Decomposed linear dynamical systems for C. elegans functional connectivity

Recent work has indicated that functional connections between C. elegans neurons can be clustered by their role in behavior, i.e. forward vs. reverse crawling vs. turning (Linderman et al. 2019). Furthermore they show how data-driven dynamical systems, e.g., the recurrent switching linear dynamical systems (rSLDS), can be learned in an unsupervised way from whole-brain recordings (Kato et al. 2015). However, these approaches ascribe a discrete overall state to the system at each time point, potentially missing different dimensions of variability in the data. For example, such models cannot flexibly describe the continuous transitions that occur during behaviors and cannot capture persistent underlying states or gradual changes.
Using decomposed linear dynamical systems (dLDS) models, it is possible to gain this additional insight into C. elegans circuit dynamics, with the benefit of linear dynamical systems’ intepretability based in sparse dynamics representations and the flexibility of continuously varying dynamics coefficients. We thus develop a dLDS model based on dictionary learning to investigate how motifs in dynamics coefficients reflect changes in behavior, e.g. from post-reversal turns to forward crawling. We show that not only does rSLDS output unrealistic periods of high oscillation between discrete behavioral state labels in all tested C. elegans examples, but that dLDS smoothly describes these time snippets in terms of the dynamics coefficients c, which can vary continuously instead of discretely switching on and off. As in Brennan et al. (2019), we see substantial inter-individual differences in the dynamics and the loading matrices between the latent states and the calcium imaging traces. Model validation on toy examples has also shown that dLDS models can flexibly model changes in speed and direction of the same subdynamic using the dynamics coefficients. Therefore, we conclude that dLDS can more robustly describe how circuits encode behavior.