The legend showing the color scale from low to high activation level is at the bottom right and was normalized to the maximum weighting coefficient seen in each map.

DISCUSSION

Muscle synergies represent a possible mechanism of simplified control of complex movements for the central nervous system (CNS) by reducing the degrees of freedom (Bernstein 1967). To investigate the presence of synergies in the primary motor cortex, we extracted synergies from VOL data and two sets of HFLD-ICMS-evoked EMG data: one from a comprehensive map of cortical sites (ALL) encompassing the entire M1 forelimb representation, and one from sites and trials that yielded stable spatial end points (END) as defined previously (Van Acker et al. 2014). We found that, depending on the monkey and the data set used, two to five synergies accounted for ≥90% of the overall variability (VAF) and that these synergies often contained coactivation of antagonistic muscles. Some synergies were similar across HFLD-ICMS- and voluntary movement-evoked data sets within each monkey. Some HFLD-ICMS-evoked synergies were similar across monkeys, but other HFLD-ICMS-evoked and all voluntary synergies were specific to a monkey’s individual data set. When the weighting coefficients for each of the ALL synergies were averaged over the three to five trials available and mapped to the cortical stimulation sites, there appeared to be regions of higher representation of some synergies, but there was no clear topographic organization across M1.

There are five primary assumptions upon which our study is based: 1) muscle activity is encoded in motor primitives or synergies; 2) synergies sum linearly to create muscle activity; 3) synergies may be encoded or partially encoded in the cortex; 4) cortical synergies can be revealed using HFLD-ICMS; and 5) the full relevant control space was represented in our data.

The idea of muscles being activated in groups, or synergies, originated by Bernstein (1967), has been a popular, yet controversial topic (Kutch and Valero-Cuevas 2012). It is well known that even simple voluntary movements involve a complex pattern of synergist muscle activation (Schieber 1995). Synergies have been investigated in multiple ways, including 1) the activity of functional muscle groups (Latash et al. 1995; Park et al. 2004; Soechting and Lacquaniti 1989); 2) the mathematical analysis of voluntary muscle activity (d’Avella and Bizzi 2005; d’Avella et al. 2006; Dominici et al. 2011; Ranganathan and Krishnan 2012; Ting et al. 2009; Torres-Oviedo and Ting 2010; Weiss and Flanders 2004); 3) the output of single or small groups of cells on muscle activity revealed with spike- and stimulus-triggered averaging of EMG activity (Cheney and Fetz 1985; Fetz et al. 1989; Kasser and Cheney 1985; Park et al. 2004); and 4) analysis of the altered muscle synergies with disease state or rehabilitation (Dimitrova et al. 2004; Pierrynowski et al. 2005; Roh et al. 2013; Ting et al. 2015). While other studies on voluntary movement synergies found some synergies were similar across subjects, none of our voluntary synergy profiles were significantly correlated across monkeys (Ajiboye and Weir 2009; d’Avella and Bizzi 2005; Dimitrova et al. 2004; Dominici et al. 2011; Overduin et al. 2008; Ting and Macpherson 2005; Torres-Oviedo and Ting 2007, 2010). Our results might be influenced by the fact that the movement sets used were somewhat different between the two monkeys or may suggest flaws in the assumptions that EMG is composed of synergies or that synergies would be meaningfully similar across subjects. Synergies decomposed mathematically from complex EMG data sets can neither prove nor disprove the actual presence of synergies; synergies can simply be declared to represent the data’s complexity in a manner better than random organization (Overduin et al. 2015). This is the first major assumption and limitation of all synergy studies.

Second, to identify cortical muscle synergies using the mathematics of NMF, we assumed that any synergies in use would sum linearly to create EMG activity. While it has been shown that cat motor cortex output of EMG and movement trajectories sum linearly in response to electrical stimulation (Ethier et al. 2006), and that optogenetic costimulation of the mouse spinal cord resulted in linear combinations of the two sites’ individual motor output (Caggiano et al. 2016), it is possible that an assumption of linearity is too simplistic to accurately model the use of encoded motor primitives. While NMF was able to identify a low number of synergies that could be combined linearly to represent 90% of the variation in each of our EMG data sets, our results cannot directly address the assumption of linearity. Only two synergies were required to account for more than 80% of the variance in each of the data sets (Fig. 4), which may be too simplistic to have realistic physical implications. Modeling studies have shown that NMF results are similar to other data decomposition algorithms and can accurately recreate neuromuscular synergies controlling reaching movements within certain limitations, such as full sampling of the control space and limited mechanical constraints on the limb (Burkholder and van Antwerp 2012; Steele et al. 2015; Trainin et al. 2007; Tresch et al. 2006).

There is still much debate about whether and where synergies may be encoded in the CNS. Our third assumption was that synergies might be at least partially encoded in the primary motor cortex. Spike- and stimulus-triggered averaging of EMG activity have identified hard-wired functional linkages from single corticomotoneuronal (CM) cells and small groups of cells to motoneuron pools of multiple muscles (Cheney and Fetz 1985; Fetz and Cheney 1980; Kasser and Cheney 1985; Lemon et al. 1986; McKiernan et al. 1998). Simple synergies exist such as single CM cells facilitating multiple synergist muscles acting at a joint (Cheney and Fetz 1985; Shinoda et al. 1981) or reciprocal inhibition of antagonist muscles encoded in the output of CM cells (Kasser and Cheney 1985). Hardwired cortical synergies can also extend to muscles acting at multiple forelimb joints (McKiernan et al. 1998).

However, there is strong evidence that synergies in amphibians are encoded in neural circuitry at the level of the brain stem and/or spinal cord, and these synergies are then recruited from higher areas (Giszter et al. 1993; Hart and Giszter 2010; Roh et al. 2011; Saltiel et al. 2001; Tresch and Bizzi 1999; Tresch et al. 1999). The evidence for spinal synergies includes cutaneous stimulation in the frog leg that evoked EMG patterns represented by four synergies (Tresch et al. 1999), chemical stimulation of the frog spinal cord that gave synergies with preferred combinations (Saltiel et al. 2001), and transections at various levels of the frog’s neuroaxis that suggested voluntary synergies may be stored in the medulla and spinal cord and coordinated by activation from higher structures (Roh et al. 2011). More recently, the frog spinal cord has been shown to have topographically organized synergies which are activated temporally in a sequence of a rostrocaudally “traveling wave” (Saltiel et al. 2015).

In mammals, optical stimulation identified a population of neurons in the mouse spinal cord that may be a critical pathway for both voluntary and reflexive synergies (Levine et al. 2014). Optogenetic stimulation-evoked mouse lumbosacral spinal synergies for isometric ankle force were shown to follow linear combinations with synergies organized topographically (Caggiano et al. 2016). However, in the cat, Mushahwar et al. (2004) found that the movements generated by spinal or muscle stimulation depended significantly on descending input, the initial state of the limb, and the stimulus intensity. They attributed spinal stimulation patterns to sensory afferents with connections to motoneuron pools (Mushahwar et al. 2004). CM cell populations also appear to be specialized for activating specific muscle synergies (Griffin et al. 2015) in that the same muscle might function as an agonist, synergist, fixator, or antagonist for different motor tasks, where each of these different functional roles is represented by different sets of CM cells. Here we showed that HFLD-ICMS-evoked EMG can be represented by linear synergies with similarities across monkeys, but no clear topographical organization of those synergies was seen across M1, suggesting that similar synergies lack a consistent spatial localization in the cortex. Overduin et al. (2012) concluded there were regions of M1 that called each synergy more specifically, which our data would support, but, also in keeping with our results, Overduin et al. (2012) also suggested that synergies may not be directly activated by M1, but instead M1 may “combine lower-level synergies into adaptive motor sequences.” More work remains to isolate synergy origins in higher vertebrates.

Our fourth assumption is that cortical synergies can be revealed using HFLD-ICMS. Since the stimulus-evoked synergy profiles were similar within monkeys, if HFLD-ICMS activates similar synergistic networks it does so regardless of whether the wrist reaches a stable spatial end point during stimulation. However, the synergy patterns extracted from our stimulus-evoked data were only somewhat similar to the voluntary synergies. This could be because the voluntary task EMG varied less than the stimulus-evoked EMG or because stimulation in the cortex does not activate natural EMG patterns. HFLD-ICMS activates muscle activity by blocking natural cortical output and replacing it with stimulus-driven output that directly translates into muscle activation or suppression (Griffin et al. 2011). While stimulation results in a variety of movements to stable end points in space, some of these movements are natural looking, while others are not (Bonazzi et al. 2013; Gharbawie et al. 2011; Graziano et al. 2005, 2002; Harrison et al. 2012; Ramanathan et al. 2006; Van Acker et al. 2014). HFLD-ICMS-evoked EMG also has a different time-scale pattern compared with that of natural movement and has been found to be tonic or phasic-tonic 72% of the time (Griffin et al. 2014). This time independence allowed one time bin (1,000 ms) to be used for the stimulus-evoked synergy calculations, while 100-ms time bins were used for voluntary synergies.

In contrast to other papers on stimulation of primary motor cortex and other cortical areas (Bonazzi et al. 2013; Gharbawie et al. 2011; Graziano et al. 2005, 2002; Harrison et al. 2012; Ramanathan et al. 2006; Stepniewska et al. 2009), we saw many movements that were not ethologically relevant (Van Acker et al. 2014). If muscles were activated as in the ALL synergy profiles (Fig. 5), synergies 2 and 4 in both monkeys would elicit cocontraction of antagonistic muscles, while the other synergies showed complex joint movements that are difficult to imagine as useful combinations. ICMS in the rat also resulted in some single joint movements that were not described in detail, which may have been similar to some of our movements (Ramanathan et al. 2006). Overduin et al. (2012, 2008) found ICMS-evoked synergies matched voluntary synergies from grasping movements in two macaques, with high levels of representation for muscles that have similar joint actions; for example, they showed a synergy with mostly hand/digit flexors.

Some of our stimulus-evoked synergies were similar across monkeys, but other synergies were specific to a monkey’s individual data set. This is consistent with other studies (Torres-Oviedo and Ting 2010) and may imply that synergies are developed by multiple mechanisms including evolutionary, developmental, or individual learning experiences. It may also imply that the motor control of reaching requires both fixed and flexible control (Santello and Lang 2015; Ting et al. 2015). However, synergy characteristics may also simply be a function of the mathematics of NMF and HFLD-ICMS may simply coactivate the hardwired length-tension relationships of agonist and antagonist muscles resulting in movements toward an end point (Van Acker et al. 2014).

When looking for accurate NMF synergies, results depend on complete sampling of the control space (Burkholder and van Antwerp 2012) as well as the number and choice of muscles included in the study (Steele et al. 2013). We recorded more muscles than typical, but, similar to most synergy studies, we cannot claim that the range of our EMG data sets was rich enough to capture all the synergies. Instead, to address the reasonability of this final assumption, we showed that the results of our ALL sites HFLD-ICMS data set reach consistency as the full set of cortical sites is approached (RS sets). The synergies reached fairly consistent profiles around 60%, but monkey X needed 80% and monkey A needed 100% of the cortical locations to fully represent all of the synergies. Therefore, we cannot say with certainty that the cortical mapping protocol and cortical sampling were sufficient to capture all NMF-derived cortical synergies. Whereas previous studies examining voluntary movement focused on sampling the movement space, in this study we focused on a complete sampling of the M1 cortex forelimb region but did not include premotor or supplementary motor areas (Graziano 2016). If the threshold was increased to 95% VAF, as in Overduin et al. (2012), our stimulation data would have yielded six to eight synergies, which is nearer to their eight or ten synergies. We expected that in the VOL data, monkey X would require more synergies to explain the more complex variable-target reach-and-grasp task, but monkey A required three synergies, while X needed only two. Our VOL data set does show a sharp break in the VAF curve for monkey X, but the other sets do not; thus some might argue that synergies cannot be correctly identified from those data sets, or that synergies do not exist in HFLD-ICMS-evoked EMG or cyclical voluntary movement-evoked EMG (Tresch et al. 2006).

Conclusion

While synergies were clearly identified using NMF, it is less clear whether the CNS and internal motor program actually use these synergies to produce voluntary movements. Comparison of HFLD-ICMS-evoked and voluntary movement-related synergies suggests that it is unlikely the CNS is utilizing the cortical synergies found in this study to create complete movements. At best, the synergies found are partial representations of those encoded within the CNS for voluntary movement. Limitations on the range of our voluntary movement data complicate a robust analysis of this issue. However, if synergies were hardwired into or activated by cortical circuitry, the stimulus-evoked synergies would be the basis for voluntary movement, and a small set of voluntary synergies were weakly similar to some of the ICMS synergies found. Of course, finding that the stimulus-evoked synergies derived from NMF are similar to those associated with voluntary movement does not prove that the CNS is actually using these synergies to produce movement. An alternative hypothesis is that patches of corticospinal neurons represent fundamental output modules defined by their target muscles as revealed with spike- and stimulus-triggered averaging of EMG activity (Cheney and Fetz 1985; Cheney et al. 1985). Activity from these fundamental modules descends from cortex to combine with other inputs to motoneurons (such as proprioceptive inputs and brainstem descending systems) (Levine et al. 2014; Shinoda et al. 1986) on the way to the periphery. Activation of cortical tissue with HFLD-ICMS is subject to physical and physiological spread of excitation that could blur relationships to underlying physiological output organization, while using low-intensity stimulus-triggered averaging to identify synergies would minimize this problem and yield results that more closely reflect the underlying output of individual CM cells (Cheney and Fetz, 1985). Future analysis of NMF synergies identified from stimulus-triggered averaged EMG should provide additional insight concerning these fundamental cortical output modules and how they might be engaged during voluntary movement.

GRANTS

This work was supported by National Institutes of Health (NIH) Grants NS-051825 and NS-064054, NIH Center Grant HD-02528, the KUMC Bio-medical Research Training Program, and the KU Madison and Lila Self Graduate Fellowship.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

S.L.A.H., C.W.L., and P.D.C. conceived and designed research; S.L.A.H. and G.M.V.A. performed experiments; S.L.A.H., G.M.V.A., C.W.L., and P.D.C. analyzed data; S.L.A.H., G.M.V.A., C.W.L., and P.D.C. interpreted results of experiments; S.L.A.H. and G.M.V.A. prepared figures; S.L.A.H. drafted manuscript; S.L.A.H., G.M.V.A., C.W.L., and P.D.C. edited and revised manuscript; S.L.A.H., G.M.V.A., C.W.L., and P.D.C. approved final version of manuscript.

APPENDIX 1: NONEGATIVE MATRIX FACTORIZATION

Each set of EMG data was factored into optimal synergies and coefficients matrices using an iterative method of NMF so that the original EMG pattern was as follows: EMG = V = W * H (Lee and Seung 2001). Here V is a (m × t) matrix of the original EMG data with t time bins or trials and m muscles, W is a (m × k) matrix with k basis column vectors, or synergies, of unit length, with representations for each muscle in each synergy, and H is a (k × t) matrix containing the weighting coefficients for each synergy for each time bin or trial in the original data set (Berry et al. 2007; Donoho and Stodden 2003; Lee and Seung 1999, 2001; Ting and Macpherson 2005; Torres-Oviedo et al. 2006). The iterative method initialized the synergies (W_{n = 0}) and weighting coefficients (H_{n = 0}) with random values and used a multiplicative update rule to iteratively minimize the root mean square residual error between the original data and the recreated data (Badeau et al. 2010; Berry et al. 2007; Lee and Seung 1999, 2001; Tresch et al. 2006). The root mean square residual error of a matrix is given as the Frobenius norm (or Euclidean distance) divided by the square root of (n*m), where n is the number of time points or trials and m the number of muscles:

$$D=\sqrt{\frac{{\displaystyle \sum {}_{i=1}^n{\displaystyle \sum {}_{j=1}^m{|v_{ij}-w_{ij}{h}_{ij}|}^2}}}{nm}}$$

The multiplicative update rule updates W_n+1 and H_n+1 as follows:

$$W_{n+1}=W_n\ast \frac{V{H}_n^T}{W_n(H_n{H}_n^T)+eps(V{H}_n^T)}$$

$$H_{n+1}=H_n\ast \frac{W_n^{T}A}{(W_n^T{W}_n)H_n+eps(W_n^{T}A)}$$

where * is element-by-element multiplication, / is element-by-element division, and the other multiplications are standard matrix multiplication. The “extra” term in the denominator, “eps(),” is the addition of a very small number (on the order of 10⁻⁹) to avoid division by zero (Berry et al. 2007; Lin 2007a). The eps function in MATLAB (MathWorks) calculates the spacing of the floating-point numbers based on the precision of the entered value, which in this case is the value in the numerator.

The multiplicative update rule is part of the gradient descent family, and has been used frequently since its introduction by Lee and Seung (Berry et al. 2007; Lee and Seung 1999, 2001). It does not converge to a local minimum theoretically, but it frequently does so in practice (Badeau et al. 2010; Berry et al. 2007; Lin 2007a). It is slower to converge than an alternating least squares method (ALS), and is theoretically more likely to produce sparse synergies, because once elements reach zero, they stay there (Berry et al. 2007; Lin 2007b). However, in our practice, the multiplicative update rule was chosen because it was much more likely to produce the number of synergies specified for analysis, and ALS would often converge to a rank lower than the one specified.

Iterations were continued until convergence was confirmed by one of three stopping mechanisms: 1) if the maximum change in the synergies and the coefficients between iterations was less than the specified tolerance (set to 1 × 10⁻⁴), 2) if the change in the root mean square residual error was less than its specified tolerance (set to 1 × 10⁻⁴), or 3) if the maximum number of iterations had been reached (Ajiboye and Weir 2009; Lee and Seung 1999, 2001; Paatero and Tapper 1994). In our case, the maximum number of iterations was set to 1,000, which was much greater than was typically needed (NMF typically converged with <100 iterations) (Berry et al. 2007; Burkholder and van Antwerp 2012).

The entire algorithm was repeated 20 times (“replicates”) with different random initialization each time using a Monte Carlo approach (Berry et al. 2007; Paatero and Tapper 1994). The W and H matrices with the least RMS residual error were chosen as the final synergy and coefficient matrices (Lee and Seung 2001; Torres-Oviedo et al. 2006; Torres-Oviedo and Ting 2010). NMF assumes that the random variations in the data caused by noise are much smaller than the variation structured by combining synergies in the neuromuscular system (d’Avella et al. 2006).