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

Muscle synergies are usually identified via dimensionality reduction techniques, such that the identified synergies reconstruct the muscle activity to a level of accuracy defined heuristically, such as 90% of the variance explained. Here, we question the assumption that the residual muscle activity not explained by the synergies is due to noise. We hypothesize instead that the residual activity is structured and can therefore influence the execution of a motor task. Young healthy subjects performed an isometric reaching task in which surface electromyography of 10 arm muscles was mapped onto estimated two-dimensional forces used to control a cursor. Three to five synergies were extracted to account for 90% of the variance explained. We then altered the muscle-force mapping via “hard” and “easy” virtual surgeries. Whereas in both surgeries the forces associated with synergies spanned the same single dimension of the virtual environment, the muscle-force mapping was as close as possible to the initial mapping in the easy surgery and as far as possible in the hard surgery. This design therefore maximized potential differences in reaching errors attributable to the residual muscle activity. Results show that the easy surgery produced much smaller directional errors than the hard task. In addition, systematic estimations of the errors for easy and hard surgeries constructed with 1 to 10 synergies show that the errors differ significantly for up to 8 synergies, which account for 98% of the variance on average. Our study therefore indicates the need for cautious interpretations of results derived from synergy extraction techniques based on heuristics with lenient levels of accuracy. Author summary The muscle synergy hypothesis states that the central nervous system simplifies motor control by grouping muscles that share common functions into modules called muscle synergies. Current techniques use unsupervised dimensionality reduction algorithms to identify these synergies. However, these techniques rely on arbitrary criteria to determine the number of synergies, which is actually unknown. An example of such criteria is that the identified synergies must be able to reconstruct the measured muscle activity to at least a 90% level of accuracy. Thus, the residual muscle activity, the remaining 10% of the muscle activity, is often disregarded as noise. We show that residual muscle activity following muscle synergy identification has a large systematic effect on movements even when the number of synergies approaches the number of muscles. This suggests that current synergy extraction techniques may discard a component of muscle activity that is important for motor control. Therefore, current synergy extraction techniques must be updated to identify true physiological synergies.

healthy subjects performed an isometric reaching task in which surface electromyography of 10 23 arm muscles was mapped onto estimated two-dimensional forces used to control a cursor. Three 24 to five synergies were extracted to account for 90% of the variance explained. We then altered the 25 muscle-force mapping via "hard" and "easy" virtual surgeries. Whereas in both surgeries the forces 26 associated with synergies spanned the same single dimension of the virtual environment, the 27 muscle-force mapping was as close as possible to the initial mapping in the easy surgery and as 28 far as possible in the hard surgery. This design therefore maximized potential differences in 29 reaching errors attributable to the residual muscle activity. Results show that the easy surgery 30 produced much smaller directional errors than the hard task. In addition, systematic estimations 31 of the errors for easy and hard surgeries constructed with 1 to 10 synergies show that the errors 32 differ significantly for up to 8 synergies, which account for 98% of the variance on average. Our 33 study therefore indicates the need for cautious interpretations of results derived from synergy 34 extraction techniques based on heuristics with lenient levels of accuracy. 35 36 Author summary: The muscle synergy hypothesis states that the central nervous system 37 simplifies motor control by grouping muscles that share common functions into modules called 38 muscle synergies. Current techniques use unsupervised dimensionality reduction algorithms to 39 identify these synergies. However, these techniques rely on arbitrary criteria to determine the 40 Introduction 49 One of the most salient problems the central nervous system (CNS) faces when generating 50 movements is the redundancy of the motor system [1]. That is, the CNS can generate an infinity 51 of different motor commands to produce the same action. This redundancy spans the length of 52 the causal chain of motor control: from neuron to muscle to joint levels. In light of the complexity 53 of this problem, the muscle synergy hypothesis posits that the CNS groups the control of 54 functionally similar muscles into modules called muscle synergies [2]. This would reduce the 55 number of variables that the CNS needs to control to produce a movement, decreasing the 56 complexity of the computations necessary for motor control [3]. 57 Direct evidence for the muscle synergy hypothesis comes from experiments in animal models [3-58 6]. These show that simultaneous stimulation of different groups of motor neurons elicits 59 movements that correspond to the superposition of the movements obtained by stimulating each 60 group of neurons separately [3,5,6]. However, most of the supporting evidence in humans is 61 indirect and comes from measurements of electromyography (EMG) from multiple muscles during 62 a variety of motor tasks [7][8][9][10][11]. Dimensionality reduction techniques, such as non-negative matrix 63 factorization, show that different muscles tend to co-activate in reliable patterns during task execution [12]. One of the interpretations of these results is that they reveal the grouping of 65 muscles into functional synergies [8,9,13,14]. An alternative interpretation, however, is that the 66 discovered patterns arise because of biomechanical constraints imposed by the task [15][16][17]. 67 This controversy notwithstanding [18], dimensionality reduction techniques for the extraction of 68 muscle synergies rely on the ability of the extracted synergies to reconstruct the originally 69 measured EMG signals accurately [19]. That is, the extracted synergies must capture a high 70 proportion of the variability in the recorded EMG, attributing the discarded or residual variability in 71 the data to measurement and process noise. This proportion is usually adjusted by making the 72 number of muscle synergies a hyper-parameter to be tuned to best fit the data [20]. A widely used 73 rule of thumb is to set the number of muscle synergies to the minimum number that accounts for 74 at least 90% of the variability in the EMG. 75 However, this method neglects the fundamental role of muscle synergies as building blocks of 76 movement, as the ability of the extracted muscle synergies to reconstruct the observed movement 77 is often ignored [19,21,22]. Indeed, the ability of muscle synergies to reconstruct measured 78 forces in an isometric task at the wrist becomes largely degraded as the number of considered 79 muscle synergies decreases [22]. This is true even when the extracted synergies capture an 80 acceptable portion of the variability in the EMG signals according to the defined heuristics. This 81 suggests that the portion of EMG variability that is not captured by the extracted muscle synergies 82 is important for a full description of the motor action. 83 Here, we therefore aimed to determine the importance of the residual EMG in the execution of a 84 motor task. We tested the null hypothesis that following extraction of muscle synergies with non-85 negative matrix factorization and using the 90% of explained variance rule to select the number 86 of synergies, the residual muscle activity is due to noise. Therefore, if our experimental data failed 87 to support this hypothesis, it would suggest that the residuals are structured and can therefore 88 influence motor performance. 89 To this end, we used the virtual surgery paradigm, which simulates tendon transfer surgeries [10]. 90 The virtual surgery alters the pulling forces of arm muscles in a virtual mapping from EMG to two-91 dimensional isometric force at the wrist, which affects performance during the reaching task. This 92 EMG-force mapping can be simplified into a synergy-force mapping by combining the pulling 93 forces for each arm muscle according to a set of previously identified muscle synergies. Given 94 that the number of muscles is necessarily larger than the number of extracted synergies, it is 95 possible to build virtual surgeries that produce identical synergy-force mappings but different 96 EMG-force mappings. We exploited this property by designing virtual surgeries that modified the 97 EMG-force mapping to two opposite extremes while producing the same synergy-force mapping. 98 The "easy" surgery modified the EMG-force mapping as little as possible with respect to the 99 baseline mapping, and the "hard" surgery modified the mapping as much as possible. The two 100 virtual surgeries were designed based on the extracted muscle synergies that account for at least 101 90% of the variability in the EMG. Consequently, the effect of the surgery on the residual portion 102 of the EMG was not specified, leading to possible differences in the effects of the easy and hard 103 surgeries on task variables. If the EMG residuals are attributable to noise, then both surgeries 104 should produce similar errors in the direction of reaching when introduced suddenly. Alternatively, 105 if the EMG residuals have a latent structure, then both surgeries should have a differential effect 106 on the residuals and on the error in the direction of reaching. We found that the sudden 107 introduction of both kinds of virtual surgeries produced largely different errors, supporting the 108 existence of a latent structure in the EMG residuals. 109

110
Subjects. Fifteen right-handed subjects (mean age, 27.9 ± 8.75 years, s.d.; thirteen males) 111 participated in the study after providing written informed consent. All procedures were approved 112 by the Ethical Review Board of the Tokyo Institute of Technology. 113 Experimental setup. Each participant sat on a racecar seat while gripping a handle located at 114 the height of the base of their sternum with their right hand. The arm posture corresponded to an 115 elbow flexion of around 90° and the elbow was supported on a stand at approximately the same 116 height as the hand. A splint was used to immobilize the hand, wrist and forearm. Participants were 117 instructed to lean on the back of the seat for the duration of the experiment. The base of the 118 handle was attached to a six axis force transducer (Dyn Pick; Wacoh-Tech Inc.) used to measure 119 isometric forces. The force transducer was mounted on a 2-D sliding rail to allow for an adjustable 120 configuration for each participant. A virtual environment was displayed on a computer screen 121 placed at the height of the participants' eyes at a distance of around 1 m. The virtual environment 122 consisted of a circular red cursor (1 cm diameter), and several ring-shaped white targets (2 cm 123 diameter) on a black background. 124 We recorded surface EMG activity from 10 muscles crossing the shoulder and elbow joints: 125 pronator teres, brachioradialis, biceps brachii long head, triceps brachii lateral head, triceps 126 brachii long head, anterior deltoid, middle deltoid, posterior deltoid, pectoralis major, and middle 127 trapezius. Active bipolar electrodes (DE 2.1; Delsys) were used to record EMG activity. EMG 128 signals were bandpass filtered (20-450 Hz) and amplified (gain 1000, Bagnoli-16; Delsys). Force 129 and EMG recordings were digitized at 2 kHz using an USB analog-to-digital converter 130 National Instruments). 131 To reduce random oscillations of the cursor caused by the stochastic nature of EMG signals, a 132 mass-spring-damper dynamics filtered the EMG signals further [10]. The mass-spring-damper 133 dynamics governed the movement of the cursor according to: 134 where p is a vector containing the x and y positions of the cursor on the screen and its derivatives 136 are indicated in dot notation, m is the system's mass, k is the stiffness, and b is the damping 137 coefficient (m = 0.05 kg, b = 100 kg/s). F(t) is the force recorded by the force transducer (during 138 force control) or the estimated force by the EMG-force mapping (during EMG control). k was 139 calculated as a function of the maximum voluntary force (MVF) (described in the next section), so 140 that targets at equal percentages of MVF required the same cursor displacement across 141

participants. 142
Experimental protocol. In all phases of the experiment, participants performed isometric force 143 tasks. These tasks required the displacement of a cursor on a visual display from a center position 144 to one of eight targets radially and uniformly distributed around the center. Participants first 145 performed a force control task and then an EMG control task (Fig 1a). In the force control task, 146 the cursor was controlled via forces applied by the arm on a load cell (force control). In the EMG 147 control task, the cursor was controlled by a linear approximation of the force derived from EMG 148 measurements of 10 arm muscles (EMG control). 149  Participants then performed an isometric reaching task by applying force with their right arm to 176 reach targets in the virtual environment. The recorded force and EMG signals during this task 177 were processed to compute the EMG-force mapping, extract muscle synergies, and construct the 178 virtual surgeries. Targets were arranged radially in eight directions and required 5, 10, 15 or 20% 179 of MVF to be reached. Each trial started by displaying the target at the central position. The central 180 position corresponded to the position of the cursor when no forces were applied. After placing the 181 cursor inside the central target for two seconds, the central target disappeared and one of the 182 radial targets appeared. After reaching each target, both the cursor and the target disappeared 183 from the screen and participants were asked to hold the applied force as steadily as possible for 184 two seconds. Next, the cursor and the central target reappeared and participants were asked to 185 move the cursor back to the center. After this, another trial began. Each target was presented 186 three times, with a total of 96 trials. Targets were presented in a randomized order. Trials were 187 repeated if participants failed to reach a target. 188 Next, cursor control was switched to EMG control without the knowledge of the participants, after 189 which participants performed the reaching task under EMG control. The first EMG control block 190 was a familiarization block, and was followed by one type of incompatible surgery, easy or hard, 191 followed by the other in a cross-over design (see Fig 1A). The order of the easy and hard surgeries 192 was pseudo-randomized such that 7 participants started with the easy surgery. Participants rested 193 for 5 minutes between surgery types. Each surgery condition consisted of three phases: baseline, 194 virtual surgery, and washout, which consisted of 6, 12, and 6 blocks, respectively. Each block 195 consisted of 24 trials: three trials for each of the eight targets at a magnitude of 10% MVF 196 randomized within target sets containing each one of the eight targets. The level of baseline noise 197 in each EMG signal was measured at the start of every block while the participant was relaxed. 198 This baseline noise was subtracted from the EMG signals measured during the corresponding 199

block. 200
Note that in this study, we focus exclusively on data recorded during the first set of eight targets 201 following the onset of each virtual surgery. Analysis of the following blocks for each surgery will 202 be covered in a separate manuscript that focuses on learning of incompatible virtual surgeries. (2) 207 where f is a two-dimensional force vector produced on the horizontal plane, m is a ten-208 dimensional vector of muscle activations, composed by normalized EMG signals recorded from 209 ten muscles simultaneously, and M is a 2 × 10 matrix that maps muscle activations to forces. M 210 was determined via linear regression of 10 EMG signals against 2D forces recorded during every 211 trial of the main force control subtask. Before performing the regression, forces were low-pass filtered (second-order Butterworth; 1 Hz cutoff) and EMG signals were band-pass filtered (second-213 order Butterworth; 5-20 Hz), rectified, and normalized. The signals were recorded from the time 214 of target go to the end of target hold. 215 Synergy extraction and number of synergies. We used non-negative matrix factorization (NMF) 216 to extract muscle synergies from the EMG signals collected during the main force control subtask: 217 where S is a 10 × N matrix that contains the identified synergies in its columns with N being the 219 number of synergies, and c is an N-dimensional vector of synergy activations. Equation 3 220 assumes perfect matrix factorization (no residual EMG activity). 221 EMG signals collected during the main force control subtask were processed in the same way as 222 described in the EMG-force mapping section. The synergy extraction procedure closely followed 223 a method previously described [10]. Synergies were extracted for all N from 1 to 10. For each 224 case, the synergy extraction algorithm was run 100 times, and the result with the highest 225 reconstruction quality R 2 of the original EMG signals was kept. Two criteria were required to select 226 N. The first was to set N as the minimum number of synergies necessary to explain at least 90% 227 of the EMG data variance. The second involved calculating the changes in slope in the R 2 curve 228 as a function of N. Linear regressions were performed on sections of the curve between N and 229 10. N was selected as the smallest value for which the mean squared error of the linear regression 230 was < 10 -4 [11]. If the two criteria did not match, N was selected as the case in which the extracted 231 synergies had the smallest number of similar preferred directions (number of adjacent directions 232 separated by less than 20°). This occurred for seven of the participants. where T is a 10 × 10 matrix that constitutes the transformation or virtual surgery. 239 Incompatible virtual surgeries are designed such that muscle activations m produced by synergy 240 combinations Sc are restricted to generate forces along only one dimension of the force space, 241 while the resulting EMG-force mapping M' spans the whole force space. Therefore, theoretically, 242 any force can still be produced by a new combination of muscle activations m', but in practice, 243 produced forces are biased towards one dimension of the plane. 244 It is important to note that the set of incompatible surgeries is infinite. This is because the number 245 of muscles used in the virtual mapping is larger than the number of muscle activity patterns found 246 using muscle synergy analysis. A previous study [10] combined randomness and difficulty 247 matching to select compatible and incompatible virtual surgeries. 248 In contrast, here we specified a series of constraints to yield only two possible virtual surgeries. 249 Specifically, we built hard TH and easy TE incompatible surgeries such that they were equivalent 250 in the force space spanned by each participant's extracted muscle synergies (Figs 1b and 1d). 251 We first note that according to equations 2, 3 and 4, forces produced during the surgery are given 252 by: 253 assuming that muscle activations are generated by combinations of synergies. This equation 255 shows that surgery T can alternatively be thought to transform the extracted synergies S into a 256 new set of synergies: 257 In order to build an incompatible surgery it is necessary to find S' such that the matrix MS' is rank 259 deficient. This guarantees that forces produced by this mapping lie in a single dimension. 260 Geometrically, this means that the forces associated with each individual synergy from S' are 261 collinear (Figs 1c and 1g). 262 Easy surgeries were built such that the angles between the column vectors of the original M 263 mapping and of the transformed mapping M' were as small as possible. In contrast, hard surgeries 264 were built by making these angles as large as possible (Fig 1b). These conditions produced M' 265 mappings that are similar or very different to the original M mapping in the case of easy or hard 266 surgeries, respectively. For this, we used a two-step optimization procedure to first obtain a 267 transformed set of synergies S', and second, to compute the incompatible surgery T. We 268 constrained S' to be equal for both the easy and hard incompatible surgeries. This ensured that 269 the only difference between both virtual surgeries is the transformed mapping MT. We chose a 270 configuration such that the individual force vectors associated to each synergy in S were rotated 271 onto a line that bisected the plane at an angle of 135° with the x-axis. Therefore, each force vector 272 conserved its magnitude, and its direction was assigned to the direction of the bisecting line that We transcribed this quadratic program into its canonical form and solved it using the quadprog 286 function in Matlab. 287 After obtaining S', we computed the incompatible surgery T by noting that 288 This is a system of equations where the elements of T are the unknowns. We note that T is a 10 290 × 10 matrix, so in this case there are 100 unknowns and 10N equations. The system is 291 overdetermined in all cases where N < 10, which in our case is guaranteed. 292 In order to find the easy virtual surgery, we used our requirements of similarity between M and M' 293 to introduce an optimization objective to arrive to a unique solution. M and M' are considered 294 similar when the angles between their corresponding column vectors are as small as possible. 295 The cosine of the angle between two vectors is proportional to the dot product of both vectors. 296 Therefore, we defined the optimization objective as 297 where hi and hi' are the column vectors of M and M', respectively. This optimization objective is 299 not bounded, so we added constraints to the magnitude of the resulting h' vectors: 300 This problem can be posed as a linear program with quadratic constraints, with equation 10 as 302 the objective, and equations 9 and 11 as equality and inequality constraints, respectively. The 303 result of this optimization procedure yields TE, the easy incompatible virtual surgery. 304 In order to compute the hard incompatible virtual surgery TH, the procedure is the same as for the 305 easy incompatible surgery. The only difference is that the optimization objective is minimized 306 instead of being maximized. In turn, this maximizes the angles between hi and hi': 307 Both linear programs with quadratic constraints were solved using the fmincon function in Matlab. 309

Data analysis
310 Task performance metric. We used the initial angular error as a metric to quantify task 311 performance during the experiment, before possible feedback corrections. The initial angular error 312 was calculated for each trial as |θtarget -θcursor|. θtarget is the direction of the target. θcursor is defined 313 as the direction of the line segment that joins the point at which the cursor exits a 2 cm diameter 314 circumference at the center of the screen and the position of the cursor 100 ms after exiting the 315 circumference. We averaged the initial angular error for the targets within sets of eight trials. We 316 only took into account targets that were not aligned with the line of action of the surgery. That is, 317 targets other than those at 135° and -45° from the horizontal on the screen. 318 EMG residual analysis. We analyzed the residual EMG signals obtained after reconstructing the 319 measured EMG signals based on the extracted muscle synergies. After synergy extraction using 320 the NMF algorithm, and extending equation 3, muscle activations can be represented as 321 where msyn is the synergy component of muscle activation, and r is the residual component of 323 muscle activation that cannot be accounted for by the extracted synergies. Consequently, the 324 forces associated with the EMG signals by the EMG-force mapping have a synergy and a residual 325 In order to decompose a given EMG sample m into its synergy and residual components (msyn 331 and r), we first computed msyn via non-negative least squared regression of S and m, which 332 yielded c. This algorithm optimizes the same cost function as the NMF algorithm. Therefore, using 333 equation 13, msyn is given by the product of S and c. Consequently, r is found by subtracting msyn 334 from m. 335 We then analyzed the effects of the surgery on both the synergy and residual components of 336 EMG. For this, we used the EMG activity that participants produced when they acquired each 337 target during the first baseline phase of the experiment. We then separated the average EMG 338 activity of each subject and target m into msyn and r. 339 We also estimated both force components Fsyn and Fres produced for each target at the onset of 340 the easy and hard virtual surgeries by substituting M by M' in equation 14. We then compared the 341 estimated force direction to the intended direction for each target to obtain an estimate of the error 342 that subjects would produce at the onset of each virtual surgery. 343

Shuffling of EMG residuals. Shuffling the residual component of different EMG signal samples 344
creates random residual components with the same statistical properties as the original residuals. 345 If the residual EMG activity can be disregarded as noise, then shuffling the residuals should have 346 no significant effect on the estimated forces with respect to pre-shuffling. On the contrary, if the 347 residuals have a structured organization, shuffling the residuals would destroy this organization. 348 Consequently, the force estimates would most likely be different from the pre-shuffling estimates. 349 We therefore shuffled the residual components of the EMG samples that we used to estimate 350 forces, and re-estimated the total forces that would be produced at the onset of the easy and hard 351 virtual surgeries. We averaged the results of 1000 different shuffling instances. 352 to the intended target during the easy surgery than during the hard surgery (Fig 2b).

370
This subject produced larger initial errors at the onset of the hard virtual surgery than at the onset of the 371 easy surgery (see targets at 45° and 90°).

373
Over all 15 participants, the mean error for the first set of targets after the onset of the surgery 374 was clearly larger for the hard surgery than for the easy surgery (hard surgery: 81.4° ± 3.8° s.e, 375 easy surgery: 54.5° ± 4.6° s.e., p < 10 -3 , paired t-test; see Fig 3B, experiment). This difference in 376 errors may appear surprising at first, given that the easy and hard surgeries had the same effect 377 on the synergy component of the force. That is, they restricted the forces associated with the synergies along one dimension. However, the synergies were only required to account for 90% 379 of the variance in EMG. Therefore, the EMG residuals appeared to generate an additional 380 component of force.   respectively. The incompatible design of the surgery can be appreciated on Fsyn, as these forces 401 lie on the 135° line of action of the virtual surgery (Fig 4, middle row).

405
Top row: Estimated forces F at each target before and after applying the hard and easy virtual surgeries.

414
Given that the EMG signals that we used to estimate forces were representative of the subjects' 415 actions during baseline, and assuming that subjects produced these EMG signals when suddenly 416 exposed to the virtual surgeries, the directions of the estimated forces after applying the virtual 417 surgery (Equation 5) also provided an estimate of the cursor error to each target (Fig 4, top row). 418 These initial error estimates were consistently higher for the hard surgery than for the easy 419 surgery (hard surgery: 82.75° ± 4.19° s.e., easy surgery: 45.57° ± 3.03° s.e., p < 10 -3 , paired t-420 test), and qualitatively matched the experimental results of the cursor error (robust regression, 421 slope = 0.47 ± 0.15 s.e., p = 0.004, R 2 = 0.25) (Fig 3a). 422 Errors following the easy and hard surgeries can be explained by the residual's structure (Fig 4,  423 bottom row). The hard surgery produced a mean rotation of Fres with respect to baseline that was 424 much larger than that produced by the easy surgery (hard surgery: 113.60° ± 10.15° s.e., easy 425 surgery: 4.42° ± 1.5° s.e., p < 10 -3 , paired t-test). Note that although we did not specify the effect 426 of the virtual surgery on the residual component of force, we found that it is stereotypical according 427 to the type of surgery. 428 Shuffling residual EMG activity revealed structure in the residuals 429 We then shuffled the residual EMG components among trials to all targets to demonstrate 430 possible structure. Initial error estimates based on the shuffled signals did not indicate a significant 431 difference in average initial error between the easy and hard virtual surgeries (easy surgery: 66.42° 432 ± 2.31° s.e., hard surgery: 72.41° ± 2.72° s.e., paired t-test, p = 0.10) (Fig 3b, shuffled residuals). 433 Furthermore, the magnitude of this error lied at an intermediate level between the errors observed 434 experimentally for the easy and hard surgeries. Importantly, the means of the estimates produced by shuffled signals were indistinguishable from estimates produced based on a null residual 436 condition, that is, exclusively using the synergy component of the EMG to produce estimates 437 (easy surgery, paired t-test, p = 0.45; hard surgery, paired t-test, p = 0.68) (Fig 3b, null and  438 shuffled residuals). 439 Estimated differences between errors for easy and hard surgeries 440 remained significant for high-dimensional synergy sets 441 We tested whether building virtual surgeries based on synergy sets with a larger N would abolish 442 the differences in initial direction error observed in the experiment. For each participant we built 443 easy and hard surgeries based on surgeries considering N = 1, …, 10 and applied the newly 444 constructed surgeries to the same set of EMG signals that we used to estimate errors after the 445 introduction of the surgery. This allowed us to estimate the errors that participants would have 446 produced if they had experienced these surgeries. 447 We found that the surgeries produced estimated differences in initial direction errors that were 448 maximal for N = 1, and gradually decreased until disappearing at N = 10 (as expected, since 449 activity from 10 muscles was recorded; Fig 5). The estimated error differences remained 450 significant up to N = 8 (p = 0.001, paired t-test). This indicates that the residual components of 451 EMG produced a differential effect on the estimated error even when high-dimensional synergy 452 sets that explained a portion of the variance that largely exceeded the heuristic rule requirements 453 (R 2 = 0.98 ± 0.0087 s.e. at N = 8) were used to build the virtual surgeries.

465
Muscle synergy extraction techniques require that combinations of the identified synergies 466 reconstruct the measured muscle activity to a heuristically defined level of accuracy, such as 467 accounting for at least 90% of the variance in the EMG. These techniques therefore attribute the 468 residual muscle activity not reconstructed by the identified synergies to noise. Here we studied 469 the importance of residual EMG activity in the execution of a virtual motor task. We designed the 470 virtual task based on a virtual surgery [10,24] and exploited the property that virtual surgeries can 471 produce equivalent muscle synergy-force mappings while resulting in different individual muscle-force mappings. We tested two different virtual surgeries that shared a common muscle synergy-473 force mapping, but differed maximally in their individual muscle-force mappings (easy and hard 474 virtual surgeries). The surgeries had the desired effect only on the portion of the EMG signals 475 explained by the extracted muscle synergies, defined to account for at least 90% of the variability 476 in the signal. Therefore, the effect on the residual EMG variability was unspecified, allowing for a 477 possible differential effect on the performance of the task. 478 We found that participants produced larger errors at the onset of the hard surgery than at the 479 onset of the easy surgery. We were able to predict this result qualitatively (Fig 3a) by estimating 480 the forces and errors that would be produced during each virtual surgery by using representative 481 EMG signals recorded during the baseline phase of the experiment and transforming the 482 estimated forces using the virtual surgeries. Importantly, this procedure also allowed us to 483 separate the recorded EMG signals and the estimated forces into their synergy and residual 484 components (Fig 4). The virtual surgeries produced the expected effects on the synergy 485 component of the EMG. However, the easy surgery barely produced any changes on the direction 486 of the forces associated with the residual component, whereas the hard surgery produced large 487 changes in the direction of these forces. Given that the total force is equal to the sum of the 488 synergy and residual components, any difference between both virtual surgeries in the estimated 489 force and error must arise from the difference in the residual components. This provides evidence 490 that the residual component of the EMG is essential for accounting for our experimental results, 491 suggesting a latent structure in the residuals. 492 We also considered the alternative case in which the residual EMG activity is composed of noise. 493 In this situation, we posited that there would be no differential effect of the easy and hard surgeries 494 on the initial error, or that this effect would be small. To test this, we used the previously 495 decomposed EMG signals and shuffled the residual components among all these EMG samples. 496 This effectively destroyed any potential structure in the residual component, as they became randomized. We found that the easy and hard surgeries did not produce significant differential 498 effects in the estimated initial error across participants when applied to the shuffled EMG signals. 499 This analysis suggests that the residual component of the EMG cannot be disregarded as purely 500 noise, and therefore demonstration of a latent structure in the residuals. 501 Dimensionality reduction techniques such as NMF are useful for extracting patterns from high-502 dimensional data sets, such as EMG recordings from multiple muscles. These techniques are 503 usually able to extract as many patterns or synergies as individual muscles. However, there is no 504 objective means for selecting the number of synergies of interest a priori given the exploratory 505 nature of the analysis and the lack of a ground truth. Therefore, heuristic rules, such as selecting 506 the number of synergies based on predefined goodness of reconstruction criteria are a common 507 practice (i.e., reconstructing the data to a given level of accuracy, or finding an elbow in the 508 goodness of reconstruction curve). These heuristic rules are necessarily ad hoc, and are tailored 509 to produce useful results in the domain of the studied problem [25]. 510 These heuristic rules ignore the role of muscle synergies in the generation of movement. That is, 511 muscle synergy extraction has mainly focused on describing muscle activity in the input space, 512 but has neglected the reconstruction of forces and movements in the task space [19,21]. A 513 number of studies have attempted the extension from input to task space in the scope of the study 514 of synergies by incorporating task-relevant constraints, such as force reconstruction, in the 515 dimensionality reduction procedure [13,26]. However, in these studies, assumptions of linearity 516 were made in the relationship between input and task spaces, that is, muscle activations and 517 forces. Alternatively, other studies took a simulation approach by using muscle synergy activity 518 derived experimentally as input to a computational biomechanical model to assess the goodness 519 of the resulting movement reconstruction [27]. However, tuning of muscle activations during the 520 simulations was necessary to obtain favorable results. Further difficulties in the use of 521 computational biomechanical models to test for reconstruction of task space variables could stem from the difficulty of measuring EMG from all muscles involved in a movement and of building 523 sufficiently accurate musculoskeletal models. 524 An alternative approach for studying the influence of extracted synergies on task-space variables 525 consists in using a virtual isometric task, such as in this study and others [10,22,28]. Virtual tasks 526 overcome the difficulty of obtaining complex biomechanical models, as the task can be defined 527 by the experimenter. This way, the physics of the system are linear and known, and can be used 528 in simulations in a straightforward way. A study using this approach showed that the 529 reconstruction of isometric forces in an EMG-controlled task using muscle synergy decomposition 530 is acceptable only when the number of synergies is equal to the number of considered muscles 531 [22]. Otherwise, the reconstruction quality quickly degrades even when the number of synergies 532 is derived from widely used heuristic rules [22]. Decreasing the number of synergies is associated 533 with larger residual components of the EMG, which we showed to play an important role in task 534 performance. Thus, our results further expand on this view, with the additional contribution of not 535 being limited by the passive reconstruction of forces, but by directly manipulating the contribution 536 of the residual component of the EMG to isometric force to highlight its importance in the execution 537 of the task. These results emphasize the need of a shift within the community in the criteria used 538 to evaluate the goodness of muscle synergies extracted through dimensionality reduction 539 methods such as NMF. 540 Our results suggest that humans produce muscle activations that cannot be fully accounted for 541 by linear combinations of low-dimensional sets of muscle synergies, as extracted via NMF. This 542 is not in conflict with the notion that the CNS uses muscle synergies as building blocks of 543 movement embedded in neural circuits, as shown by numerous animal studies [3][4][5][6]. Additionally, 544 synergy control in a myoelectric task (isolating msyn from r) has been shown to produce cursor 545 trajectories similar to control through individual muscles, showing that synergy control may be 546 useful for myoelectric interface applications [28]. However, this does not necessarily imply that the CNS is limited to a small number of muscles synergies structurally defined in neural circuits. 548 It is entirely possible that the CNS readily learns and exploits a large number of task-dependent 549 muscle synergies, which may vary from individual to individual [29]. In this sense, the role of 550 muscle synergies could be viewed as a source of flexibility in the repertoire of possible motor 551 commands and stability in movement execution, as opposed to a way to simplify the control of 552 movement by limiting the number of control inputs [30]. 553 The main limitation of our study is that our experimental design was originally conceived to test 554 motor learning of virtual surgeries (to be presented in a follow-up manuscript). Therefore, the 555 virtual surgery trials were presented in blocks after transitions from baseline trials. This could have 556 induced a small amount of learning in the trials immediately following the perturbation or engaged 557 exploratory behaviors due to the saliency of the perturbation. These undesired factors could 558 explain why our initial error estimates do not match the experimental data more closely. A more 559 appropriate design would randomly introduce catch trials for each surgery type among baseline 560 trials to reduce learning effects. Nonetheless, because the observed differential effect between 561 both virtual surgeries was large and robust, the results of a study that addresses these limitations 562 would probably not be very different from our current results. 563 Overall, our results indicate that current muscle synergy identification techniques wrongly attribute 564 the fraction of unexplained variability in the EMG signals to noise. Our study is not able to discern 565 whether the structure of the residual component of the EMG is due to the inadequacy of an 566 additive linear model of muscle synergies, additional muscle synergies left out of the analysis by 567 the 90% variance or other heuristic rules, or due to other possible sources like individual muscle 568 control. However, it is clear that studies that aim to infer neural structures through EMG recordings 569 should carefully consider the role of the residual component of the EMG signals. 570