Extended Data Fig. 5. Recurrent linear dynamical systems uncover conserved latent dynamics that switch between two states if sufficient populations are recorded to recreate regenerative dynamics.

(a) The latent structure of 2 separate recordings may have similar qualitative characteristics but differ in the permutation and scaling of the latent axes (top). We linearly re-align the two recordings (see Methods) such that the trajectories overlap one another (bottom). Left shows the phase aligned average of the two latent components for two recordings before (top) and after (bottom) linear alignment. The phase-colored trajectories through the unaligned (top) and aligned (bottom) latent spaces. Dots are 10 ms time bins. (b) Increasing the number of states (that is, partitions of the latent space with different dynamics) for the rSLDS does not change the qualitative structure of the latent trajectories and does not increase the number of recordings that are regenerative (37 for K = 2, and for K = 3). (c) Allowing no partitions (K = 1) of the latent space prevents any recording from being regenerative (example shown top), despite oscillatory structure in the observed data. Rather, single state LDSs form decaying spirals. Panel organized as in Fig. 4f-i. (d) Percent of units that can be removed while maintaining regenerative simulated diaphragm activity. Sequential removal of units in order of respiratory coherence or firing rate: random, lowest coherence first, highest coherence first, highest firing rate first (One-way RM ANOVA F_{(2.337,84.13)}=318, p < 0.0001; Tukey’s multiple comparisons post-hoc test lines are p < 0.0001, ***p < 0.001; n = 37 recordings). Box is median +/- one quartile, whiskers are 1.5 * IQR. e) The trajectories of all regenerative recordings, and (f) non-regenerative recordings, aligned to a reference recording. Markers are colored by phase as in (a).