Switching autoregressive low-rank tensor (SALT) models for neural dynamics

Switching linear dynamical system (SLDS) models are widely used for modeling neural and behavioral time series data [Glaser et al., 2020, Linderman et al., 2019, Nair et al., 2022, Zoltowski et al., 2022]. These models characterize high-dimensional time series through a combination of discrete and continuous latent states. The continuous states offer a low-dimensional representation of the data, while the discrete state defines how the continuous states evolve over time. Together, they provide quantitative and interpretable summaries of multidimensional time series. Unfortunately, fitting these models requires a complex and computationally expensive optimization.
We propose an alternative approach called switching autoregressive low-rank tensor (SALT) models. SALT models decompose observed data into a similar representation, defined through discrete states and continuous dynamics, with simpler optimization. Indeed, we can show that SALT models closely approximate SLDS models. The key technical insight is that when you integrate over the continuous states of an SLDS you obtain an autoregressive hidden Markov model (ARHMM) with low-rank dynamics tensors. Rather than fitting the SLDS, we fit the low-rank ARHMM directly.
We demonstrate SALT models via an application to whole-brain recordings of male C. elegans during mating [Susoy et al., 2021]. Mating is the worms’ most complex behavior: it requires several distinct sub-behaviors to be enacted in a coherent sequence, where each sub-behavior involves continuous control of movement. We show that SALT extracts discrete states that reflect expert annotations of behavioral state. We also inspect the learned low-rank tensors, which provide an alternative route for investigating the continuous dynamics within each behavioral state. These analyses highlight the value of SALT models as an alternative to switching linear dynamical systems, offering a similar decomposition while using simpler and computationally cheaper inference procedures.