Title: Learning Nonlinear Brain Dynamics: van der Pol Meets LSTM,
Abstract: Many real-world data sets, especially in biology, are produced by highly
multivariate and nonlinear complex dynamical systems. In this paper, we focus
on brain imaging data, including both calcium imaging and functional MRI data.
Standard vector-autoregressive models are limited by their linearity
assumptions, while nonlinear general-purpose, large-scale temporal models, such
as LSTM networks, typically require large amounts of training data, not always
readily available in biological applications; furthermore, such models have
limited interpretability. We introduce here a novel approach for learning a
nonlinear differential equation model aimed at capturing brain dynamics.
Specifically, we propose a variable-projection optimization approach to
estimate the parameters of the multivariate (coupled) van der Pol oscillator,
and demonstrate that such a model can accurately represent nonlinear dynamics
of the brain data. Furthermore, in order to improve the predictive accuracy
when forecasting future brain-activity time series, we use this analytical
model as an unlimited source of simulated data for pretraining LSTM; such
model-specific data augmentation approach consistently improves LSTM
performance on both calcium and fMRI imaging data.