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

Abstract

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.

Existence of n-cycles and border-collision bifurcations in piecewise-linear continuous maps with applications to recurrent neural networks

“…Attractors and border-collision bifurcations of the skew tent map, which as will be shown below is tightly related to the system studied here under certain conditions, have been studied in depth in [1,25]. Very recently, Patra [22] investigated the coexistence of a period-2 orbit, a period-3 orbit, and an unstable chaotic orbit for some parameter values of a 3D PWL normal form map (see also [14] for similar numerical observations in PLRNNs inferred from data).…”

“…Piecewise linear recurrent neural networks (PLRNNs), which build on so-called 'rectified linear units (ReLU)', φ(z) = max(z, 0), as the network's nonlinear activation function, are one example of such maps. In general, RNNs are the standard these days in machine learning for processing sequential, time-series information, due to their success in domains rich in temporal structure like natural language processing [15,27], prediction of con-sumer behavior [16], movement trajectories [19], or identification of dynamical systems from experimental data [14]. ReLU-based RNNs are particularly popular as they allow for highly efficient inference and training algorithms that exploit their piecewise linear structure [6,14,17,21].…”

“…In general, RNNs are the standard these days in machine learning for processing sequential, time-series information, due to their success in domains rich in temporal structure like natural language processing [15,27], prediction of con-sumer behavior [16], movement trajectories [19], or identification of dynamical systems from experimental data [14]. ReLU-based RNNs are particularly popular as they allow for highly efficient inference and training algorithms that exploit their piecewise linear structure [6,14,17,21]. To understand the representational and computational capabilities of these systems in various application areas indicated above, it is important to study them more systematically from a mathematical, dynamical systems perspective.…”

“…To understand the representational and computational capabilities of these systems in various application areas indicated above, it is important to study them more systematically from a mathematical, dynamical systems perspective. In particular, periodic motion forms an integral part of many natural data domains (like speech or movement signals), and periodic orbits of various orders have been extracted from brain signals recently using PLRNNs [14].…”

Analytic Model of fMRI BOLD Signals for Separable Metrics of Neural and Metabolic Activity

“…Furthermore, in relation to the challenges of characterizing fMRI BOLD signal as neural and metabolic components, a challenge is that under nominal external stimuli driven responses, the brain function can transition from a linear to a nonlinear response, in which the measured signal is not a linear superposition of impulse functions, but rather is attenuated in amplitude and transient dynamics. Consequently, modeling techniques of BOLD signals have been extended to include such nonlinear features [19,32,7], and deployed for example to analyze fMRI data of the visual cortex [20] and for dynamic modeling of applications in connectivity [25].…”