{"title": "Modularity in the motor system: decomposition of muscle patterns as combinations of time-varying synergies", "book": "Advances in Neural Information Processing Systems", "page_first": 141, "page_last": 148, "abstract": null, "full_text": "Modularity in the motor system: decomposition\n\nof muscle patterns as combinations of\n\ntime-varying synergies\n\nAndrea d\u2019Avella and Matthew C. Tresch\nDepartment of Brain and Cognitive Sciences\n\nMassachusetts Institute of Technology, E25-526\n\nCambridge, MA 02139\n\n davel, mtresch\n\n@ai.mit.edu\n\nAbstract\n\nThe question of whether the nervous system produces movement through\nthe combination of a few discrete elements has long been central to the\nstudy of motor control. Muscle synergies, i.e. coordinated patterns of\nmuscle activity, have been proposed as possible building blocks. Here we\npropose a model based on combinations of muscle synergies with a spe-\nci\ufb01c amplitude and temporal structure. Time-varying synergies provide\na realistic basis for the decomposition of the complex patterns observed\nin natural behaviors. To extract time-varying synergies from simultane-\nous recording of EMG activity we developed an algorithm which extends\nexisting non-negative matrix factorization techniques.\n\n1 Introduction\n\nIn order to produce movement, every vertebrate has to coordinate the large number of de-\ngrees of freedom in the musculoskeletal apparatus. How this coordination is accomplished\nby the central nervous system is a long standing question in the study of motor control.\nAccording to one common proposal, this task might be simpli\ufb01ed by a modular organiza-\ntion of the neural systems controlling movement [1, 2, 3, 4]. In this scheme, speci\ufb01c output\nmodules would control different but overlapping sets of degrees of freedom, thereby de-\ncreasing the number of variables controlled by the nervous system. By activating different\noutput modules simultaneously but independently, the system may achieve the \ufb02exibility\nnecessary to control a variety of behaviors.\n\nSeveral studies have sought evidence for such a modular controller by examining the pat-\nterns of muscle activity during movement, in particular looking for the presence of muscle\nsynergies. A muscle synergy is a functional unit coordinating the activity of a number of\nmuscles. The simplest model for such a unit would be the synchronous activation of a set\nof muscles with a speci\ufb01c activity balance, i.e. a vector in the muscle activity space. Using\ntechniques such as the correlation between pairs of muscles, these studies have generally\nfailed to provide strong evidence in support of such units. However, using a new analysis\nthat allows for simultaneous combinations of more than one synergy, our group has recently\nprovided evidence in support of this basic hypothesis of the neural control of movement.\n\n\u0001\n\fWe used a non-negative matrix factorization algorithm to examine the composition of mus-\ncle activation patterns in spinalized frogs [5, 6]. This algorithm, similarly to that developed\nindependently by others [7], extracts a small number of non-negative 1 factors which can be\ncombined to reconstruct a set of high-dimensional data.\n\nHowever, this analysis assumed that the muscle synergies consisted of a set of muscles\nwhich were activated synchronously. In examinations of complex behaviors produced by\nintact animals, it became clear that muscles within a putative synergy were often activated\nasynchronously. In these cases, although the temporal delay between muscles was nonzero,\nthe dispersion around this delay was very small. These observations suggested that the ba-\nsic units of motor production might involve not only a \ufb01xed coordination of relative muscle\nactivation amplitudes, but also a coordination of relative muscle activation timings. We\ntherefore have developed a new algorithm to factorize muscle activation patterns produced\nduring movement into combinations of such time-varying muscle synergies.\n\n2 Combinations of time-varying muscle synergies\n\nwith a speci\ufb01c time course in the activity of each muscle. In discrete time, we can represent\nin muscle activity\nspace. The data set which we consider here consists of episodes of a given behavior, e.g. a\nset of jumps in different directions and distances, or a set of walking or swimming cycles.\n\nWe model the output of the neural controller as a linear combination of muscle patterns\neach pattern, or time-varying synergy, as a sequence of vectors\u0001\u0003\u0002\u0005\u0004\u0007\u0006\nIn a particular episode\b , each synergy is scaled by an amplitude coef\ufb01cient\t\u000b\n and time-\n\n . The sequence of muscle activity for that episode is then given by:\nshifted by a delay\u0004\n\n(1)\n\n\u0002\r\u0004\u0007\u0006\u000f\u000e\n\n\u0013\u0015\u0014\n\n\u0002\u0005\u0004\u0017\u0016\u0018\u0004\n\nthat minimize the reconstruction error\n\nFig. 1 illustrates the model with an example of the construction of a muscle pattern by\ncombinations of three synergies. Compared to the model based on combinations of syn-\n\n3 Iterative minimization of the reconstruction error\n\n\u0012 ) for each episode and each synergy.\n\nall parameters. In fact, with synchronous synergies there is a combination coef\ufb01cient for\neach time step and each synergy, whereas with time-varying synergies there are only two\n\nchronous muscle synergies this model has more parameters describing each synergy (\u0019\u001b\u001a\u000f\u001c\nvs.\u0019\n, with\u0019 muscles and\u001c maximum number of time steps in a synergy) but less over-\n\u0012 and\u0004\u0007\n\nparameters (\t\u001d\n\nFor a given set of episodes, we search for the set of\n,\u001e\n\u001d\u001e\n\u0013\u001f\u0014! \" \" \n\u0002\u0005%&\u0006(')'*'\n\u000e$#\n\u0002\u00071-%&\u0006 and\u0004\nof coef\ufb01cients\t\n\u000e65\u000b7\n\u0013\u001f\u0014:9\n\n\u0002\u0005\u001c+\u0016-,.\u00060/ , of maximum duration\u001c\n243\n\n1The non-negativity constraint arises naturally in the context of motor control from the fact that\n\ufb01ring rates of motoneurons, and consequently muscle activities, cannot be negative. While it is con-\nceivable that a negative contribution on a motoneuronal pool from one factor would always be can-\ncelled by a larger positive contribution from other factors, we chose a model based on non-negative\nfactors to ensure that each factor could be independently activated.\n\nnon-negative time-varying synergies\ntime steps and the set\n\n\u0002\r\u0004\u0007\u0006;\u0016\n\n\u0013\u001f\u0014\n\n\u0002\r\u0004\u0017\u0016<\u0004\n\n\f\n\n\u0010\n\u0011\n\u0012\n\t\n\n\u0012\n\u0001\n\u0012\n\n\u0012\n\u0006\n\u0012\n\u0001\n\u0012\n\u0010\n\u0012\n\u0001\n\u0012\n\u0001\n\u0012\n\u0012\n\n\u0012\n\n\u000e\n\u0011\n\n2\n\n3\n2\n\n3\n\u0011\n8\n\f\n\n\u0010\n\u0011\n\u0012\n\t\n\n\u0012\n\u0001\n\u0012\n\n\u0012\n\u0006\n9\n3\n\f1\n\ny\ng\nr\ne\nn\ny\nS\n\n2\n\ny\ng\nr\ne\nn\ny\nS\n\n3\n\ny\ng\nr\ne\nn\ny\nS\n\n1\n\n2\n\n3\n\n4\n\n5\n\n1\n\n2\n\n3\n\n4\n\n5\n\n1\n\n2\n\n3\n\n4\n\n5\n\nl\n\ns\ne\nc\ns\nu\nM\n\n1\n\n2\n\n3\n\n4\n\n5\n\n10\n\n20\n\n30\n\n40\n\n50\nTime\n\n60\n\n70\n\n80\n\n90\n\n100\n\nT\n1\n\nT\n2\n\nT\n3\n\nC\n1\n\nC\n2\n\nC\n3\n\nfor\n\n.\n\n1-\u001c\n\nscaled and shifted components (top right, broken lines) are summed together.\n\nAfter initializing synergies and coef\ufb01cients to random positive values, we minimize the\nerror by iterating the following steps:\n\nFigure 1: An example of construction of a muscle pattern by the combinations of three\ntime-varying synergies. In this example, each time-varying synergy (left ) is constituted by\na sequence of 50 activation levels in 5 muscles chosen as samples from Gaussian functions\nwith different centers, widths, and amplitudes. To construct the muscle pattern (top right,\nshaded area), the activity levels of each synergy are \ufb01rst scaled by an amplitude coef\ufb01cient\n(\n\n\u0012 , represented in the bottom right by the height of an horizontal bar) and shifted in time\n\u0012 , represented by the position of the same bar). Then, at each time step, the\nby a delay (\u001c\n\u0002\u0002\u0001\nwith\u0001\n1. For each episode, given the synergies\u001e\ndelays\u0004\n2. For each episode, given the synergies and the delays\u0004\n\ufb01cients\t\n3. Given delays and scaling coef\ufb01cients, update the synergy elements\u0001\n\n\u0001\u0004\u0003-% or\n\u0012 , \ufb01nd the\n\u0012 using a nested matching procedure based on the cross-correlation of the\n\u0012 , update the scaling coef-\n\u0012 by gradient descent\n\n\u0012 and the scaling coef\ufb01cients\t\n\u0016\t\b\u000b\n\r\f\u000f\u000e\n\u0016\t\b\u0012\u0011\u0013\f\u0015\u0014\u0017\u0016\u0019\u0018\n\n243\n\nHere and below, we enforce non-negativity by setting to zero any negative value.\n\nsynergies with the data (see 3.1 below).\n\nby gradient descent\n\n\u0005\u0007\u0006\n\n\u0002\u0002\u0001\n\n\n\u0012\n\u0006\n\u000e\n%\n\u0001\n\n\u000e\n7\n2\n3\n\n\u0012\n\u0010\n\u000e\n\u0001\n\u0012\n\u0006\n\u0005\n\u0001\n\u0012\n\u0010\n\u000e\n\f3.1 Matching the synergy delays\n\nTo \ufb01nd the best delay of each synergy in each episode we use the following procedure:\n\ni. Compute the sum of the scalar products between the s-th data episode and the i-th\n\nsynergy time-shifted by\u0004\nor scalar product cross-correlation at delay\u0004 , for all possible delays.\n\n\u0016<\u0004\u0007\u0006\n\n\u0002\u0005\u0004\u0007\u0006\n\n\u0002\u0002\u0001\n\n\u0002\u0002\u0001\n\nii. Select the synergy and the delay with highest cross-correlation.\niii. Subtract from the data the selected synergy (after scaling and time-shifting by the\n\n(2)\n\nselected delay).\n\niv. Repeat the procedure for the remaining synergies.\n\n4 Results\n\nWe tested the algorithm on simulated data in order to evaluate its performance and then\napplied it to EMG recordings from 13 hindlimb muscles of intact bullfrogs during several\nepisodes of natural behaviors [8].\n\n4.1 Simulated data\n\nWe \ufb01rst tested whether the algorithm could reconstruct known synergies and coef\ufb01cients\nfrom a dataset generated by those same synergies and coef\ufb01cients. We used two different\ntypes of simulated synergies. The \ufb01rst type was generated using a Gaussian function of\ndifferent center, width, and amplitude for each muscle. The second type consisted of syn-\nergies generated by uniformly distributed random activities. For each type, we generated\nsets of three synergies involving \ufb01ve muscles with a duration of 15 time steps. Using these\nsynergies, 50 episodes of duration 30 time steps were generated by scaling each synergy\n\n\u0002\u0005\u0004\u0007\u0006 with random coef\ufb01cients\t\n3 of less than,\n\nIn \ufb01gure 2 the results of a run with Gaussian synergies are shown. Using as a convergence\nfor 20 iterations, after 474 iterations the solution\ncriterion a change in\nhad\n. Generating and reconstructed synergy activations are shown side by side\non the left, in gray scale. Scatter plots of generating vs. reconstructed scaling coef\ufb01cients\nand temporal delays are shown in the center and on the right respectively. Both synergies\nand coef\ufb01cients were accurately reconstructed by the algorithm.\n\n\u0012 and shifting it by random delays\u0004\n%\u0004\u0003\u0006\u0005\n\n\u0007\b\u0007\n\t\f\u000b\n\n\u0012 .\n\nIn table 1, a summary of the results from 10 runs with Gaussian and random synergies\nis presented. We used the maximum of the scalar product cross-correlation between two\nnormalized synergies (see eq. 2) to characterize their similarity. We compared two sets\nof synergies by matching the pairs in each set with the highest similarity and computing\n) between these pairs. All the synergy sets that we reconstructed\nthe mean similarity (\n(\n). We also compared the gen-\n, and\n\n) had a high similarity with the generating set (\n\nerating and reconstructed scaling coef\ufb01cients\t.\n\ndelays\u0004\nafter compensating for possible lags in the synergies (\u001c\u001b\n\n\u0012 using their correlation coef\ufb01cient\n\u0012 by counting the number of delay coef\ufb01cients that were reconstructed correctly\n3 ) but with synergies slightly different from\n\n). The match in scaling coef-\n\ufb01cients and delays was in general very good. Only in a few runs with Gaussian synergies\nwere the data correctly reconstructed (high\nthe generating ones (as indicated by the lower\n).\ning coef\ufb01cients (lower\n\n) and consequently not perfectly match-\n\n\u000e\u0016\u0015\u0017\u0011\u0019\u0018\n\n\u000e\u0010\u000f\u0012\u0011\u0014\u0013\n\n\u0013\u001e\u001d\n\nand\n\n\u0012\u0011\n\n\u001a\f\u0013\n\n\u001f \u001a\n\n\u0013\"!\n\n\u001f\u0005#\u001b\n\n\u0013\u001e\u001d\f$\n\n\n\n\u0012\n\u000e\n\u0011\n\u0010\n\f\n\n\u0006\n5\n\u0001\n\u0012\n\u0001\n\u0012\n\n\u0002\n\u0002\n3\n\u000e\n%\n'\n\n8\n\u0002\n\n\u0011\n8\n\n8\n!\n\f1\n\ny\ng\nr\ne\nn\ny\nS\n\n1\n2\n3\n4\n5\n\n1\n2\n3\n4\n5\n\n1\n2\n3\n4\n5\n\n2\n\ny\ng\nr\ne\nn\ny\nS\n\n3\n\ny\ng\nr\ne\nn\ny\nS\n\n2\n\n0\n\n2\n\n0\n\n0\n\n0\n\n1\n\n16\n\n0\n2\n16\n\n0\n\n0\n16\n\n0\n\n2\n\n15\n\n5\n\n10\nWgen\n\n5\n\n10\nWrec\n\n0\n15 0\n\n1\nCgen vs. Crec\n\n0\n\n0\n\nTgen vs. Trec\n\n16\n\n16\n\n16\n\n\u000e\u0016\u0015\u0017\u0011\u0019\u0018\n\nFigure 2: An example of reconstruction of known synergies and coef\ufb01cients from simu-\nlated data. The \ufb01rst column (\n) shows three time-varying synergies, generated from\nGaussian functions, as three matrices each representing, in gray scale, the activity of 5\n) shows the three\nmuscles (rows) over 15 time steps (columns). The second column (\nsynergies reconstructed by the algorithm: they accurately match the generating synergies\n(except for a temporal shift compensated by an opposite shift in the reconstructed delays).\nThe third and fourth columns show scatter plots of generating vs. reconstructed scaling\ncoef\ufb01cients and delays in 50 simulated episodes. Both sets of coef\ufb01cients are accurately\nreconstructed in almost all episodes.\n\n\u000f\u0012\u0011\u0014\u0013\n\n4.2 Time-varying muscle synergies in frog\u2019s muscle patterns\n\nWe then applied the algorithm to EMG recordings of a large set (\nkicks, a defensive re\ufb02ex that frogs use to remove noxious stimuli from the foot. Each kick\nconsists of a fast extension followed by a slower \ufb02exion to return the leg to a crouched\nposture. The trajectory of the foot varies with the location of the stimulation on the skin\nand, as a consequence, the set of kicks spans a wide range of the workspace of the frog.\nCorrespondingly, across different episodes the muscle activity patterns in the 13 muscles\nthat we recorded showed considerable amplitude and timing variations that we sought to\nexplain by combinations of time-varying synergies.\n\n, ) of hindlimb\n\nAfter rectifying and integrating the EMGs over 10 ms intervals, we performed the opti-\n\u0002\u0005\u0004 . We chose the maximum\nduration of each synergy to be 20 time steps, i.e. 200 ms, a duration larger than the duration\nof a typical muscle burst observed in this behavior. We repeated the procedure 10 times for\n\nsynergies, with\n\n'*')'\n\n\u0001\u0003\u0002\n\nmization procedure with sets of\neach\n\n.\n\n\u000e\n\u000e\n,\n,\n\u000e\n\fmax\n\nmedian\n\nmin\n\nmax\n\nmedian\n\nmin\n\n\u0011\u0019\u000f\n561\n451\n297\n\n\u0011\u0019\u000f\n555\n395\n208\n\nGaussian synergies\n\n0.9989\n0.9952\n0.9874\n\n0.9996\n0.9990\n0.8338\n\n\u001f \u001a\n0.9983\n0.9974\n0.2591\n\nRandom synergies\n\n0.9999\n0.9998\n0.9998\n\n1.0000\n1.0000\n1.0000\n\n\u001f \u001a\n0.9996\n0.9990\n0.9984\n\n\u001f\u0005\n\u001f\u0005\n\n\u0013\u001e\u001d\n\u0013\u001e\u001d\n\n0.9467\n0.9233\n0.3133\n\n0.9867\n0.9800\n0.9733\n\nTable 1: Comparison between generated and reconstructed synergies and coef\ufb01cients for\n10 runs with Gaussian and random synergies. See text for explanation.\n\n\u000b\n\t\n\n(median%\n(median%\n\nto%\n\nIn \ufb01gure 3 the result of the extraction of four synergies with the highest\nconvergence criterion of a change in\nafter 100 iterations with a \ufb01nal\nwere in general very similar to this set, as indicated by a mean similarity (\n\nis shown. The\nfor 20 iterations was reached\n\u0004 . The synergies extracted in the other nine runs\n) ranging from\n\n3 smaller than,\u001d%\n\n\u0003\u0001\n\n\u0007\b\u0007\nshown in \ufb01gure 3 was not properly matched.\n\n\u0002 ) and a correlation between scaling coef\ufb01cients ranging from%\n\n\u0007\b\u0007\n\u0002 ). In the case with the lowest similarity, only one synergy in the set\n\nto%\n\n\u0002\u0003\u0002\n\nThe four synergies captured the basic features of the muscle patterns observed during dif-\nferent kicks. The \ufb01rst synergy, recruiting all the major knee extensor muscles (VI, RA, and\nVE), is highly activated in laterally directed kicks, as seen in the \ufb01rst kick shown in \ufb01g-\nure 3, which involved a large knee extension. The second synergy, recruiting two large hip\nextensor muscles (RI and SM) and an ankle extensor muscle (GA), is highly activated in\ncaudally and medially directed kicks, i.e. kicks involving hip extension. The third synergy\ninvolves a speci\ufb01c temporal sequencing of several muscles: BI and VE \ufb01rst, followed by\nRI, SM, and GA, and then by AD and VI at the end. The fourth synergy has long activation\npro\ufb01les in many \ufb02exor muscles, i.e. those involved in the return phase of the kick, with a\nspeci\ufb01c temporal pattern (long activation of IP; BI and SA before TA).\n\nWhen this set of EMGs was reconstructed using different numbers of muscle synergies,\nwe found that the synergies identi\ufb01ed using N synergies were generally preserved in the\nsynergies identi\ufb01ed using N+1 synergies. For instance, the \ufb01rst two synergies shown in\n\u0004 . Therefore, increasing\nthe number of synergies allowed the data to be reconstructed more accurately (as seen by a\nhigher\n\n\ufb01gure 3 were seen in all sets of synergies, from\n\n3 ) but without a complete reorganization of the synergies.\n\nto\n\n5 Discussion\n\nThe algorithm that we introduced here represents a new analytical tool for the investigation\nof the organization of the motor system. This algorithm is an extension of previous non-\nnegative matrix factorization procedures, providing a means of capturing structure in a set\nof data not only in the amplitude domain but also in the temporal domain. Such temporal\nstructure is a natural description of motor systems where many behaviors are character-\nized by a particular temporal organization. The analysis applied to behaviors produced by\nthe frog, as described here, was able to capture signi\ufb01cant physiologically relevant char-\nacteristics in the patterns of muscle activations. The motor system is not unique, however,\nin having structure in both amplitude and temporal domains and the techniques used here\ncould easily be extended to other systems.\n\n\n\u0012\n8\n\u0002\n3\n\n\u0011\n\u0013\n!\n\u001b\n8\n$\n\n8\n!\n\n\u0012\n8\n\u0002\n3\n\n\u0011\n\u0013\n!\n\u001b\n8\n$\n\n8\n!\n\u0002\n3\n\u0002\n\u0002\n3\n\u000e\n%\n'\n\u000b\n\n\u0011\n%\n'\n'\n'\n\u0007\n'\n'\n\u0004\n\u0007\n'\n\u0007\n\u000e\n\u0001\n\u000e\n\u0002\n\f1\n\ny\ng\nr\ne\nn\ny\nS\n\nRI\nAD\nSM\nST\nIP\nVI\nRA\nGA\nTA\nPE\nBI\nSA\nVE\nRI\nAD\nSM\nST\nIP\nVI\nRA\nGA\nTA\nPE\nBI\nSA\nVE\nRI\nAD\nSM\nST\nIP\nVI\nRA\nGA\nTA\nPE\nBI\nSA\nVE\nRI\nAD\nSM\nST\nIP\nVI\nRA\nGA\nTA\nPE\nBI\nSA\nVE\n\n2\n\ny\ng\nr\ne\nn\ny\nS\n\n3\n\ny\ng\nr\ne\nn\ny\nS\n\n4\n\ny\ng\nr\ne\nn\ny\nS\n\nl\n\ns\ne\nc\ns\nu\nM\n\nRI\n\nAD\n\nSM\n\nST\n\nIP\n\nVI\n\nRA\n\nGA\n\nTA\n\nPE\n\nBI\n\nSA\n\nVE\n\n10\n\n20\n\nTime\n\n30\n\n5\n\n10\n\n20\n\n25\n\n15\nTime\n\nT\n1\n\nT\n3\n\nT\n2\n\nT\n4\n\n5\n\n10\n\n15\n\n20\n\nC\n1\nC\n2\nC\n3\nC\n4\n\nFigure 3: Reconstruction of recti\ufb01ed and integrated (10 ms) EMGs for two kicks by time-\nvarying synergies. Left: four extracted synergies constituted by activity levels (in gray\nscale) for 20 time steps in 13 muscles: rectus internus major (RI), adductor magnus (AD),\nsemimembranosus (SM), ventral head of semitendinosus (ST), ilio-psoas (IP), vastus inter-\nnus (VI), rectus anterior (RA), gastrocnemius (GA), tibialis anterior (TA), peroneous (PE),\nbiceps (BI), sartorius (SA), and vastus externus (VE) [8]. Top right: the observed EMGs\n(thin line and shaded area) and their reconstruction (thick line) by combinations of the four\nsynergies, scaled in amplitude (\n\n\u0012 ) and shifted in time (\u001c\n\n\u0012 ).\n\nOur model can be naturally extended to include temporal scaling of the synergies, i.e.\nallowing different durations of a synergy in different episodes. Work is in progress to\nimplement an algorithm similar to the one presented here to extract time-varying and time-\nscalable synergies. We will also address the issue of how to identify time-varying muscle\nsynergies from continuous recordings of EMG patterns, without any manual segmentation\ninto different episodes. A possibility that we are investigating is to extend the approach\nbased on a sparse and overcomplete basis used by Lewicki and Sejnowski [9]. Finally,\nfuture work will aim to the development of a probabilistic model to address the issue of the\ndimensionality of the synergy set in terms of Bayesian model selection [10].\n\nAcknowledgments\n\nWe thank Zoubin Ghahramani, Emanuel Todorov, Emilio Bizzi, Sebastian Seung, Simon\nOverduin, and Maura Mezzetti for useful discussions and comments.\n\n\n\fReferences\n\n[1] E. Bizzi, P. Saltiel, and M. Tresch. Modular organization of motor behavior. Z Naturforsch [C],\n\n53(7-8):510\u20137, 1998.\n\n[2] F. A. Mussa-Ivaldi. Modular features of motor control and learning. Curr Opin Neurobiol,\n\n9(6):713\u20137, 1999.\n\n[3] W. J. Kargo and S. F. Giszter. Rapid correction of aimed movements by summation of force-\n\n\ufb01eld primitives. J Neurosci, 20(1):409\u201326, 2000.\n\n[4] Z. Ghahramani and D. M. Wolpert. Modular decomposition in visuomotor learning. Nature,\n\n386(6623):392\u20135, 1997.\n\n[5] M. C. Tresch, P. Saltiel, and E. Bizzi. The construction of movement by the spinal cord. Nature\n\nNeuroscience, 2(2):162\u20137, 1999.\n\n[6] P. Saltiel, K. Wyler-Duda, A. d\u2019Avella, M. C. Tresch, and E. Bizzi. Muscle synergies encoded\nwithin the spinal cord: evidence from focal intraspinal nmda iontophoresis in the frog. Journal\nof Neurophysiology, 85(2):605\u201319, 2001.\n\n[7] D. D. Lee and H. S. Seung. Learning the parts of objects by non-negative matrix factorization.\n\nNature, 401(6755):788\u201391, 1999.\n\n[8] A. d\u2019Avella. Modular control of natural motor behavior. PhD thesis, MIT, 2000.\n[9] M. S. Lewicki and T. J. Sejnowski. Coding time-varying signals using sparse, shift-invariant\nIn M. S. Kearns, S. A. Solla, and D. A. Cohn, editors, Advances in Neural\n\nrepresentations.\nInformation Processing Systems 11. MIT Press, 1999.\n\n[10] L. Wasserman. Bayesian model selection and model averaging. Journal of Mathematical Psy-\n\nchology, 44:92\u2013107, 2000.\n\n\f", "award": [], "sourceid": 1974, "authors": [{"given_name": "A.", "family_name": "D'avella", "institution": null}, {"given_name": "M.", "family_name": "Tresch", "institution": null}]}