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Learning interpretable representations of neural dynamics at a populationlevel is a crucial first step to understanding how observed neural activityrelates to perception and behavior. Models of neural dynamics often focus oneither low-dimensional projections of neural activity, or on learning dynamicalsystems that explicitly relate to the neural state over time. We discuss howthese two approaches are interrelated by considering dynamical systems asrepresentative of flows on a low-dimensional manifold. Building on thisconcept, we propose a new decomposed dynamical system model that representscomplex non-stationary and nonlinear dynamics of time series data as a sparsecombination of simpler, more interpretable components. Our model is trainedthrough a dictionary learning procedure, where we leverage recent results intracking sparse vectors over time. The decomposed nature of the dynamics ismore expressive than previous switched approaches for a given number ofparameters and enables modeling of overlapping and non-stationary dynamics. Inboth continuous-time and discrete-time instructional examples we demonstratethat our model can well approximate the original system, learn efficientrepresentations, and capture smooth transitions between dynamical modes,focusing on intuitive low-dimensional non-stationary linear and nonlinearsystems. Furthermore, we highlight our model's ability to efficiently captureand demix population dynamics generated from multiple independent subnetworks,a task that is computationally impractical for switched models. Finally, weapply our model to neural "full brain" recordings of C. elegans data,illustrating a diversity of dynamics that is obscured when classified intodiscrete states.

Decomposed Linear Dynamical Systems (dLDS) for learning the latent components of neural dynamics