Learning Oscillatory Brain Dynamics: van der Pol Meets LSTM

German Abrevaya, Alexandr Aravkin, Guillermo Cecchi, Pablo Polosecki, Irina Rish

Abstract: Motivated by the challenging problem of modeling nonlinear dynamics of brain activations in calcium imaging, we propose a novel approach for learning a nonlinear differential equation model: a variable-projection optimization approach for estimating the parameters of the multivariate (coupled) van der Pol oscillator. To the best of our knowledge, we are the first to propose a successful approach to learning such nonlinear dynamical model, and to demonstrate that it can accurately capture nonlinear dynamics of the brain data. Furthermore, in order to further improve the predictive accuracy when forecasting future brain activity, we use the learned analytical van der Pol model to generate large amounts of simulated data for LSTM pretraining, since brain imaging datasets are often limited in size, and since the generic LSTM model may benefit from the oscillator prior imposed via such pretraining. Indeed, the proposed combination of the analytical approach with the general-purpose statistical LSTM model improves the performance of both methods.

TL;DR: We propose an optimization approach for learning a coupled oscillatory van der Pol model, which captures nonlinear dynamics in brain imaging data; when combined with generic LSTM model, the hybrid aproach improves both LSTM and van der Pol.

Keywords: brain imaging, van der Pol oscillator, learning nonlinear dynamics, variable-projection method, LSTM