Tractable Dendritic RNNs for Identifying Unknown Nonlinear Dynamical Systems

Manuel Brenner, Leonard Bereska, Jonas Magdy Mikhaeil, Florian Hess, Zahra Monfared, Po-Chen Kuo, Daniel Durstewitz

Keywords: chaos, dendritic computation, piecewise linear, recurrent neural network, variational inference, interpretability, tractability, basis expansion

Abstract: In many scientific disciplines, we are interested in inferring the nonlinear dynamical system underlying a set of observed time series, a challenging task in the face of chaotic behavior and noise. Previous deep learning approaches toward this goal often suffered from a lack of interpretability and tractability. In particular, the high-dimensional latent spaces often required for a faithful embedding, even when the underlying dynamics lives on a lower-dimensional manifold, can hamper theoretical analysis. Motivated by the emerging principles of dendritic computation, we augment a dynamically interpretable and mathematically tractable piecewise-linear (PL) recurrent neural network (RNN) by a linear spline basis expansion. We show that this approach retains all the theoretically appealing properties of the simple PLRNN, yet boosts its capacity for approximating arbitrary nonlinear dynamical systems in comparatively low dimensions. We introduce two frameworks for training the system, one based on fast and scalable variational inference, and another combining BPTT with teacher forcing. We show that the dendritically expanded PLRNN achieves better reconstructions with fewer parameters and dimensions on various dynamical systems benchmarks and compares favorably to other methods, while retaining a tractable and interpretable structure.

One-sentence Summary: To approximate dynamical systems we augment an interpretable and tractable piecewise-linear recurrent neural network by a dendrite-inspired basis expansion.