Nonlinear Langevin equations for wandering patterns in stochastic neural fields

TL;DR

This paper develops a nonlinear Langevin equation framework to analyze how additive noise influences wandering patterns like fronts and bumps in stochastic neural fields, revealing stable and unstable phase-locking behaviors.

Contribution

It introduces a novel nonlinear Langevin equation approach to describe noise-induced wandering in neural field patterns, including cases with complex coupling and stability properties.

Findings

Wandering of stimulus-locked fronts follows an Ornstein-Uhlenbeck process.

Mutually coupled bumps can be modeled by multivariate OU processes.

Large deviations can cause phase-slip events in bump positions.

Abstract

We analyze the effects of additive, spatially extended noise on spatiotemporal patterns in continuum neural fields. Our main focus is how fluctuations impact patterns when they are weakly coupled to an external stimulus or another equivalent pattern. Showing the generality of our approach, we study both propagating fronts and stationary bumps. Using a separation of time scales, we represent the effects of noise in terms of a phase-shift of a pattern from its uniformly translating position at long time scales, and fluctuations in the pattern profile around its instantaneous position at short time scales. In the case of a stimulus-locked front, we show that the phase-shift satisfies a nonlinear Langevin equation (SDE) whose deterministic part has a unique stable fixed point. Using a linear-noise approximation, we thus establish that wandering of the front about the stimulus-locked state is given by an Ornstein-Uhlenbeck (OU) process. Analogous results hold for the relative phase-shift between a pair of mutually coupled fronts, provided that the coupling is excitatory. On the other hand, if the mutual coupling is given by a Mexican hat function (difference of exponentials), then the linear-noise approximation breaks down due to the co-existence of stable and unstable phase-locked states in the deterministic limit. Similarly, the stochastic motion of mutually coupled bumps can be described by a system of nonlinearly coupled SDEs, which can be linearized to yield a multivariate OU process. As in the case of fronts, large deviations can cause bumps to temporarily decouple, leading to a phase-slip in the bump positions.