Fig. 4: Effect of noise on simulated data.

a Schematic of motor modules using three different models (from top to bottom) spatial modularity (\({m}^{s}\left(t\right)={\sum }_{i}{c}_{i}^{s}\left(t\right){w}_{i}\)), temporal modularity (\({m}^{s}\left(t\right)={\sum }_{i}{c}_{i}(t){w}_{i}^{s}\)) and space-by-time modularity (\({m}^{s}\left(t\right)={\sum }_{i}{\sum }_{j}{c}_{i}\left(t\right){a}_{{ij}}^{s}{w}_{j}\)). The outputs of the first (blue), second (magenta), and third (green) modules were summed together to generate overall muscle activation (\({m}^{s}\left(t\right)\), black envelope). The simulated example of muscle activity profiles was computed using constant basic patterns (\({c}_{i}^{s}\left(t\right)\)) for the spatial modularity model, constant synergies (\({w}_{i}^{s}\)) for the temporal modularity model across different strides (s), and the identity matrix for the activation coefficients (\({a}_{{ij}}^{s}\)) for the space-by-time modularity model. b Effect of noise on the percent of variance accounted for (VAF, upper) by the reconstruction of simulated EMGs using three different decomposition methods (from left to right, spatial, temporal, and space-by-time decomposition) and on the similarity measures (middle) and their slope (lower): average similarity of the activation patterns across strides for spatial decomposition, average similarity of synergies across strides for temporal decomposition and average diagonality of the activation coefficients matrix across strides for space-by-time decomposition (from left to right). The simulated EMGs were constructed, by an adaptation of the method proposed by Tresch et al.⁴⁸, using the corresponding equation for each model to calculate noiseless data value \({m}^{s}\left(t\right)\) and adding signal-dependent noise with \({SD}=\eta \cdot {m}^{s}\left(t\right)\), where η is the slope of the relationship between the SD and noiseless data value. We constructed sets of 8 EMGs for 7 strides starting from 3 modules composed of constant synergies and basic patterns as in panel a. Each simulation was performed using increasing values of η for 100 times (averages across simulations and confidence intervals—as shaded areas—are shown in the graphs). Note that VAF was significantly affected by noise, making it difficult to distinguish the exact number of modules (N = 3) used to generate the simulated EMGs for higher levels of noise for each decomposition method, while the similarity measures showed a substantial reduction going from 3 to 4 modules regardless of the level of noise. Red lines represent the results from structureless data obtained by randomly shuffling the simulated EMG samples (see Methods).