Abstract
It remains an open question how neural responses in motor cortex relate to movement. We explored the hypothesis that motor cortex reflects dynamics appropriate for generating temporally patterned outgoing commands. To formalize this hypothesis, we trained recurrent neural networks to reproduce the muscle activity of reaching monkeys. Models had to infer dynamics that could transform simple inputs into temporally and spatially complex patterns of muscle activity. Analysis of trained models revealed that the natural dynamical solution was a lowdimensional oscillator that generated the necessary multiphasic commands. This solution closely resembled, at both the singleneuron and population levels, what was observed in neural recordings from the same monkeys. Notably, data and simulations agreed only when models were optimized to find simple solutions. An appealing interpretation is that the empirically observed dynamics of motor cortex may reflect a simple solution to the problem of generating temporally patterned descending commands.
This is a preview of subscription content
Access options
Subscribe to Journal
Get full journal access for 1 year
$59.00
only $4.92 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Tax calculation will be finalised during checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
from$8.99
All prices are NET prices.
References
 1
Evarts, E.V. Relation of pyramidal tract activity to force exerted during voluntary movement. J. Neurophysiol. 31, 14–27 (1968).
 2
MussaIvaldi, F.A. Do neurons in the motor cortex encode movement direction? An alternative hypothesis. Neurosci. Lett. 91, 106–111 (1988).
 3
Sanger, T.D. Theoretical considerations for the analysis of population coding in motor cortex. Neural Comput. 6, 29–37 (1994).
 4
Todorov, E. Direct cortical control of muscle activation in voluntary arm movements: a model. Nat. Neurosci. 3, 391–398 (2000).
 5
Hatsopoulos, N.G. Encoding in the motor cortex: was evarts right after all? Focus on 'motor cortex neural correlates of output kinematics and kinetics during isometricforce and armreaching tasks'. J. Neurophysiol. 94, 2261–2262 (2005).
 6
Scott, S.H. Inconvenient truths about neural processing in primary motor cortex. J. Physiol. (Lond.) 586, 1217–1224 (2008).
 7
Aflalo, T.N. & Graziano, M.S.A. Relationship between unconstrained arm movements and singleneuron firing in the macaque motor cortex. J. Neurosci. 27, 2760–2780 (2007).
 8
Kalaska, J.F. From intention to action: motor cortex and the control of reaching movements. Adv. Exp. Med. Biol. 629, 139–178 (2009).
 9
Georgopoulos, A.P., Schwartz, A.B. & Kettner, R.E. Neuronal population coding of movement direction. Science 233, 1416–1419 (1986).
 10
Shenoy, K.V., Sahani, M. & Churchland, M.M. Cortical control of arm movements: a dynamical systems perspective. Annu. Rev. Neurosci. 36, 337–359 (2013).
 11
Fetz, E. Are movement parameters recognizably coded in the activity of single neurons? Behav. Brain Sci. 15, 679–690 (1992).
 12
Pearce, T.M. & Moran, D.W. Strategydependent encoding of planned arm movements in the dorsal premotor cortex. Science 337, 984–988 (2012).
 13
Schwartz, A.B. Direct cortical representation of drawing. Science 265, 540–542 (1994).
 14
Fetz, E.E. Cortical mechanisms controlling limb movement. Curr. Opin. Neurobiol. 3, 932–939 (1993).
 15
Rokni, U. & Sompolinsky, H. How the brain generates movement. Neural Comput. 24, 289–331 (2012).
 16
Lillicrap, T.P. & Scott, S.H. Preference distributions of primary motor cortex neurons reflect control solutions optimized for limb biomechanics. Neuron 77, 168–179 (2013).
 17
Morrow, M.M. & Miller, L.E. Prediction of muscle activity by populations of sequentially recorded primary motor cortex neurons. J. Neurophysiol. 89, 2279–2288 (2003).
 18
Schieber, M.H. & Rivlis, G. Partial reconstruction of muscle activity from a pruned network of diverse motor cortex neurons. J. Neurophysiol. 97, 70–82 (2007).
 19
Ajemian, R. et al. Assessing the function of motor cortex: singleneuron models of how neural response is modulated by limb biomechanics. Neuron 58, 414–428 (2008).
 20
Buys, E.J., Lemon, R.N., Mantel, G.W. & Muir, R.B. Selective facilitation of different hand muscles by single corticospinal neurones in the conscious monkey. J. Physiol. (Lond.) 381, 529–549 (1986).
 21
Churchland, M.M. & Cunningham, J.P. A dynamical basis set for generating reaches. Cold Spring Harb. Symp. Quant. Biol. Published online, doi:10.1101/sqb.2014.79.024703 (7 April 2015).
 22
Churchland, M.M., Cunningham, J.P., Kaufman, M.T., Ryu, S.I. & Shenoy, K.V. Cortical preparatory activity: representation of movement or first cog in a dynamical machine? Neuron 68, 387–400 (2010).
 23
Churchland, M.M. et al. Neural population dynamics during reaching. Nature 487, 51–56 (2012).
 24
Sussillo, D. & Abbott, L.F. Generating coherent patterns of activity from chaotic neural networks. Neuron 63, 544–557 (2009).
 25
Tanji, J. & Evarts, E.V. Anticipatory activity of motor cortex neurons in relation to direction of an intended movement. J. Neurophysiol. 39, 1062–1068 (1976).
 26
Weinrich, M., Wise, S.P. & Mauritz, K.H. A neurophysiological study of the premotor cortex in the rhesus monkey. Brain 107, 385–414 (1984).
 27
Pruszynski, J.A., Omrani, M. & Scott, S.H. Goaldependent modulation of fast feedback responses in primary motor cortex. J. Neurosci. 34, 4608–4617 (2014).
 28
Todorov, E. & Jordan, M.I. Optimal feedback control as a theory of motor coordination. Nat. Neurosci. 5, 1226–1235 (2002).
 29
Scott, S.H. Optimal feedback control and the neural basis of volitional motor control. Nat. Rev. Neurosci. 5, 532–546 (2004).
 30
Doya, K. Bifurcations in the learning of recurrent neural networks. Proc. IEEE Int. Symp. Circuits Syst. 6, 2777–2780 (1992).
 31
Sussillo, D. & Barak, O. Opening the black box: lowdimensional dynamics in highdimensional recurrent neural networks. Neural Comput. 25, 626–649 (2013).
 32
Mante, V., Sussillo, D., Shenoy, K.V. & Newsome, W.T. Contextdependent computation by recurrent dynamics in prefrontal cortex. Nature 503, 78–84 (2013).
 33
Fetz, E.E. & Cheney, P.D. Postspike facilitation of forelimb muscle activity by primate corticomotoneuronal cells. J. Neurophysiol. 44, 751–772 (1980).
 34
Rathelot, J.A. & Strick, P.L. Muscle representation in the macaque motor cortex: an anatomical perspective. Proc. Natl. Acad. Sci. USA 103, 8257–8262 (2006).
 35
Churchland, M.M., Yu, B.M., Ryu, S.I., Santhanam, G. & Shenoy, K.V. Neural variability in premotor cortex provides a signature of motor preparation. J. Neurosci. 26, 3697–3712 (2006).
 36
Churchland, M.M., Afshar, A. & Shenoy, K.V. A central source of movement variability. Neuron 52, 1085–1096 (2006).
 37
Kaufman, M.T., Churchland, M.M., Ryu, S.I. & Shenoy, K.V. Cortical activity in the null space: permitting preparation without movement. Nat. Neurosci. 17, 440–448 (2014).
 38
Olshausen, B.A. & Field, D.J. Emergence of simplecell receptive field properties by learning a sparse code for natural images. Nature 381, 607–609 (1996).
 39
Tanaka, H. & Sejnowski, T.J. Computing reaching dynamics in motor cortex with Cartesian spatial coordinates. J. Neurophysiol. 109, 1182–1201 (2013).
 40
Todorov, E. Cosine tuning minimizes motor errors. Neural Comput. 14, 1233–1260 (2002).
 41
Maier, M.A., Shupe, L.E. & Fetz, E.E. Dynamic neural network models of the premotoneuronal circuitry controlling wrist movements in primates. J. Comput. Neurosci. 19, 125–146 (2005).
 42
Jaeger, H. & Haas, H. Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication. Science 304, 78–80 (2004).
 43
Maass, W., Joshi, P. & Sontag, E.D. Computational aspects of feedback in neural circuits. PLOS Comput. Biol. 3, e165 (2007).
 44
Laje, R. & Buonomano, D.V. Robust timing and motor patterns by taming chaos in recurrent neural networks. Nat. Neurosci. 16, 925–933 (2013).
 45
Hennequin, G., Vogels, T.P. & Gerstner, W. Optimal control of transient dynamics in balanced networks supports generation of complex movements. Neuron 82, 1394–1406 (2014).
 46
Churchland, M.M. & Shenoy, K.V. Temporal complexity and heterogeneity of singleneuron activity in premotor and motor cortex. J. Neurophysiol. 97, 4235–4257 (2007).
 47
Kaufman, M.T. et al. Roles of monkey premotor neuron classes in movement preparation and execution. J. Neurophysiol. 104, 799–810 (2010).
 48
Kaufman, M.T., Churchland, M.M. & Shenoy, K.V. The roles of monkey M1 neuron classes in movement preparation and execution. J. Neurophysiol. 110, 817–825 (2013).
 49
Rifai, S. et al. Learning invariant features through local space contraction. Preprint at http://arxiv.org/abs/1104.4153 (2011).
 50
Martens, J. & Sutskever, I. Learning recurrent neural networks with hessianfree optimization. Proc. 28th Int. Conf. Mach. Learn. (2011).
Acknowledgements
We thank L. Abbott and S. Ganguli for insightful conversations. We thank S.I. Ryu for electrode array implantation surgical assistance, M. Mazariegos and J. Aguayo for surgical assistance and veterinary care, B. Otskotsky for IT support, and S. Eisensee, B. Davis and E. Castaneda for administrative assistance. This work was supported by The Searle Scholars Program (M.M.C.), The Sloan Foundation (M.M.C.), The McKnight Foundation (M.M.C.), The Grossman Charitable Trust (M.M.C.), US National Institutes of Health (NIH) Director's New Innovator Award DP2 NS083037 (M.M.C.), an NIH R01 MH9333802 grant (M.M.C.), Burroughs Wellcome Fund Career Awards in the Biomedical Sciences (M.M.C., K.V.S.), a National Science Foundation graduate research fellowship (M.T.K.), an NIH TRO1 Award R01NS076460 (K.V.S.), an NIH Director's Pioneer Award 8DP1HD075623 (K.V.S.) and a DARPA REPAIR Award N6600110C2010 (K.V.S.).
Author information
Affiliations
Contributions
D.C.S. and M.M.C. conceived the study and designed the simulations. D.C.S. performed the simulations and analyses. M.M.C. and M.T.K. designed and performed the experiments. D.C.S. and M.M.C. wrote the manuscript. K.V.S. supervised the study, analyses and manuscript writing.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Integrated supplementary information
Supplementary Figure 1 Feedback and spinal cord model variants.
In the main manuscript, we focused on a single recurrent model that directly output EMG. However, the interaction of the motor cortex with other systems is central to the production of movement. To determine whether results depend on the exact form of our simplified model, we tested additional models of two varieties. First, we tested models that incorporate (simulated) sensory feedback. Second, we tested models that incorporate transformations (e.g., spinal circuitry) between motor cortex and muscle activity. (AD) State space visualizations for monkey J for 4 model variants (same conventions as Figure 6). Each model has a similar architecture to the regularized dynamical model, but with the additions detailed below. (EF) Summary canonical correlations of model variants and neural data for monkey J. Model details are given below, but the key observation is that the regularized model population response continued to resemble the neural data regardless of the addition of simulated feedback or downstream transformations. This is reflected both in the oscillations about a single fixed point (AD) and the high canonical correlations. Why does this similarity persist? In the case of feedback, that feedback simply becomes part of the overall dynamics. Once trained in the presence of feedback the network depends on that feedback: its removal would have a profound effect. Yet the overall dynamics remain similar for a straightforward reason: the cost function is still encouraging dynamics that produce the empirical EMG in the simplest way possible. That solution remains similar regardless of whether the dynamics depend on feedback or not. In the case of downstream transformations, the network continues to find a similar solution so long as the ‘basic building blocks’ of EMG remain similar. Consider the case where the model is asked to output muscle synergies (from which EMG can be composed) rather than EMG directly. This constraint changes the model very little: it was already attempting to produce EMG as simply as possible and was thus already encouraged to exploit the redundancy across muscles.
Lowpass Feedback Model Variant We wished to incorporate reasonable sources of feedback into the model. In the absence of afferent recordings, we assumed that feedback is a temporally transformed (filtered) version of the output command. There are many possible such transformations, but we concentrated on two. First, we assumed that sensory feedback is proportional to muscle force / stretch, which is a lowpass filtered version of the electrical activation. To emulate this we filtered the EMG with a lowpass filter having the following parameters: 100order minimum phase FIR filter with 1Hz pass band with a 2Hz stopband of 10db attenuation. The filter “group delay” of the resulting filtered EMG was 140ms at 0.5Hz, testing larger delays of peripheral feedback to cortex. The filtered EMG was used as additional input to the model. The remaining parameters were identical to those of the respective regularized dynamical model and EMG fits were just as good (normalized error 7%).
Highpass Feedback Model Variant Another reasonable hypothesis is that sensory feedback is selectively sensitive to changes in muscle force, in which case it should be modeled as a highpass filtered version of the EMG activity. We therefore filtered the EMG with a highpass filter having the following parameters: 126order minimum phase FIR filter with a 2Hz pass band and a 1Hz stopband with 20db attenuation. The filter “group delay” of the resulting filtered EMG was 25ms at 3Hz, testing smaller delays of peripheral feedback to cortex. Simulations were otherwise as described above for the lowpass model. Fits to the EMG remained excellent (normalized error 7%).
Muscle Synergies Model Variant  The muscle synergies model variants were trained to output the top P PCA projections of the EMG for each monkey. Specifically, for both monkeys the following steps were followed: PCA was performed on the M × CT matrix of the meancentered EMG. The projections of the EMG onto the resulting top P PCs were used as training data for the regularized models. All inputs were the same between the muscle synergy models and the regularized dynamical models. All other training parameters were the same between the muscles synergy variant and the respective regularized dynamical models. The value for P was set to 4 for monkey J and 5 for monkey N. Shown in C is a model that is optimized to generate muscle synergies instead of EMG (normalized error 5%).
Spinal Cord Model Variant  The spinal cord models used two RNNs. The first RNN simulated the motor cortex and was trained to generate the top 3 muscle synergies. These muscle synergies were computed in the same way as for the muscle synergy model variants, except that the number of synergies was 3 for both monkeys. A second network then used these outputted muscle synergies of the simulated motor cortex models to dynamically generate the full Mdimensional EMG. Aside from producing the 3 muscle synergies as output, the parameters for the simulated motor cortex RNN were identical to those of the regularized dynamical model in the main paper. The parameters for the simulated spinal cord model for both monkeys were as follows: # inputs = 3, N = 100, α = 2 × 10^{–6}, β = 0.015, and γ= 2 × 10^{–6}. The simulated motor cortex RNN is analyzed in Supp. Figure 1D. Shown in D is a model that is optimized with an additional spinal cord module, which receives muscle synergies and outputs EMG. The statespace plot shows the M1 module activity of the M1spinal cord system (normalized error 2%).
(E) Summary canonical correlations between models shown in panels AD and neural data. All model variants have as high or higher mean canonical correlation with the data as regularized dynamical model.
(F) The mean CC with the neural data for the respective models is 0.75, 0.76, 0.78, and 0.74.
Supplementary Figure 2 Example PSTHs from monkey N, and the regularized and complicated models for monkey N.
(A) Example PSTHs from five neurons from monkey N (format as in Figure 1B). These examples were chosen to illustrate the range of responses, including neurons with strong preparatory activity (rows one and four), neurons with a broad rise in activation during the movement period (middle row), neurons with oscillatory activity during the movement period (rows two and four), a neuron with a narrow bump of activation (bottom row). Vertical and horizontal scale bars indicate 20 spikes per second and 200ms.
(B) Example PSTHs chosen from the regularized model for monkey N. Examples were chosen to highlight the similarities between neural and model responses, and are representative of the patterns exhibited by the model units.
(C) Example PSTHs from five units from the complicated model for monkey N. The PSTHs of the complicated models did not bear any obvious resemblance to those of the neural data. Vertical and horizontal scale bars indicate 20 spikes per second and 200ms.
Supplementary Figure 3 jPCA projections of the population responses for monkey N and the regularized model for monkey N.
(A) jPCA projections of the neural population response for monkey N. Each trace shows the evolution of the neural state over 500ms. Traces start 180ms before movement onset; at the moment the relatively stable preparatory state (circles) transitioned to the movement period trajectory (same conventions as Figure 3). The three projections correspond to the largest magnitude complex eigenvalue pairs of the matrix M_{skew}, found when fitting the data with ẋ = M_{skew}x (Experimental Procedures). These eigenvalues correspond to frequencies of 2.6Hz, 1.6Hz, and 1.2Hz (left to right) with a quality of fit (R^{2}) for the optimal purely oscillatory linear system of 0.77.
(B) jPCA projections for the regularized model of monkey N. Same conventions as in panel A. Frequencies are 2.2Hz, 1.3Hz, and 0.8 Hz. The linear system, ẋ = M_{skew}x P has a quality of fit (R^{2}) of 0.79.
Supplementary Figure 4 Canonical correlations analysis for monkey N.
(A, B) CCA projections (canonical variables) of the neural population response (A) and the regularized model for monkey N (B). In each row is shown the canonical variables 1, 2, 5, 9, and 10 out of 10 (conventions as in Figure 4). The vertical scale bar indicates 1 arbitrary unit while the horizontal scale bar indicates 200ms.
(C, D) Canonical variables of the neural population response (C) and the complicated model for monkey N (D) (format as in A and B).
Supplementary Figure 5 Comparison of simulated and neural population responses for monkey N.
(A) Summary Canonical Correlations (same conventions as in Figure 5.) (B) The average of the canonical correlations between the models and the data.
Supplementary Figure 6 Monkey N and regularized model statespace visualizations.
(A) 3D visualization of the neural data during the movement period for monkey N. The projection is comprised of the first jPC plane (Supp. Figure 3A, left panel) and an additional dimension that captures variance from the crosscondition mean. Each trace is colorcoded to show one of the 27 reach conditions. For all conditions the trajectory during the preparatory period is colored blue. Time shown is 400ms before to 220ms after movement onset.
(B) Analogous 3D visualization of the regularized model for monkey N (same conventions as in panel A). Additionally, the single, conditionindependent fixed point of the model, which organizes the dynamics of movement generation, is shown with an orange ‘x’. Time shown is 800ms before to 220ms after movement onset.
Supplementary Figure 7 Frequency analysis of neural data and regularized model for monkey N.
(A) Eigenvalue analysis of the neural data. Shown on the line of stability (‘Inf’ – neither decaying nor growing) are the purely imaginary eigenvalues associated with the jPCA analysis of the neural data in Figure 3A (blue squares). Also shown are the top eigenvalues of an unconstrained linear fit to the neural data (blue triangles).
(B) The complex eigenvalue spectrum of the linearized system around the fixed point in the regularized model for monkey N (red ‘x’) based on a structural analysis of the weight matrix. Highlighted with red numbers / letters are those modes of the linearized system that have a slow decay. Shown along the line of stability (‘Inf’) are the purely imaginary eigenvalues associated with the jPCA analysis of the regularized model data (green squares). Grey lines show the connection between the jPCA analysis and the structural analysis, as given by a subspace angle analysis of eigenvectors in panel C.
(C) On the vertical axis are listed the 3 oscillatory planes found by jPCA (corresponding to the green numbered modes in panel B). Color indicates the minimum subspace angle (the minimum angle between the corresponding planes). For comparison, the minimum subspace angle between two randomly chosen planes in a N = 300D space is 84 ± 2 degrees (mean and std dev., black arrow labeled ‘chance’). Thus, a minimum subspace angle of 30 to 40 degrees indicates highly overlapping subspaces. In the present case jPC plane 1 overlaps heavily with mode 2 (the fastest frequency), jPC plane 2 overlaps most strongly with oscillatory mode 3 and jPC plane 3 overlaps most strongly with oscillatory mode 1.
Supplementary Figure 8 Regularization impacts similarity to data and model robustness (monkey N).
(A) Average canonical correlation, as training progresses, between the regularized model and the neural data from monkey N. Black shows the mean canonical correlation of 0.39 between the neural data and the data from the untrained model with correct inputs injected. This provides a baseline for comparing mean canonical correlation. As training goes forward with heavy regularization (blue) the model becomes more and more similar to the neural data, ending with a mean canonical correlation of 0.65 for this model (blue arrow). When trained to generate EMG without any regularization, the model has a mean canonical correlation with the data of 0.48 (red arrow).
(B) The normalized error of the network output as a function of training for the untrained regularized and complicated models (same conventions as in panel A). Note that the error decreases very quickly for the regularized model, while in panel A, the mean canonical correlation continues to increase over a much longer period of training, demonstrating that simplicity also makes the model more similar to the data, and not simply fitting the EMG.
(C) Perturbation test of the initial conditions for the regularized and complicated models analyzed in panel A (blue and red arrows, respectively). The inputs were randomly perturbed according to a normalized percentage of the input strength (as given on horizontal axis) 50 times. The network was then run and the mean normalized EMG error of the outputs was averaged (vertical axis). Error bars show standard deviation. The vertical axis is truncated at 100% error.
(D) A structural perturbation test of the recurrent connectivity matrix in equation (1) for the regularized and complicated models analyzed in panel A (blue and red arrows, respectively). The connectivity matrix was randomly perturbed 50 times according to a normalized percentage of the mean absolute connection strength (as given on horizontal axis). The perturbed network was then run and the mean normalized EMG error of the outputs was averaged (vertical axis). Error bars show standard deviation. The vertical axis is truncated at 100% error.
Supplementary Figure 9 jPCA projections for the complicated models of monkey J and monkey N.
(A) jPCA projections for the complicated model of monkey J. Same conventions as in Figure 3. Frequencies are 1.3Hz, 0.8Hz, and 0.6Hz. The linear system, ẋ = M_{skew}x has a quality of fit (R^{2}) of 0.35.
(B) jPCA projections for the complicated model of monkey N. Frequencies are 1.2Hz, 0.9Hz, and 0.8Hz. The linear system, ẋ = M_{skew}x has a quality of fit (R^{2}) of 0.37.
Supplementary Figure 10 The relative strength of preparatory activity to movement activity correlates with model unit output strength.
There is a well known anterior to posterior gradient in PMd / M1: preparatory activity is strongest in more anterior regions, while more posterior regions are more strongly dominated by movementperiod activity. This gradient runs opposite another wellknown gradient: more posterior regions have stronger and more direct connections to the spinal cord. While the model was not designed with such an anatomical gradient, it is nevertheless reasonable to inquire whether the units that are more strongly ‘preparatory tuned’ tend to project less strongly onto the output while the units that are more ‘movement tuned’ tend to project more strongly onto the output. To do so we measured the ratio of preparatory activity to movement activity in each unit. Specifically, for all units and all times we calculated the total variance of preparatory activity (600 to 400 before movement onset) across conditions and the total variance of perimovement period activity (200 to 400 ms after movement onset) across conditions. We then took the time point for each unit when these numbers were maximal, subtracted the movement variance from the preparatory variance, and normalized over the total variance. This gave a measure of the units’ activity that varied between preparatoryactivity dominated (+1) or movementactivity dominated (1). We then took this measure and correlated it with the absolute strength of each unit’s projection onto the output units (i.e. how strongly that unit influence the muscle activity directly). The correlation for the regularized model for monkey J was 0.25 (pvalue of 0.007) and for the regularized model for monkey N was 0.27 (pvalue of 0.0002). A scatter plot of these two measures is shown (blue dots are individual units). For illustration, PSTHs are shown for two example units near the extremes (red dots in scatter, conventions as in Figure 2). The black line is the linear fit.
We also performed an additional analysis examining possible differences between those units with relatively stronger versus weaker output connections. We found that the population of neurons with stronger than average connections exhibited slightly higherfrequency oscillations (by 14% and 7% for the two monkeys) relative to the population with weaker than average connections. While it is hard to draw any firm conclusions, this result is in agreement with our recent report that the M1 population has slightly higherfrequency oscillations than the PMd population.
Supplementary Figure 11 Further comparisons between models and data, that include EMG.
(A, B) Average canonical correlations between the neural data and various datasets, as in Figure 5B (same conventions, EMG in orange). In this analysis the number of canonical correlations was first reduced to a number of principal components equal to the number of muscles in our recorded population (7 for monkey J and 8 for monkey N). In this way, we could also include the EMG versus neural data canonical correlations as another benchmark against which we could measure the similarity of the regularized model to the neural data. Note that this analysis is thus based on fewer principle components than the analysis in Figure 5 and Supplementary Figure 5. In both monkeys, (A) monkey J, (B) monkey N, the average canonical correlation between the regularized model and the neural data was higher than all other models studied, including also the EMG, thus showing that the similarity between the neural data and the regularized models does not arise simply by generating EMG.
(C, D) Comparison of the eigenvalue spectra between the neural data, the regularized dynamical model, and the EMG. PCA was run on the EMG and the eigenvalue spectrum plotted (orange). For the neural data and model data. For both the neural data and the regularized dynamical model, the PC eigenvalue spectra were generated on a number randomly selected neurons / units equal to the number of EMG each model output. These eigenvalue spectra were then averaged over 100 random neuron/unit selections. Finally, the eigenvalue spectra were normalized so that the eigenvalue for the first PC had a value of 1.0. In this way, the dimensionality of the datasets can be seen in the falloff of the spectra. This analysis shows that even though the regularized dynamical model was trained to generate EMG, it contains more dynamical content than just the EMG. (C) monkey J, (D) monkey N.
Supplementary information
Supplementary Text and Figures
Supplementary Figures 1–11 (PDF 9926 kb)
Rights and permissions
About this article
Cite this article
Sussillo, D., Churchland, M., Kaufman, M. et al. A neural network that finds a naturalistic solution for the production of muscle activity. Nat Neurosci 18, 1025–1033 (2015). https://doi.org/10.1038/nn.4042
Received:
Accepted:
Published:
Issue Date:
Further reading

Probing the structure–function relationship with neural networks constructed by solving a system of linear equations
Scientific Reports (2021)

Measurement, manipulation and modeling of brainwide neural population dynamics
Nature Communications (2021)

Multiscale lowdimensional motor cortical state dynamics predict naturalistic reachandgrasp behavior
Nature Communications (2021)

Independent generation of sequence elements by motor cortex
Nature Neuroscience (2021)

Longterm stability of cortical population dynamics underlying consistent behavior
Nature Neuroscience (2020)