An important problem in time-series analysis is modeling systems with time-varying dynamics. Probabilistic models with joint continuous and discrete latent states offer interpretable, efficient, and experimentally useful descriptions of such data. Commonly used models include autoregressive hidden Markov models (ARHMMs) and switching linear dynamical systems (SLDSs), each with its own advantages and disadvantages. ARHMMs permit exact inference and easy parameter estimation, but are parameter intensive when modeling long dependencies, and hence are prone to overfitting. In contrast, SLDSs can capture long-range dependencies in a parameter efficient way through Markovian latent dynamics, but present an intractable likelihood and a challenging parameter estimation task. In this paper, we propose _switching autoregressive low-rank tensor_ SALT models, which retain the advantages of both approaches while ameliorating the weaknesses. SALT parameterizes the tensor of an ARHMM with a low-rank factorization to control the number of parameters and allow longer range dependencies without overfitting. We prove theoretical and discuss practical connections between SALT, linear dynamical systems, and SLDSs. We empirically demonstrate quantitative advantages of SALT models on a range of simulated and real prediction tasks, including behavioral and neural datasets. Furthermore, the learned low-rank tensor provides novel insights into temporal dependencies within each discrete state.

Neuropixels (NP) probes are dense linear multi-electrode arrays that have rapidly become essential tools for studying the electrophysiology of large neural populations. Unfortunately, a number of challenges remain in analyzing the large datasets output by these probes. Here we introduce several new methods for extracting useful spiking information from NP probes. First, we use a simple point neuron model, together with a neural-network denoiser, to efficiently map spikes detected on the probe into three-dimensional localizations. Previous methods localized spikes in two dimensions only; we show that the new localization approach is significantly more robust and provides an improved feature set for clustering spikes according to neural identity (``spike sorting"). Next, we apply a Poisson denoising method to the resulting three-dimensional point-cloud representation of the data, and show that the resulting 3D images can be accurately registered over time, leading to improved tracking of time-varying neural activity over the probe, and in turn, crisper estimates of neural clusters over time. The code to reproduce our results and an example neuropixels dataset is provided in the supplementary material.