  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.
