Learning Nonlinear Brain Dynamics: van der Pol Meets LSTM

TL;DR

This paper introduces a novel nonlinear dynamical model using coupled van der Pol oscillators for brain activity analysis, demonstrating improved interpretability and comparable or better prediction accuracy than LSTM, especially with hybrid modeling.

Contribution

The paper presents a new coupled van der Pol oscillator model with a hybrid VDP-LSTM approach for brain dynamics, enhancing interpretability and prediction performance on calcium imaging data.

Findings

VDP accurately models neural dynamics with 0.82-0.94 correlation.

Coupling matrix reveals meaningful brain interactions.

Hybrid VDP-LSTM outperforms individual models in prediction.

Abstract

Many real-world data sets, especially in biology, are produced by complex nonlinear dynamical systems. In this paper, we focus on brain calcium imaging (CaI) of different organisms (zebrafish and rat), aiming to build a model of joint activation dynamics in large neuronal populations, including the whole brain of zebrafish. We propose a new approach for capturing dynamics of temporal SVD components that uses the coupled (multivariate) van der Pol (VDP) oscillator, a nonlinear ordinary differential equation (ODE) model describing neural activity, with a new parameter estimation technique that combines variable projection optimization and stochastic search. We show that the approach successfully handles nonlinearities and hidden state variables in the coupled VDP. The approach is accurate, achieving 0.82 to 0.94 correlation between the actual and model-generated components, and interpretable, as VDP's coupling matrix reveals anatomically meaningful positive (excitatory) and negative (inhibitory) interactions across different brain subsystems corresponding to spatial SVD components. Moreover, VDP is comparable to (or sometimes better than) recurrent neural networks (LSTM) for (short-term) prediction of future brain activity; VDP needs less parameters to train, which was a plus on our small training data. Finally, the overall best predictive method, greatly outperforming both VDP and LSTM in short- and long-term predictive settings on both datasets, was the new hybrid VDP-LSTM approach that used VDP to simulate large domain-specific dataset for LSTM pretraining; note that simple LSTM data-augmentation via noisy versions of training data was much less effective.