Identifying nonlinear dynamical systems via generative recurrent neural networks with applications to fMRI

A major tenet in theoretical neuroscience is that cognitive and behavioral processes are ultimately implemented in terms of the neural system dynamics. Accordingly, a major aim for the analysis of neurophysiological measurements should lie in the identification of the computational dynamics underlying task processing. Here we advance a state space model (SSM) based on generative piecewise-linear recurrent neural networks (PLRNN) to assess dynamics from neuroimaging data. In contrast to many other nonlinear time series models which have been proposed for reconstructing latent dynamics, our model is easily interpretable in neural terms, amenable to systematic dynamical systems analysis of the resulting set of equations, and can straightforwardly be transformed into an equivalent continuous-time dynamical system. The major contributions of this paper are the introduction of a new observation model suitable for functional magnetic resonance imaging (fMRI) coupled to the latent PLRNN, an efficient stepwise training procedure that forces the latent model to capture the ‘true’ underlying dynamics rather than just fitting (or predicting) the observations, and of an empirical measure based on the Kullback-Leibler divergence to evaluate from empirical time series how well this goal of approximating the underlying dynamics has been achieved. We validate and illustrate the power of our approach on simulated ‘ground-truth’ dynamical systems as well as on experimental fMRI time series, and demonstrate that the learnt dynamics harbors task-related nonlinear structure that a linear dynamical model fails to capture. Given that fMRI is one of the most common techniques for measuring brain activity non-invasively in human subjects, this approach may provide a novel step toward analyzing aberrant (nonlinear) dynamics for clinical assessment or neuroscientific research.Author summary: Computational processes in the brain are often assumed to be implemented in terms of nonlinear neural network dynamics. However, experimentally we usually do not have direct access to this underlying dynamical process that generated the observed time series, but have to infer it from a sample of noisy and mixed measurements like fMRI data. Here we combine a dynamically universal recurrent neural network (RNN) model for approximating the unknown system dynamics with an observation model that links this dynamics to experimental measurements, taking fMRI data as an example. We develop a new stepwise optimization algorithm, within the statistical framework of state space models, that forces the latent RNN model toward the true data-generating dynamical process, and demonstrate its power on benchmark systems like the chaotic Lorenz attractor. We also introduce a novel, fast-to-compute measure for assessing how well this worked out in any empirical situation for which the ground truth dynamical system is not known. RNN models trained on human fMRI data this way can generate new data with the same temporal structure and properties, and exhibit interesting nonlinear dynamical phenomena related to experimental task conditions and behavioral performance. This approach can easily be generalized to many other recording modalities.