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. (A-D) 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. (E-F) 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 (A-D) 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.

Low-pass 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 low-pass filtered version of the electrical activation. To emulate this we filtered the EMG with a low-pass filter having the following parameters: 100-order minimum phase FIR filter with 1Hz pass band with a 2Hz stop-band 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%).

High-pass 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 high-pass filtered version of the EMG activity. We therefore filtered the EMG with a high-pass filter having the following parameters: 126-order minimum phase FIR filter with a 2Hz pass band and a 1Hz stop-band 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 low-pass 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 mean-centered 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 M-dimensional 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 state-space plot shows the M1 module activity of the M1-spinal cord system (normalized error 2%).

(E) Summary canonical correlations between models shown in panels A-D 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.