Extended Data Fig. 6. Nonlinear DPAD extracted distinct low dimensional latent states from neural activity for all datasets, which were more behaviorally relevant than those extracted using nonlinear NDM.

a, The 3D reach task. b, The latent state trajectory for 2D states extracted from spiking activity using nonlinear DPAD, averaged across all reach and return epochs across sessions and folds. Here only optimization steps 1-2 of DPAD are used to just extract 2D behaviorally relevant states. c, Same as b for 2D states extracted using nonlinear NDM (special case of using just DPAD optimization steps 3-4). d, Saccadic eye movement task. Trials are averaged depending on the eye movement direction. e, The latent state trajectory for 2D states extracted using DPAD (extracted using optimizations steps 1-2), averaged across all trials of the same movement direction condition across sessions and folds. f, Same as d for 2D states extracted using nonlinear NDM. g-i, Same as d-f for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum. j-l, Same as d-f for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position. Overall, in each dataset, latent states extracted by DPAD were clearly different for different behavior conditions in that dataset (b,e,h,k), whereas NDM’s extracted latent states did not as clearly dissociate different conditions (c,f,i,l). Of note, in the first dataset, DPAD revealed latent states with rotational dynamics that reversed direction during reach versus return epochs, which is consistent with the behavior roughly reversing direction. In contrast, NDM’s latent states showed rotational dynamics that did not reverse direction, thus were less congruent with behavior. In this first dataset, in our earlier work⁶, we had compared PSID and a subspace-based linear NDM method and, similar to b and c here, had found that only PSID uncovers reverse-directional rotational patterns across reach and return movement conditions. These results thus also complement our prior work⁶ by showing that even nonlinear NDM models may not uncover the distinct reverse-directional dynamics in this dataset, thus highlighting the need for dissociative and prioritized learning even in nonlinear modeling, as enabled by DPAD.