Modularity and Compositionality in Motor Control: Acknowledging

The central nervous system (CNS) may produce coordinated motor outputs via the combination of motor modules representable as muscle synergies. Identi ﬁ cation of muscle synergies has hitherto relied on applying factorization algorithms to multimuscle electromyographic data (EMGs) recorded during motor behaviors. Recent studies have attempted to validate the neural basis of the muscle synergies identi ﬁ ed by independently retrieving the muscle synergies through CNS manipulations and analytic techniques such as spike-triggered averaging of EMGs. Experimental data have demonstrated the pivotal role of the spinal premotor interneurons in the synergies ’ organization and the presence of motor cortical loci whose stimulations offer access to the synergies, but whether the motor cortex is also involved in organizing the synergies has remained unsettled. We argue that one dif ﬁ - culty inherent in current approaches to probing the synergies ’ neural basis is that the EMG generative model based on linear combination of synergies and the decomposition algorithms used for synergy identi ﬁ cation are not grounded on enough prior knowledge from neurophysiology. Progress may be facilitated by constraining or updating the model and algorithms with knowledge derived directly from CNS manipulations or recordings. An investigative framework based on evaluating the relevance of neurophysiologically constrained models of muscle synergies to natural motor behaviors will allow a more sophisticated understanding of motor modularity, which will help the community move forward from the current debate on the neural versus nonneural origin of muscle synergies.


CURRENT APPROACHES TO REVEALING THE NEURAL BASIS OF MOTOR COORDINATION
In one of his first contributions to the philosophy of the brain, Roger Sperry (1) argued eloquently that "the principal function of the nervous system is the coordinated innervation of the musculature. . . [and] the sole product of brain function is motor coordination" (p. 297). Understanding the neural mechanism of motor coordination has remained a central, hard question in neuroscience (2)(3)(4). To be viable, this mechanism must tractably and robustly coordinate the activities of thousands of motor units within the hundreds of muscles (5) and, at the same time, efficiently generate diverse motor behaviors relevant to survival. One solution for the central nervous system (CNS) to achieve both control tractability and just enough behavioral diversity is to compute motor commands via the combination of preorganized low-level units called motor primitives or motor modules (6,7). Data gathered in the last two decades have further demonstrated that motor modules can be represented as muscle synergies-profiles that constrain how groups of muscles are activated together spatiotemporally as discrete units (Refs. 8-12 and others).
In the literature, the determination of the identity of the muscles belonging to a muscle synergy and the synergy's time course of activation has relied on analyses of multimuscle electromyographic activities (EMGs) based on the application of dimensionality reduction algorithms. In a typical, but by no means unique, muscle synergy analysis, EMGs collected into a matrix M as column vectors are decomposed by an algorithm into two matrices, C and W, such that where vectors w i , columns of matrix W, are the muscle synergies; vectors c i , columns of matrix C, are the synergies' temporal coefficients; and R is the residual unexplained by the model. Note that the above formulation presupposes each muscle synergy as a time-invariant vector. Other models of muscle synergies, such as those based on time-varying bases, have been proposed (e.g., Refs. 8,13). Regardless, the W and C are invariably identified from experimental EMGs with an algorithm whose formulation matches the muscle synergy model. Different algorithms, including the popular nonnegative matrix factorization (NMF) (14,15) and independent component analysis (ICA) (16,17), have been employed (18). They differ in their assumptions and implementations, but, importantly, all of them enforce linearity in the synergies' combination as a constraint. The algorithm extracts the statistical regularities from the EMG variability and represents these embedded data structures as muscle synergies. The synergies are then presumed to be representations of motor modules utilized by the CNS for motor control because they describe the EMG variability well. Naturally, the more EMG variability from more behaviors that can be described by the less number of synergies, the stronger the inference of them being genuine neuromotor controllers is.
At this point, it should be helpful to clarify that in this review the term "muscle synergy," or simply "synergy," specifically refers to any group of muscles identified from experimental EMGs based on the spatiotemporal regularities of the muscles' activities and according to a certain model of how the muscle groups are organized and combined to generate the EMGs. This definition of muscle synergy or synergy should not be confused with other definitions of synergy in the motor control literature. A muscle synergy identified by the algorithm may or may not correspond to a motor module, which refers to the hypothesized discrete neuromotor control unit that functions as a neural constraint on motor output. The "muscle synergy hypothesis" is the proposition that there exist motor modules representable as muscle synergies that are defined by a certain model. Validating the neural basis of a muscle synergy, i.e., attributing the provenance of a muscle synergy to the structure and/or functions of a neuronal network, is a test of the muscle synergy hypothesis and the validity of the model that defines the synergies' representation. Muscle synergies and motor modules exemplify motor compositionality-the general idea that motor actions are composed of elementary building blocks that may be defined at different levels of the motor hierarchy (19) and may or may not have a CNS origin.
To what extent the muscle synergies extracted from EMGs represent neural modules for movement control has remained controversial. As mere data regularities, muscle synergies can alternatively indicate a set of bases that spans the subspace of motor commands compatible with the execution of the experimental motor tasks or regularities of feedback-driven activities (e.g., those from stretch reflex) that arise from fixed patterns of muscle length changes as dictated by the muscles' anatomical arrangement (20). These interpretations are especially reasonable when EMGs for synergy extraction are collected from a small number of tasks with limited within-task variability. But to what extent task and biomechanical constraints alone are sufficient to account for muscle synergies has remained unclear, given that for many tasks the dimensionality of the space of compatible muscle patterns appears to be quite large (21,22). Muscle synergies could also be epiphenomena of other motor control principles that may or may not be concurrently implemented by the nervous system in conjunction with control based on motor modules. Some proposed principles that may account for synergies include optimization of task-relevant cost functions (4,23), global optimization of multiple criteria (24), task-specific control that exploits the natural limb dynamics (25), representations learned from prior sensorimotor experiences (26) including functionally good-enough habits (27), or regulations of low-level limb mechanics such as internal joint stresses (28). The neural bases of these proposed principles have mostly remained obscure (but see Ref. 29). Since the observed EMG covariations could be a by-product or genuine reflection of any of the above constraints or control schemes, at present there is no consensus in the field as to how much, or what kind of, EMG variability must be described by the synergy model to suggest that the extracted muscle synergies represent neural constraints on movement (30,31).
The potential non-CNS origins of muscle synergies and the interpretive difficulties adumbrated above have inspired a series of recent studies that aim to directly evaluate whether behavioral muscle synergies represent neural constraints on movement (32). Generally speaking, in these studies muscle synergies are likewise extracted from the behavioral EMGs by factorization algorithms, but EMGs are also collected either when the nervous systems of the same subjects are manipulated or when the activities of specific CNS loci are recorded. The manipulations can include stimulations delivered to parts of the motor or afferent systems or lesions such as spinal transection; the neural recordings can involve neuronal spike trains or electroencephalography. Crucially, these additional recordings are further analyzed to produce a second estimate of the synergies and their activation coefficients. The neurally based synergies may be derived, for instance, by spike (33)-or stimulus (34)-triggered averaging or by the NMF (35); the temporal coefficients may be estimated, for instance, by finding the trajectory of neural activities in a certain subspace within the activity space of the neural population. If these estimates reproduce those derived from the behavioral EMGs, the behavioral muscle synergies are inferred to have a neural basis because they can be independently retrieved by manipulating or recording from the nervous system. The nature of the manipulation or recording may further allow additional inferences such as where in the CNS the muscle synergies are organized. Figure 1 summarizes the logic behind the abovedescribed approaches of revealing the neural basis of muscle synergies.
Here, we argue that the idea of motor modularity grounded on the combination of muscle synergies as specified by the model in Eq. 1 remains a useful and advantageous concept for studying the hard question of how the CNS achieves motor coordination. From the experimentalists' point of view, decomposing complex spatiotemporal muscle activities into discrete muscle synergies (W) and their temporal coefficients (C) is attractive, because both the W and the C may be separately accessed and validated through different experimental techniques in motor neuroscience. Below, we illustrate the value of the synergy concept by reviewing what we have learned about the neural representation of muscle synergies across multiple CNS levels, from single neurons to neuronal networks to whole CNS regions. Experimental findings obtained from multiple animal models and humans have provided compelling demonstrations of the neural basis of many behavioral muscle synergies. Recent data have suggested that the structuring of the synergies (W) across muscles and the dynamics of their activities (C) across time may originate from anatomically separable neuronal networks, even though it remains challenging to pinpoint where in the CNS these networks are located. Also, it seems clear that many muscle synergies undergo plastic alterations during motor development and learning as the neuro-musculoskeletal properties and behavioral requirements change. Almost all studies reviewed here have explicitly or implicitly adopted the logic summarized in Fig. 1 while taking advantage of the system simplification offered by the muscle synergy concept. We then present a critique of this current paradigm by analyzing how it can fail to demonstrate the neural basis of muscle synergies and pointing out some limitations emanating from the linearity of the synergy model and the assumptions behind the synergy identification algorithms. To facilitate progress in our understanding of motor coordination, we conclude by outlining an alternative approach that focuses on constraining the muscle synergy model and synergy identification methods with neurophysiological knowledge and applying the constrained model and methods to describing behavioral data.
Importantly, we discuss below only recent advances, without any exhaustive review of earlier relevant works. Readers are referred to other comprehensive reviews (Refs. 7, 19, 31, 32, 36-47 and others) for perspectives on many important earlier studies.

THE NEURAL BASIS OF MUSCLE SYNERGIES IN THE CNS
Representations at the Single Neuronal Level Revealed by Spike-Triggered Averaging Inasmuch as the motoneuron is the "final common pathway" for neural control of movement (48,49), different types of premotor neurons (i.e., neurons that project to and establish synaptic contacts with motoneurons) exert facilitatory and inhibitory effects on the ongoing activities of the motoneurons and the muscle fibers they activate. Importantly, the premotor neurons' patterns of projection are usually divergent. Each premotor neuron typically innervates the motoneuronal pools of multiple muscles, widely known as the neuron's "muscle field." Individual corticomotoneuronal (50,51), rubromotoneuronal (51), reticulospinal (52,53), and spinal premotor interneurons (PreM-INs) (9, 54) as well as the dorsal root ganglion neurons (55) all have projections to multiple motoneuronal pools. A key in vivo experimental technique that can reveal this divergent connectivity is the spike-triggered averaging of EMG (SpTA) (50,56). In SpTA, activities of CNS neurons and the EMGs from multiple muscles are recorded while the subject performs a motor behavior. The experimenter can then examine whether the action potentials of a recorded cell may facilitate or suppress the activity of any of the recorded muscles-the neuron's postspike effect (PspE)-after averaging the postspike EMGs corresponding to thousands of action potentials. Upon finding the PspEs in the muscles, the set of facilitated or suppressed muscles are identified as the target muscles (the "muscle field") and the neuron as the premotor neuron for those muscles.
A premotor neuron with a divergent muscle field should be a candidate neuronal encoder of a muscle synergy (W) with muscles similar to the muscle field, even though in previous works the link between SpTA-derived muscle fields and EMG-derived muscle synergies has seldom been explicitly mentioned. Previous studies from one of the present authors and his colleagues have established the link between the muscle fields and muscle synergies (33,54). By using SpTA on spinal PreM-INs in monkeys performing a grasping task, they found that the PreM-INs have a divergent facilitatory effect on multiple hand muscles (2.5 ± 1.9 muscles per neuron) (54). Subsequently, they showed that the muscle fields of PreM-INs were not uniformly distributed across hand muscles but rather distributed as clusters that corresponded to the muscle synergies factorized from the behavioral EMGs by the NMF algorithm (33

If Yes
Infer neural organization of muscle synergies Behavioral Neurally-derived Figure 1. Schematic that summarizes current approaches to revealing the neural basis of muscle synergy. In a muscle synergy analysis, multichannel electromyograms (EMGs) are recorded during natural motor behaviors or specific motor tasks, and an algorithm that assumes a certain generative model for the EMG is applied to the experimental EMGs to identify the muscle synergies and their activation parameters. To validate the synergies' neural basis, the central nervous system (CNS) of the same experimental subjects is either manipulated or recorded to independently retrieve the muscle synergies. This validation analysis may entail the same EMG generative model and/or additional models related to the manipulation or recording modality. If the muscle synergies derived from behavioral EMGs match those derived from neural manipulation/recording, we can infer that the set of behavioral muscle synergies identified by the algorithm has a "neural basis." FA, factor analysis; Grad. Desc., gradient descent; ICA, independent component analysis; NMF, nonnegative matrix factorization; Rec., recording; SpTA, spike-triggered averaging of EMG; Stim., stimulation; W, muscle synergy.
computed the similarity between the muscle fields extracted from the PreM-INs and the three muscle synergies extracted from the EMGs (12 muscles) ( Fig. 2A   simple reflexes are complex enough to demand activations of more than one neuronal cluster, but the higher centers again send a simple command to a few PreM-IN clusters representing different Ws so that, with the linear summation of the outputs from the activated Ws, the final motor output can be generated. Evidence for this model of synergy encoding and selection has been recently shown in the primate forelimb control system (57).

Representation by single neuron
It is important to point out here that, although many studies in the literature have assigned the spinal cord as the site of organization of muscle synergies (reviewed in Refs. 32, 39 and others), it is possible that other areas in the CNS are also involved in this organization. Our understanding of the neural basis of muscle synergies (W) will be certainly expanded if comparable SpTA analyses that explore the link between the W and the premotor neurons in other motor centers, such as those in the rubro-and corticomotoneuronal systems, are performed. Indeed, the rubromotoneurons appear to possess wider muscle fields with slightly different muscles than those of the corticomotoneurons (58,59), suggesting that the red nucleus and motor cortical areas may differentially contribute to the organization of different muscle synergies. Preliminary observations from Takei et al. (60) suggest that the muscle fields of the spinal PreM-INs and corticomotoneurons also differ.

Representations at the Neural Network Level Revealed by Stimulation
An obvious disadvantage of the SpTA technique discussed above is that it demands invasive experiments, thus making its use for exploring human motor control difficult. Also, a well-executed SpTA is technically demanding even for a small collection of neurons, thus rendering this technique less useful for revealing representations at the neuronal ensemble level. To overcome these limitations, eliciting motor patterns by stimulating the CNS with electrical, magnetic, or optical means and comparing the retrieved patterns with behavioral EMGs represent another approach for revealing the neural basis of muscle synergies.
If individual neurons across different regions of the CNS contribute to the organization of multiple muscle synergies, it is natural to suppose that for some synergies each neuron may organize the synergy not just by itself but by forming a network with other neighboring and/or distant neurons. Activating the "synergy access points" (61) by electrical microstimulation may uncover the muscle synergies represented in the network. Indeed, multiple recent studies have demonstrated the existence of such access points in the motor cortical areas including the primary motor cortex (M1). Reasoning along the lines above, Overduin et al. (35) identified, from two monkeys, a total of 18 synergies from the EMGs of a grasping task and retrieved 12 of them (67%) by applying long-train intracortical microstimulations (ICMSs) to the M1 of the same animals. Similarly, Amundsen Huffmaster et al. (62) recorded EMGs from two monkeys during reaching and extracted from these data a total of five muscle synergies, three of which (60%) correlated significantly with a synergy from ICMS. These recent data have provided the neural basis for some of the factorization-derived behavioral synergies, in the sense that the synergies' muscular compositions (W) could be independently verified by stimulating specific motor cortical loci.
In contrast to results from SpTA, it remains difficult, however, to ascertain from these studies whether the Ws of the synergies are structured by motor cortical networks (as suggested by Ref. 165) or elsewhere in the CNS. A stimulation delivered to the cortex could conceivably activate neurons that recruit or modify downstream synergies, neurons that organize the entire synergy, neurons that are just part of the synergy-organizing network (see Ref. 34), axons originating from elsewhere that pass through the point of stimulation, or a combination of the above. Data from cortical electrical stimulations alone without concurrent manipulations of the downstream motor centers or the use of additional approaches (e.g., detailed anatomical dissections of the stimulated neurons) likely cannot conclusively settle the loci of origin of the muscular couplings.
The above limitation notwithstanding, several investigators have attempted to probe the cortical representation of muscle synergies in humans by taking advantage of the noninvasiveness of transcranial magnetic stimulation (TMS). Yarossi et al. (63) applied TMS to M1 of a subject while EMGs of hand and forearm muscles were recorded. They then developed a multilayer neural network model that forwardly maps the TMS stimulation parameters to the EMGs (64). Interestingly, when the network outputs were arranged to predict the EMGs via an additional "pre-EMG" layer that represented the EMG-derived muscle synergies, the model's EMG prediction became more accurate than when the synergy layer was absent. If we assume that the structures of artificial neural networks can be modeled to roughly correspond to the anatomy and physiology of biological neuronal networks, as has been recently argued (Ref. 65; see Fig. 2E), these preliminary results suggest that the structures of the muscle synergies being examined may originate from the layer of premotor neurons that directly activates the motoneurons. It is certainly possible that the synergies for different sets of muscles and behaviors are organized at different levels of the motor system (Fig. 2E). Future dissections of the neural sources of muscle couplings will likely demand clever experimental design combined with computational modeling.

Representations Established by Causality Analysis
Thus far, we have reviewed recent works that address the neural basis of muscle synergies by correlating outputs from the neural and muscular domains, but other studies have attempted to establish a causal relationship between a specific neural structure and the organization of certain muscle synergies by evaluating how the synergies are altered by manipulating the neural structure. Specifically, the classic manipulation is to lesion the CNS loci of interest and examine whether it may affect the synergies' muscular compositions. Recent data from animal models have provided strong support for the prominent role of the spinal cord in structuring some of the hindlimb muscle synergies, a notion previously argued for mostly by using the frog (6,9,10,12) and mouse (66,67). By comparing the synergies of intact animals with those observed after complete spinal transection, multiple groups have directly assessed the extent to which supraspinal structures influence the spatial structures of the synergies. Desrochers et al. (68) recorded EMGs from the cat during tied-and split-belt locomotion, before and after spinal transection, and identified muscle synergies by applying a cluster analysis onto the on-and off-set times of the EMG bursts (69,70). The muscles composing the synergies remained the same across spinal transection, and even across different gait speeds and different left-right speed differences on the split-belt treadmill. Such robustness of the identified muscle synergies argues that they represent some fundamental spinally encoded muscle groupings that underpin locomotor muscle patterns (71) (but see Plasticity of Muscle Synergies). Yang et al. (72) further found that in the rat hindlimb locomotor muscle synergies after adult spinalization were similar to those in adult rats spinalized as neonates, raising the possibility that the muscle synergies observed in spinalized adults represent a set of robust, "default-mode" modules for locomotion that are specified early in life and preserved into adulthood. A full confirmation of this fascinating hypothesis will demand a robust assessment of the neonatal locomotor synergies without influences both from postnatal development of the supraspinal system and from any plasticity of the spinal circuits triggered by spinalization. Overall, the above recent findings are consistent with the idea that spinal neuronal circuits determined early in development contribute pivotally to the structuring of locomotor muscle synergies, but as we further discuss below (Plasticity of Muscle Synergies), for healthy adults supraspinal circuits may have a prominent role in their organizations as well.
The next conceivable question is how supraspinal and spinal circuits contribute to muscle coordination in nonlocomotor behaviors. Chvatal et al. (73) addressed this question in the cat by examining the effect of spinalization on the automatic postural response-a stereotyped motor behavior. Importantly, they found that the postural synergies were absent after spinalization, in clear contrast with the locomotor synergies (68,72). Although this finding does not preclude the involvement of spinally organized synergies in these postural tasks (74), it clearly points to the necessary involvement of supraspinal neurons for the access and/or organization of the muscle synergies for behaviors beyond locomotion (75). These feline studies demonstrate that experiments relying on a simple, single lesion are not adequate in revealing possible contributions from CNS loci at multiple levels to the organization of muscle synergies (see Fig. 2E).
Whereas the above-reviewed studies establish muscle synergies' representation through demonstrations of their invariances after lesions, other studies have attempted to causally link recruitment of the muscle synergies to their sensory inputs by dynamically manipulating the afferent activities. In the frog hindlimb, vibration-controlled muscle spindle activity in a single muscle, curarized to eliminate its own EMG, could time-shift or scale the amplitudes of different groups of muscles independently (76). Remarkably, one of the modulated groups, consisting of iliopsoas, rectus anterior, and sartorius, was recruited also as a synergy by cutaneous feedback that signaled collision of the ankle with an obstacle during wiping, so that the trajectory could be corrected by this muscle group to avoid the obstacle (77). In fact, the same synergy appeared to be deployed for frog natural behaviors such as jumping and swimming (78,79). Subsequently, in a realistic biomechanical model of the frog hindlimb, recruitment of the same experimentally identified muscle synergies reproduced the wiping trajectories observed in frogs before and after deafferentation (80), thus directly suggesting that these synergies can generate behaviors. The above studies argue strongly that the muscle groups identified are stable motor modules underpinned by networks accessible by proprioceptive and cutaneous afferents, and susceptible to their manipulation to produce behaviors. Cutaneous modulation of muscle synergies has recently been demonstrated in humans (81).
For other data from humans, Cheung, Bizzi, and colleagues (82) performed an experiment conceptually analogous to the animal lesion experiments reported above, but on the upper limb of stroke survivors with cortical damage. Chronic stroke survivors with mild-to-moderate motor impairment resulting from a unilateral lesion that involved the motor cortical areas were asked to perform seven different movements related to activities of daily living, using either the unaffected (or less affected) or stroke-affected upper limb. They found that most, but not all, of the muscle synergies for these movements from the two arms appeared identical despite kinematic differences between the two arms. These results are consistent with the notion that the muscle synergies for these relatively simple movements are organized either downstream of the cortex in the brain stem or spinal cord or in the cortical areas undisturbed by the stroke lesions. In a subsequent work by Roh et al. (83) that studied severely impaired chronic stroke survivors-presumably subjects with larger lesions than those studied by Cheung et al.-performing an isometric force-matching task, two synergies involving elbow extensors and flexors were observed in both stroke survivors and healthy control subjects, consistent with the downstream organization of these synergies. Readers are referred to Cheung et al. (39) for a more comprehensive discussion of poststroke muscle synergies in humans.
For the human lower limb, some intriguing data have recently emerged from survivors of spinal cord injury (SCI). Cheng et al. (84) found in two patients with complete SCI that spinal cord stimulation could elicit muscle synergies that allowed the patients to achieve unassisted weight-bearing standing. Interestingly, these post-SCI synergies retrieved by stimulation did not resemble those used by healthy subjects for the same task. The observed post-SCI synergies could certainly be muscle couplings acquired after the injury as a result of spinal plasticity. Alternatively, it is possible that some functional lower limb synergies are normally represented in the spinal cord, but in neurologically intact adults either the spinal circuits for these spinal synergies are completely inhibited and taken over by supraspinally organized synergies or they are part of a larger network that includes spinal and supraspinal components, responsible for shaping the normal synergies. These possibilities are all compatible with the observation from survivors of incomplete SCI that the degree of post-SCI synergy changes correlates with the severity of the injury (85)(86)(87)(88)(89). Certainly, additional data from patients and unimpaired individuals will be needed to validate or rule out these possibilities.
Overall, the recent efforts summarized in the above sections have produced much evidence in support of the idea that the muscle synergies observed in motor behaviors have concrete neural underpinnings, but for many behaviors precisely locating the neuronal networks or regions responsible for structuring the synergies' muscle couplings has remained challenging. For certain simple movements of the fore (upper)limb of both animals and humans, some synergies appear to be encoded within the spinal cord. For the hindlimb of animal models, the synergies for some behaviors (e. g., wiping of the frog) are certainly entirely spinal in origin. Even though there is a general agreement that the spinal cord is likely the only major locus for organizing the locomotor synergies in animal models, the sufficiency of the spinal cord for organizing some other behavioral synergies used by neurologically intact adult animals is less clear. In particular, the expression of postural synergies necessitates an intact brain stem. For the human lower limb, although some forms of spinal muscle synergies should exist, the extent to which they participate in natural motor behaviors is even less certain.
For future studies, the most convincing demonstration of a causal relationship between a neuronal network and the synergy's organization would come from the deletion [analogous to the spontaneous deletions observed by Giszter and Kargo (90) in the frog, Stein and Daniels-McQueen (91) in the turtle, and Lafreniere-Roula and McCrea (92) in the cat] or recruitment of a specific, behaviorally relevant muscle synergy after the selective suppression or activation of a discrete, well-defined neuronal network. As a first step toward this goal, recent studies have employed optogenetic and molecular tools to obtain precise characterizations of the molecular identities and neurotransmitter statuses of the synergyencoding neurons-a sine qua non of selectively controlling their activities during natural motor behaviors using, for instance, optogenetics. Levine et al. (67) identified three candidate genes expressed in synergy-encoding interneurons, including Tfab2b, a transcription factor expressed in $23% of the PreM-INs of a muscle, and Satb1 and Satb2, two factors related to nuclear and chromatin organization, expressed in $13% of the PreM-INs. Whether these genes may serve as definitive molecular markers of the synergy encoders and what precise cellular functions these proteins may serve all remain to be determined. Interestingly, the premotor interneurons marked by Tfab2b or Satb1/2 expression could either be inhibitory or excitatory. On the other hand, using the Thy1-channelrhodopsin (ChR2) strain of transgenic mice, Caggiano et al. (66) concluded that the set of spinal interneurons (not necessarily the last-order interneurons that synapse with motoneurons) whose optogenetic activations could retrieve the force-field primitives are excitatory, on the grounds that previous results have shown that markers of inhibitory neurotransmitters are absent in Thy1-ChR2 þ neurons (93). These two observations lead naturally to the question of the relative contributions of excitatory and inhibitory neurons to the premotor network that organizes the synergies. One possibility is that the muscle activation profile of a synergy (W) emerges as excitatory and inhibitory last-order interneurons coactivate and cosuppress different sets of muscles, respectively, while the higher-order interneurons control these last-order neurons via excitatory commands. In any case, it is likely that the coordinated muscular responses arising from the recruitment of a muscle synergy represent the output of a complex network composed of different types of interacting inhibitory and excitatory neurons. It is also likely that the same muscle synergy may arise from many possible neuronal network configurations (94). At present, it remains unclear how the neuronal network responsible for organizing a single, specific muscle synergy can be selectively controlled.

Organizing vs. Driving: The Synergies' Temporal Coefficients
We have reviewed so far results pertaining to how the CNS organizes muscle synergies (W), i.e., the spatial coordinative structure across muscles. As we have argued, premotor neurons and their associated neuronal networks may organize this structure. Although the spinal interneuronal networks have been most frequently implicated as the synergies' primary neural underpinnings, other premotor neurons at other CNS loci, including the motor cortical areas and other subcortical centers, could also organize the synergies. Our next question is, how are the organized muscle synergies driven to produce the required dynamic trajectory of the limb end point? What is the neural basis of the synergies' temporal coefficients (C)?
As we have already pointed out, the activities of the premotor neurons and the CNS networks that involve them follow certain temporal dynamics during motor behaviors, and their activities may themselves be the neural substrate of the temporal coefficients. This possibility implicitly assumes that the temporal coefficient (C) is generated within the same level that organizes the spatial structure of the synergy (W) (e.g., the spinal PreM-INs). This mechanism could be expected from a previous model in which each muscle synergy is treated as a discrete functional unit that is activated by synchronous recruitment of the spinal interneurons that organize the synergy (31,43), so that the C emerges from the intrinsic network dynamics of the recruited interneurons. Some recent experimental findings, however, have not supported this assumption. Takei et al. (33) found that the firing patterns of individual spinal PreM-INs, when compared with the C, did not show any distinct cluster of similarity (Fig.  2B), in contrast to the clear clusters seen for the similarity between the same PreM-INs' muscle fields and the W (Fig.  2A). This result suggests that the W and the C may not be represented at the same PreM-IN level to comparable extents. Takei et al. then proceeded to characterize the temporal dynamics of all PreM-INs as a neural ensemble and found an activation trajectory from the population that is similar to the dynamics of the C (Fig. 2, C and D). This finding implies that the synergies' temporal coefficients could be supplied by neurons upstream of the PreM-INs, a layer that has a global influence on how individual PreM-INs should be activated. In other words, the organizing and driving of the muscle synergies may originate from neurons in different levels that may be anatomically localizable to different CNS regions.
This proposal is illustrated in Fig. 2E, which depicts a hypothetical neural mechanism that controls four muscle synergies. The Ws of these synergies, activating four overlapping sets of muscles (S 1 to S 4 ), can be represented by four distinct clusters of premotor neurons (W 1 to W 4 ) projecting to their respective muscle sets as muscle fields. Also, these W-encoding premotor networks may reside in different CNS regions (e.g., M1, the red nucleus in the midbrain, reticular formation, and spinal cord) and potentially involve the sensory afferents in their organizations (95). Note that in this scheme W 1 to W 4 represent nonoverlapping sets of premotor neurons, so that their input-output relations are well posed and their upstream controller-the layer containing their corresponding drivers, C 1 to C 4 -can recruit and drive each of them discretely (Fig. 2E, black arrows from the Cs to Ws). For example, to generate a movement that requires S 1 and S 2 , the controller can activate just W 1 and W 2 for obtaining the summation of S 1 and S 2 activities. In contrast to W 1 to W 4 , the upstream drivers C 1 to C 4 may be underpinned by overlapping neuronal populations spanning multiple regions including the motor cortical areas (96). This overlapping is justified for the following reason. Since Takei et al. (33) could not find distinct clusters of similarity when matching the temporal activities of individual PreM-INs to the C of the synergies (Fig. 2B) but the time course of the PreM-IN ensemble matched the C well (Fig. 2D), probably each premotor neuron is driven by a mixture of temporal drives. If we further assume that the C-to-W channels for the synergies are anatomically segregated from each other, such mixing of drives may arise because the neural networks representing C 1 to C 4 are intermingled to some extent by default. Interestingly, merging of muscle synergies has been reported in both human stroke survivors (97,98) and marathoners running with high energetic efficiency (99). Such merging could in principle result from the gradual synchronization of multiple drivers made possible by their shared inputs and the intermingling of their representations [see also Cheung et al. (99), their Fig. 7C]. We hasten to emphasize that the neural correlates of the temporal drives in this scheme remain obscure at present. We do not know whether the networks respectively encoding the C and W of a synergy reside in the same or different CNS regions, or whether they are represented by the same neuronal type.
The hypothesis delineated above should be compatible with an existing model of the central pattern generator (CPG) in the spinal cord (100,101). This model assumes a two-level CPG with a rhythm generating network (RG) and a pattern formation network (PF). The PF consists of spinal interneurons that project to multiple motoneuron pools, and the interneurons are themselves interconnected. The RG consists of reciprocal connections between the flexor and extensor half centers, and, importantly, it resides in a layer upstream of the PF, functioning to supply rhythmic activity to the PF layer. We propose that the neural organization of muscle synergies may be equivalent to a generalized twolevel CPG, with the muscle synergies (W) corresponding to the PF network and the temporal coefficients (C) to the RG network. Understanding in what ways the dynamics of the drives for discrete and rhythmic movements are similar or different (102,103), and how a generalized RG must be structured to handle a variety of motor tasks, will demand substantial future experimental and modeling efforts. Deciphering the neural basis of the synergies' temporal coefficients remains a hard problem.

Plasticity of Muscle Synergies
Through our discussions above, we have assumed that the set of muscle synergies is a time-invariant organization that remains stable over a long time, but the human neuro-musculoskeletal system changes continuously during development, aging, motor training, or disease progression, and any traumatic injury would alter this system drastically. It is reasonable to expect that to ensure successful execution of the desired behaviors with the altered plant, this modular organization must be changed, or at least fine-tuned, during motor development (104), motor adaptation (105-108), acquisition of novel motor skills (109,110), or recovery from injury (95,111,112). A recent finding from Yang et al. (72), already cited above, is intriguing for suggesting the potential involvement of supraspinal input in the plastic reshaping of spinal synergies during typical motor development. In addition to the default-mode synergies expressed in adult rats after adult spinal transection, the synergies extracted from normal adult rats (i.e., the pretransection synergies) deviated modestly, but significantly, from the default-mode synergies. This result suggests that the supraspinal system, which matures after the spinal circuits, may also organize some of the muscle synergies used by normal adults. Furthermore, they may sculpt how the synergies are expressed in a way that does not alter the default synergy-encoding spinal circuits because the default synergies could still be observed after spinal transection of the adults (Fig. 2E, magenta arrows). Conceivably, such adjustments of the synergies can be underpinned, for instance, by presynaptic facilitation or inhibition of the synapses between the synergy-encoding interneurons and motoneurons mediated through axoaxonic synapses from descending axons, in a way similar to the presynaptic modulations of afferent input to the spinal cord and subsequent reflex output during voluntary movement (113). We speculate that the extent and direction of this supraspinally driven synergy fine-tuning are dependent on the individual's sensorimotor experience during development, and this fine-tuning contributes to the observed intersubject variability of muscle synergies (114). This hypothesis is not inconsistent with the finding in human complete SCI survivors that synergies elicited by spinal stimulations were different from those used by healthy individuals, as discussed above (see Representations Established by Causality Analysis). But, as already acknowledged, a full confirmation of this idea of supraspinal fine-tuning of spinal synergies will demand additional data that rule out the possibility that the defaultmode and post-SCI synergies are products of postinjury spinal plasticity.
We note that the small deviations of synergies after spinalization described by Yang et al. (72) appear to disagree with the results of Desrochers et al. (68), also cited above, who found that feline locomotor synergies persisted after spinalization. This could be due either to the difference between the rat and the cat or to the different algorithms used by these authors to identify synergies. Specifically, if posttransection synergy changes involve only fine adjustments of the balance between the active muscle components within each W but not the muscle groupings, the method of identifying Ws by relying on the muscles' on-and off-set times utilized by Desrochers et al. may not be able to reveal such modest changes.
Additional insights on the developmental plasticity of muscle synergies in humans have been recently provided by Sylos-Labini et al. (115), who compared the lower limb synergies of kicking and stepping in neonates with the synergies of walking in older children. On one hand, the temporal activation patterns (the Cs) of neonatal locomotor-like kicking were not associated with any stable muscle synergies (the Ws) in the kicks but resembled the Cs observed during gait of preschoolers. On the other hand, the Ws of neonatal stepping did not resemble those of neonatal kicks but could be fractionated to account for the Ws of preschooler gait. Indeed, fractionation of muscle synergies has also been observed during child-to-adult development of running (99). Thus, maturation of the motor patterns for walking necessitates complex reconfiguration of the locomotor circuitry that reassigns the early Cs present in the kicks to drive the Ws fractionated from the early Ws present in neonatal stepping. This apparent flexibility of the motor system to deploy a C profile to drive another W, muscle subsets within a W for fractionation, or a collection of Ws for merging (discussed above) further underscores the idea that the W and the C are represented by distinct neuronal networks. How early sensorimotor experiences interact with the dynamics of the intrinsic developmental program and the growth of the musculoskeletal system to determine the maturation of the circuits for W and C is a research direction that warrants more investigations (116).
Finally, we note that the motor system can acquire new muscle synergies during motor skill learning, especially for skills that cannot be adequately accomplished by deploying preexisting synergies (105). Ballet dancers, over decades of training, tune their synergies for walking to accomplish difficult balance maneuvers (110). Human runners after training also attain increased energetic efficiency of running by merging specific collections of pretraining muscle synergies, possibly achieved when the merged Ws are reassigned to be all driven by one of the C profiles (99). The exact mechanism responsible for this learning has not been established, but during early skill learning the motor system may discover the direction of synergy change by exploiting and modulating the intrinsic variability of the synergies and of their temporal activations and subsequently drive this change by reinforcing the synergy patterns that lead to reward-producing actions (117) in a manner analogous to how reinforcement learning relies on action exploration (118).

A CRITIQUE OF CURRENT APPROACHES
As reviewed above, much of our present knowledge of the neural representations of muscle synergies has come from studies that validate EMG-derived muscle synergies by additional CNS manipulations and/or neural recordings. Inferences on the synergies' neural organizations have relied on assessments of how similar the behavioral synergies are to those retrieved from the additional recordings (Fig. 1). In many studies the reported levels of synergy similarity-at least according to the similarity measures employed-are indeed significant, but certainly less than perfect. We already noted that in Overduin et al. (35) and Amundsen Huffmaster et al. (62) only $60-70% of the behavioral synergies could be matched to ICMS-elicited synergies, but even within the matched synergy pairs, the degree of similarity appears quite variable (Overduin et al., scalar product of 0.75-0.93, mean of 0.84 ± 0.05; Amundsen Huffmaster et al., Pearson's correlation coefficient of 0.56-0.76, mean of 0.63 ± 0.11). Likewise, for the temporal activations of the muscle synergies studied in Takei et al. (33) the match between the synergy activation and neural population trajectories is highly significant, but at a modest R 2 of 61% (Fig. 2, C and D). Some of the differences can be reasonably attributed to noise, insufficient sampling, other uncontrollable experimental variables, the lack of spatiotemporal specificity, and other limitations of the stimulation technologies, including our lack of knowledge of the exact stimulation parameters that mimic the natural synaptic inputs during behaviors and our inability to target specific neuronal types most relevant to the behaviors studied. Alternatively, the similarity metrics used in the above evaluations may not correctly reflect the correspondence between the synergy sets. Given any specific manipulation or recording modality, it is also unclear which level of similarity is sufficient for a reliable inference on neural organization.
At this point, it would be useful to contemplate the following question: Suppose all motor commands to motoneurons and muscles are indeed generated by neurally encoded motor modules, what factors may contribute to any observed differences between the factorization-derived and neurally derived synergies? Any of the following six scenarios may lead to such differences (Fig. 3A, #1 to #6). There may be a lack of movement variability during the collection of the behavioral EMGs, and thus the extracted synergies reflect task constraints rather than neural constraints (#1) (20,31). Or the manipulated or recorded CNS loci play no or only a partial role in the execution of the motor tasks examined (#4). An equally likely possibility is that the linear model of EMG generation assumed by current formulations of muscle synergies is not a good enough representation of the actual motor modules implemented by neuronal networks (#2) or the algorithm employed to extract the synergies from the behavioral EMGs is not performing optimally for the adopted EMG generation model (#3) (18,119,120). Finally, it is possible that the analytic methodologies and any additional models employed for retrieving the neurally derived synergies, such as spike-or stimulus-triggered averaging, are not as reliable as expected (#5 and #6). In any experiment that aims to reveal the neural basis of muscle synergies, it is extremely challenging to simultaneously control for all of the above potential sources that would contribute to a mismatch between the factorization-derived and neurally derived synergies.

Potential Limitations from Linearity
Of the six factors listed above that would hamper the validation of the synergies' neural basis, the potential limitation coming from the linear generative model for the EMGs (Eq. 1) (Fig. 3A, #2) has been less discussed in previous reviews. Apart from the mathematical convenience of linearity and the ready availability of linear factorization algorithms (see Potential Limitations from Algorithmic Assumptions), neurophysiological results obtained thus far without relying on statistical data decomposition do support the validity of the linear model. When motor modules are represented as spatial fields of isometric limb-end point forces, two dissimilar force fields elicited separately by stimulating distinct CNS loci can be linearly summed to predict the force field elicited from costimulation of the same loci. This well-known result has been demonstrated in species ranging from the spinalized frog (6,121,122), mouse (66), and rat (123) to the monkey (57). During frog hindlimb wiping, corrective ankle trajectory elicited upon obstacle impact could be accounted for by linearly summing a cutaneously evoked corrective field with the underlying sequence of fields (77), thus suggesting that linear summation of fundamental force fields can construct a wide variety of motor behaviors, a conclusion also supported by simulation studies (124,125). Importantly, subsequent EMG recordings have established that force-field summation observed during costimulation and corrective responses is underpinned by linear summation of distinct muscle groups (57,77,121,126,127). Indeed, phasic alteration of proprioceptive activity during frog wiping induced independent adjustments of either the amplitude or onset time of premotor activity bursts that drive distinct muscle groups without disrupting the muscle groups themselves (76), thus supporting the preservation of linearity even after afferent manipulation. Consistent with the force field and EMG results above, at the motoneuronal level synaptic integration over the junction between the premotor neurons and the motoneuronal pools is, to a first approximation, reasonably linear over a wide input range (128)(129)(130)(131)(132). These results argue that the linearity of muscle synergy models used in EMG decomposition is grounded on experimental findings, at least for the behaviors and species investigated in the above-mentioned studies.
It remains surprising how linearity arises from the highly nonlinear neuro-musculoskeletal system. What remains unclear is whether linearity may break down in certain situations, in specific behaviors, species, or sets of muscles. Given that input-output nonlinearities are present at all levels of the motor system, it would not be surprising for linearity to be found inadequate in describing motor outputs under some circumstances. For instance, at the level of individual motoneurons, nonlinear relations between inputs and outputs could originate from the persistent inward currents (PICs) (133) mediated by specific voltage-gated cation channels in the motoneuronal dendrites. The PIC exerts multiple nonlinear effects on motoneuronal discharges, including amplification of synaptic inputs, sustained firing after termination of input, derecruitment of the motoneurons occurring at lower input levels than those for recruitment ("derecruitment hysteresis"), and decreased sensitivity to input currents (reviewed in Ref. 134). Nonlinearities may also arise at the motoneuronal pool level when Henneman's "size principle" of motor unit recruitment order (135,136) is violated. When the principle holds, smaller motor units for a muscle are recruited before the larger units; as such, although the relationship between the input synaptic current to the pool and the muscle force produced is overall sigmoidal in shape, it is still approximately linear over a wide interval of midrange currents (137). This linear force increase is believed to reflect both motor unit recruitment and rate modulation of the units already recruited as input current increases (130). Although the size principle is believed to hold in most circumstances (reviewed in Refs. 138,139), in some situations differential recruitment of specific motor units, mediated by spinal and supraspinal circuits and possibly by motoneuronal intrinsic properties like the PIC, appears possible so that specific task demands could be met (e.g., Refs. 140-142). At the neuronal network level, Capaday and van Vreeswijk (143) have attempted to understand the emergence of linear motor-output summation with a nonlinear network model. Through modeling, these authors found that linearity breaks down when local disinhibition (achieved through inhibition of inhibitory neurons) exceeds a certain threshold (see their Fig. 8) Figure 3. An alternative conceptual scheme for approaching the neural basis of muscle synergy. A: even if all motor commands are indeed generated by combination of neurally encoded muscle synergies, multiple factors in current approaches (Fig. 1) can contribute to observed differences between behavioral and neurally derived synergies. Here, we highlight in red (#1 to #6) the potential sources that can contribute to such mismatches. See A CRITIQUE OF CURRENT APPROACHES for a detailed explanation. CNS, central nervous system; EMG, electromyogram. B: to facilitate progress in our understanding of how the CNS achieves motor coordination, we propose an alternative conceptual scheme for approaching the neural basis of muscle synergy. The data from CNS recording and/or manipulation may be used to directly constrain the generative model for EMG and/ or the algorithm used to extract muscle synergies from the behavioral EMGs (magenta arrows). This way, the scientific question becomes, "given a certain EMG generation model, to what extent are the muscle synergies encoded in a CNS region relevant to the production of the behaviors examined as per the model?". See A CRITIQUE OF CURRENT APPROACHES for the full argument. ADLs, activities of daily living; movt., movement.
theoretically, in a nonlinear neuronal network linearity can emerge or break down under different conditions.
As anticipated by these nonlinearities inherent in motoneuronal physiology, limitations of the linear model could indeed be found in the literature. The "winner-take-all" phenomenon, in which the force field from costimulation resembles one of the separately evoked fields instead of their linear summation, was observed in the spinalized frog (19.5% of fields from costimulation in Ref. 122), mouse (22.4% in Ref. 66), and the anesthetized monkey (17.0% in Ref. 57). In a force-field study performed on the decerebrate cat (144), winner-take-all even prevailed in all (n = 6) of the costimulation fields obtained (but note that 4 of 6 fields in this work involved 1 contralateral locus while all other works involved only ipsilateral loci). Although the extent of winner-take-all may vary depending on the animal preparation, the spinal segments stimulated, method of stimulation, loci of stimulation, anesthesia, and other factors [see Table 1 of Yaron et al. (57) for a systematic comparison of previous papers], nonlinear summation of force-field vectors may be a feature of motor modularity that operates concurrently with the well-known linear summation.
Besides winner-take-all, the magnitude of the EMG or force vectors can also show nonlinear summation. In the leg of spinalized frog, Lemay et al. (121) calculated, for all force responses derived from costimulation, the magnitude ratio between the costimulation and summated responses and found that 46% of vectors had a ratio of either <0.8 (indicating supralinear summation, $29% of responses) or >1.2 (indicating sublinear summation, $17%) (see their Fig. 7). More recently, Yaron et al. (57) found that in the monkey cervical spinal cord instances of nonlinearity in magnitude summation were surprisingly more frequent than in previous reports (e.g., Refs. 6, 122, 145) and were nonuniformly distributed across the forelimb muscles. For example, in an elbow muscle the EMG magnitude during costimulation (Fig. 4, B and D, red) was similar to the linear sum (orange), but in another wrist muscle (Fig. 4, A and C) the costimulation magnitude was much larger. In their population analysis, the extent of nonlinear summation, quantified by a scaling index (SI), was indeed different across different muscle groups (Fig. 4E). Although the SI for the wrist and digital muscles was consistently large (5.9-fold facilitation), the median SI for the group of proximal shoulder muscles was close to zero (0.09), suggesting an almost linear summation. This observation clearly indicates that supralinear summation occurs with a bias toward the distal finger and wrist muscles. Yaron et al. (57) further proposed a potential neural mechanism underlying both the linear and nonlinear components when two spinal motor outputs for the distal muscles are summed. In this model (Fig. 4F) The above hypothesis is reasonable for larger animals such as the primates, because an increased total number of cervical spinal neurons may imply an increased number of interneurons with branching structures analogous to those of the INc's in the model (146). What then may be the functional advantage of supralinear force-magnitude summation in the distal forelimb muscles, especially in the nonhuman primates and possibly humans? Whereas linear summation of force direction could simplify the planning of hand paths in space, the spinal mechanism for supralinear magnitude amplification can have an independent role. For example, bringing a cup of water to the mouth is achieved by the same overall kinematics whether the cup is full or empty, but the force required to support the weight of the cup depends on how much water it contains. To accommodate a changed cup weight with ease, separately changing the commands sent to each module is one strategy the CNS could utilize, but this requires the coherence between modules to be maintained or else the resulting kinematics will be altered. This strategy is especially disadvantageous for the efficient control of the hand and wrist muscles. Coordinated activities of these muscles demand the recruitment of a larger number of modules owing to their anatomical and biomechanical complexity, and maintaining coherent activities of a higher number of degrees of freedom in parallel could be computationally overwhelming. Hence, boosting the muscle forces across an already-computed path through a separate channel of interneurons may avoid the risk of producing an unintended alternation of movement path. Indeed, cortical coding of movement direction has been shown to occur earlier than that of movement amplitude in the premotor, parietal, and primary motor cortices (147). Because most corticospinal axons from the motor and parietal cortices terminate in the intermediate spinal layers (148), cortical commands could independently control movement amplitude and direction (Fig. 4F, blue, red, and cyan arrows) by activating different sets of spinal interneurons.
Overall, although these new data agree to a first approximation with the original finding of linear force-field summation produced by costimulation, they also found nonlinear facilitation of force magnitude and occasional winner-takeall interactions of modules occurring simultaneously. It is not known whether these nonlinearities may be specific to the control of certain behaviors, but these results clearly show a potential limitation of the linear models used for identifying the neural basis of muscle synergies. For instance, because of these nonlinearities, the neuronal activities found to represent input drives to individual muscle synergies (e.g., inputs a and b in Fig. 4F) would not match the temporal coefficients of the corresponding synergies extracted from the EMGs.

Potential Limitations from Algorithmic Assumptions
Beyond the constraint of linearity, a related limitation in current approaches of revealing the synergies' neural basis comes from the potential unwarranted assumptions behind the algorithm chosen for analysis (Fig. 3A, #3). Even though Tresch et al. (18) have demonstrated that multiple factorization methods [except principal component analysis (PCA)] all returned similar muscle synergies from behavioral EMGs, all algorithms tested in that study assume linearity, which may be a strong-enough constraint that dictates how the extracted synergies should appear (see Potential Limitations from Linearity). Likewise, other assumptions specific to each algorithm may be unrealistic. The PCA, for instance, assumes orthogonality of the basis vectors identified (i.e., the muscle synergy vectors from PCA must be perpendicular to each other). Although this assumption carries the advantage of permitting negative muscle components that may be interpreted as inhibitory drives, thus far there has been no experimental evidence that supports orthogonality of muscle synergies. For muscle synergy analysis, the PCA is probably more useful for identifying the lower-dimensional EMG subspace within which the higher-dimensional data lie (i.e., the subspace spanned by the collection of PCA basis vectors with the highest corresponding eigenvalues) than for identifying the individual muscle synergies themselves. In this are not considered outliers. The notches represent 5% confidence intervals around the medians. ÃÃÃ P < 0.0001. Wrst/Fingr, wrist and finger muscles; Elbw, elbow muscles; Shldr, shoulder muscles; Top two, the lateral head of the triceps brachii and the spinal part of the deltoid muscle. F: a hypothetical neuronal circuit that may account for supralinear summation of EMG vector magnitude. Single-site stimulation activates interneurons for a module (e.g., INa, blue) and the neuronal axons projecting to them ("a"). As a result, the motoneurons and muscles innervated by INa (Syn-a, denoting muscle synergy a) are recruited during stimulation. This muscle synergy creates a specific EMG balance within the recruited muscles, which is represented as an EMG vector (EVa, blue arrow). Costimulation in the primate and/or cervical spinal cord activates interneurons for 2 modules (INa, blue, and INb, cyan) and the neuronal axons projecting to them ("a" and "b" regard, the EMG reconstructed by the selected principal components may be treated as denoised data suitable for further downstream analysis, with the assumption that the data variance in the removed components is indeed random noise irrelevant to the motor behavior in question. Unlike the PCA, the popular NMF does not assume orthogonality but instead assumes nonnegativity of both the W and C matrices. Although a nonnegative combination has the advantage of making the extracted W conducive to being sparse and thus physiologically and functionally interpretable (15), it does not explicitly model the potential contributions of any inhibitory components in either the W or the C. This aspect of the NMF renders it a less-than-perfect algorithm, especially given the potential involvement of inhibitory neurons in the organization and activation of muscle synergies. In the literature there is scattered evidence, obtained without using a decomposition algorithm, for muscle synergies (W) with inhibitory components. Ethier et al. (126) studied the cat primary motor cortex and observed that, for a muscle, stimulation of a cortical locus resulted in an inhibitory response relative to its prestimulation background activity. When this locus was costimulated with another locus that produced an excitatory response in that muscle, the resulting EMG could be well predicted by subtracting the degree of inhibition (relative to background) from the separately evoked excitatory response [see their Even though the rectified EMG is by definition nonnegative, any negative muscle components in the Ws should in principle be recoverable if enough data variance is provided to the decomposition algorithm, and if sufficient information on how the negative components interact with the positive components in other muscle synergy vectors is retained after any nonlinear thresholding of EMGs. Tresch et al. (18) showed that in simulated EMGs none of the commonly used factorization algorithms performed well in identifying the negative components in W, but this reduced performance is likely in part due to information lost from EMG thresholding in the simulated data. It is conceivable that new decomposition algorithms that are optimized to detect inhibitory components and are suitably constrained with prior neurophysiological knowledge (see Constraining Models with Neurophysiological Knowledge) would perform much better in identifying the negative muscle synergy components.

Constraining Models with Neurophysiological Knowledge
Here, to facilitate our understanding of muscle synergies and to partially overcome the above limitations, we would like to put forth the following conceptual scheme-one that should be complementary to the approaches hitherto taken (Fig. 1)-for how we should think about the muscle synergies' neural basis. Instead of treating the additional neural and/or EMG recordings derived from neural manipulations as just data for validating the behavioral synergies' neural basis, these additional data and/or the "neural" synergies identified from them may be directly used to constrain or update both the synergy combination model of EMG generation and the algorithm used for identifying synergies from the behavioral EMGs (Fig. 3B). This way, any synergy extraction algorithm and its underlying EMG generative model will automatically be grounded on information derived from the nervous system. Importantly, this move would amount to changing the scientific question from "Do the muscle synergies factorized from the behavioral EMGs have a neural basis?" (Fig. 1) to "Given an EMG generation model, to what extent are the muscle coordinative patterns encoded in a CNS region relevant to the production of the behaviors examined?" (Fig. 3B). If there is a good match between the behavioral synergies derived from the neurally constrained model and those from CNS manipulations, one can infer that 1) the model is valid and 2) the manipulated or recorded CNS region is indeed relevant to the motor coordination of the examined behavior.
How should a neurally constrained model of motor modules be formulated? Whether the existing linear models and factorization algorithms may be suitably modified to accommodate additional neural constraints or whether completely new models are required will depend on the nature of the neurophysiological and behavioral data collected, the plausible control policy suggested by the data, and the levels of mechanistic details and potential nonlinearities to be modeled. At the very least, even if the current basic NMF-or ICAbased linear models are to be further pursued, the properties of the W and C parameters to be estimated from behavioral EMGs could be further constrained by additional neural data. For example, with the linear NMF model, one could first estimate neurally derived muscle synergies by stimulating or recording from a CNS region. These neural synergies could then be used to explain the behavioral EMGs via the NMF update rules (by fixing the neural synergies as W while updating the C matrix, or vice versa) to test whether the CNS-derived synergies can describe or even predict the behavioral EMGs without the need of synergy modification or extra synergies. For human behavioral EMGs, although this approach could be easily applied to constrain the search of Ws through concurrently recorded electroencephalographic or magnetoencephalographic data, one could also impose algorithmic constraints by using information derived from the level of individual neurons of other model organisms using, for instance, SpTA (with the caveat that the neuromusculoskeletal systems of different species are different). Other properties acquired from additional neurophysiological experiments, such as the expected sparseness of the muscle synergy vectors and the degree of independence between the Cs of multiple synergies in the CNS region being analyzed, or even the expected distribution of EMG noise (119,120,149), may also be incorporated into the model as prior knowledge through modification or rederivation of the update rules for estimating W and C. Beyond the level of single neurons, if the experiment involves recording spikes of a neuronal ensemble, the dynamics of neural activities derived after some dimensionality reduction of the recordings (150,151) may constrain the temporal structure of the Cs through some variants of the classic NMF algorithm (152).
As more concepts and detailed knowledge on the anatomy and biophysics of the mammalian sensorimotor circuits are accumulated, linear models may no longer be deemed sufficient to accommodate all relevant neurally derived constraints. It is conceivable that neural network models with built-in nonlinearities and a modular architecture (perhaps similar to those described in Refs. 63 and 64) can be pretrained with neural data to reflect biological aspects of the network's connectivity specified by the genome and typical developmental experiences (153); they can then be tested on their control tractability and their ability to describe or predict behavioral EMGs (see Ref. 65 for an analogous example performed on the visual system). Very recently, activities of the motoneurons along the body segments of Caenorhabditis elegans during locomotion have been successfully modeled with nonlinear differential equations that incorporate motoneuronal couplings mediated by proprioceptive and descending drives (154). It remains to be explored whether nonlinear models may be useful, or even tractable from the control perspective, for understanding the considerably more complex mammalian motor system (e.g., Ref. 143).
In our view, the above approach of thinking about the neural basis of muscle synergies has the advantage of making the scientific question more circumscribed by focusing on testing whether a relationship between a CNS structure and the generation of a motor behavior exists given a model. If multiple EMG generation models or algorithms are compared in the analysis, this approach may also facilitate model selection by restricting the viable models to only those compatible with the neural data [with goodness of fit evaluated by metrics suitable for nonlinear models (155)]. Note that our proposed approach does not automatically render the muscle synergy hypothesis immune to being disproved (156,157). If, for instance, a mismatch between the behavioral and neurally derived synergies is still observed for a CNS region whose exclusive role in coordinating the motor behavior examined has been otherwise established by other means, the synergy model employed would be invalidated, and thus require updating. Our approach, however, will lead to models that both describe the behavioral and neural data more accurately and are better suited for testing new hypotheses that are more neurophysiologically oriented, specific, and useful than the original muscle synergy hypothesis. For example, when comparing the synergies elicited by stimulating two neural populations with those derived from a motor behavior, how a model must be parametrized or modified to accommodate the neural data while explaining the behavioral data may generate new hypotheses concerning the distinct roles of the two neural populations in structuring and activating the motor modules for the behavior in question.
We note that the conceptual framework for thinking about the neural basis of muscle synergies outlined above may also be analogously applied to elucidate any plausible neural basis of other motor structures identified from behavioral or neural data, such as the proposed elementary temporal motifs embedded within complex movement trajectories (158,159).

FUTURE DIRECTIONS
In the past few years, great strides have been made in our quest for the neural basis of muscle synergies. We have emphasized above our current understanding of the roles of the spinal cord, brain stem, and motor cortical areas in the activation, organization, and fine-tuning of muscle synergies, but recent data have also suggested the potential involvement of the basal ganglia (37,81,160) and the cerebellum (161) in the synergies' spatiotemporal organization. With the ever-improving new technologies for selective neuronal stimulations (162), large-scale neural ensemble recordings (151), neural circuit dissection (93,163), and machine learning-based tracking of kinematics (164) and motor behaviors (160), we will soon be able to gather large volumes of data that would demand substantial data analysis and computational modeling for deriving new insights in the principles of neural control of movement (157). Such data can likely be productively used to constrain models of motor modularity and inform future developments of synergy identification algorithms. Furthermore, these novel technologies may permit the realization of previously impossible experiments so that new data can be produced to specifically constrain and refine models of motor control. It is hoped that the investigative framework proposed here, based on evaluating the relevance of neurophysiologically constrained models of muscle synergies to a behavior, will facilitate the emergence of a more sophisticated and nuanced understanding of motor modularity, so that we can move forward from the current debate on the neural versus nonneural origin of muscle synergies.