Noise-Induced Precursors of State Transitions in the Stochastic Wilson–Cowan Model

Abstract

The Wilson–Cowan neural field equations describe the dynamical behavior of a 1-D continuum of excitatory and inhibitory cortical neural aggregates, using a pair of coupled integro-differential equations. Here we use bifurcation theory and small-noise linear stochastics to study the range of a phase transitions—sudden qualitative changes in the state of a dynamical system emerging from a bifurcation—accessible to the Wilson–Cowan network. Specifically, we examine saddle-node, Hopf, Turing, and Turing–Hopf instabilities. We introduce stochasticity by adding small-amplitude spatio-temporal white noise, and analyze the resulting subthreshold fluctuations using an Ornstein–Uhlenbeck linearization. This analysis predicts divergent changes in correlation and spectral characteristics of neural activity during close approach to bifurcation from below. We validate these theoretical predictions using numerical simulations. The results demonstrate the role of noise in the emergence of critically slowed precursors in both space and time, and suggest that these early-warning signals are a universal feature of a neural system close to bifurcation. In particular, these precursor signals are likely to have neurobiological significance as early warnings of impending state change in the cortex. We support this claim with an analysis of the in vitro local field potentials recorded from slices of mouse-brain tissue. We show that in the period leading up to emergence of spontaneous seizure-like events, the mouse field potentials show a characteristic spectral focusing toward lower frequencies concomitant with a growth in fluctuation variance, consistent with critical slowing near a bifurcation point. This observation of biological criticality has clear implications regarding the feasibility of seizure prediction.

Critical slowing as a biomarker for seizure susceptibility

“…Critical slowing is a marker of decreased resilience as a system approaches a tipping point. Prior to this study, critical slowing had been theorized to occur in the brain as seizure susceptibility increased [3,14,16,18]. However, there has been little evidence of critical slowing prior to seizures demonstrated in humans.…”

“…Theoretical predictions from dynamical systems models suggest that critical transitions are preceded by increases in signal variance and autocorrelation prior to state changes. There are numerous models that can describe state changes and there are many possible paths leading to a state change [16]. It may be possible to detect the occurrence of critical slowing prior to a seizure [3], and we describe here a model to demonstrate how critical slowing may occur with the view that this concept may hold for a wider class of models.…”

“…It has been hypothesized that a rapid transition from normal brain activity to an epileptic seizure is such a critical transition [3,4,[14][15][16][17]. However, there is insufficient empirical evidence to test this hypothesis in humans, because of the lack of long-term recordings as well as intra-patient and inter-patient variability.…”

“…Numerous computational models of epilepsy suggest that critical slowing occurs prior to seizures [14,16,18]. Mathematical analyses of dynamic systems, combined with simulations, enable classification of bifurcations and critical transitions.…”

Seizure Prediction: Science Fiction or Soon to Become Reality?

“…Seizure likelihood can be evaluated by measuring how close the brain is to a critical transition point. There are distinct patterns of cortical activity that indicate the brain is approaching a critical point [71]; however, detection of these patterns requires accurate estimation of the brain's current state from EEG. It is also important that the approach to critical transition can be detected with enough time to enable corrective action (e.g.…”