  Learning interpretable representations of neural dynamics at a population
level is a crucial first step to understanding how observed neural activity
relates to perception and behavior. Models of neural dynamics often focus on
either low-dimensional projections of neural activity, or on learning dynamical
systems that explicitly relate to the neural state over time. We discuss how
these two approaches are interrelated by considering dynamical systems as
representative of flows on a low-dimensional manifold. Building on this
concept, we propose a new decomposed dynamical system model that represents
complex non-stationary and nonlinear dynamics of time series data as a sparse
combination of simpler, more interpretable components. Our model is trained
through a dictionary learning procedure, where we leverage recent results in
tracking sparse vectors over time. The decomposed nature of the dynamics is
more expressive than previous switched approaches for a given number of
parameters and enables modeling of overlapping and non-stationary dynamics. In
both continuous-time and discrete-time instructional examples we demonstrate
that our model can well approximate the original system, learn efficient
representations, and capture smooth transitions between dynamical modes,
focusing on intuitive low-dimensional non-stationary linear and nonlinear
systems. Furthermore, we highlight our model's ability to efficiently capture
and demix population dynamics generated from multiple independent subnetworks,
a task that is computationally impractical for switched models. Finally, we
apply our model to neural "full brain" recordings of C. elegans data,
illustrating a diversity of dynamics that is obscured when classified into
discrete states.
