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Many natural systems, especially biological ones, exhibit complex multivariate nonlinear dynamical behaviors that can be hard to capture by linear autoregressive models. On the other hand, generic nonlinear models such as deep recurrent neural networks often require large amounts of training data, not always available in domains such as brain imaging; also, they often lack interpretability. Domain knowledge about the types of dynamics typically observed in such systems, such as a certain type of dynamical systems models, could complement purely data-driven techniques by providing a good prior. In this work, we consider a class of ordinary differential equation (ODE) models known as van der Pol (VDP) oscil lators and evaluate their ability to capture a low-dimensional representation of neural activity measured by different brain imaging modalities, such as calcium imaging (CaI) and fMRI, in different living organisms: larval zebrafish, rat, and human. We develop a novel and efficient approach to the nontrivial problem of parameters estimation for a network of coupled dynamical systems from multivariate data and demonstrate that the resulting VDP models are both accurate and interpretable, as VDP's coupling matrix reveals anatomically meaningful excitatory and inhibitory interactions across different brain subsystems. VDP outperforms linear autoregressive models (VAR) in terms of both the data fit accuracy and the quality of insight provided by the coupling matrices and often tends to generalize better to unseen data when predicting future brain activity, being comparable to and sometimes better than the recurrent neural networks (LSTMs). Finally, we demonstrate that our (generative) VDP model can also serve as a data-augmentation tool leading to marked improvements in predictive accuracy of recurrent neural networks. Thus, our work contributes to both basic and applied dimensions of neuroimaging: gaining scientific insights and improving brain-based predictive models, an area of potentially high practical importance in clinical diagnosis and neurotechnology.

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.