Fig. 5: Overall neural activity magnitude is conserved for multi-finger movements.

A Magnitude of the recorded neural activity (Euclidean norm of the high-dimensional activity) during the hold period after removing the condition invariant signal (red dots) for an example set of two finger-group movements (thumb vs. all other fingers). Distance from the origin corresponds to the magnitude of neural activity, with different movement conditions represented along different directions. Predictions from linearly summing the neural activity of single finger-group movements are indicated in green. The magnitude of the recorded neural activity for two finger-group movements was lower than what would be expected from the sum of the corresponding single finger-group parts. B A comparison of recorded activity magnitude (red) and predictions from a linear model (black) for 80 combinatorial movements of four finger-groups. Each point is a distinct movement combination. Movements are grouped by the number of fingers either flexed or extended (x-axis). A linear model using a one-hot encoding of kinematics (8 dimensions, one dimension for each combination of finger and movement direction) predicts that magnitude will increase as more fingers move, but the recorded activity magnitudes appear relatively constant. The linear model under-predicts the neural activity magnitude for a small number of moving finger-groups, as the linear model is fit to all movement conditions, and there are a large number of conditions with multiple finger-groups moving. Statistical significance of the difference in means between the data and model fit for each group of conditions with a particular number of fingers moving is evaluated with a two-sided t-test. Significance level is indicated by the number of stars (‘*’), with n stars indicating p < 10⁻ⁿ. C A linear-nonlinear model (with 10 hidden units) better captures the magnitude of neural activity (compared to (B)). The difference in means was not statistically significant (two-sided t-test). D The linear-nonlinear model schematic which applies a tanh(.) non-linearity to map kinematics to predicted neural activity patterns; this nonlinearity should allow the model to capture activity magnitudes more accurately via saturation. A and B denote linear maps from kinematics to hidden unit inputs and hidden unit activity to the observed neural activity respectively. E Fraction of variance explained across the 80 combinatorial movements by the linear-nonlinear model with an increasing number of hidden layers (red). A low-rank linear model was used for comparison (gray). Linear-nonlinear models outperform the linear model, and performance saturates at a small number of hidden dimensions of ~10. Cross-validated performance was averaged across 100 resamplings of trials used for training and evaluation (shaded regions indicate standard deviation). F Similar analysis as (E), but testing generalization across movements—models were trained on a random subset of 64 movements (out of 80) and tested on the remaining movements. Since the performance varies based on which movements are partitioned into the test set, the performance change was measured compared to the linear-nonlinear model with the highest number of hidden units, and averaged across resamplings. Linear-nonlinear model generalizes better than a linear model to novel movement conditions. G Cumulative distribution of hidden unit activity in the linear-nonlinear model for conditions with different numbers of moving finger-groups across 100 resamplings. Note the greater saturation with a larger number of moving finger-groups. Source data are provided in the Source Data file.