Ghosts of bump attractors in stochastic neural fields: Bottlenecks and extinction

TL;DR

This paper investigates how additive noise influences the stability and extinction of neural bump solutions near a saddle-node bifurcation, deriving amplitude equations to predict bump dynamics and extinction times.

Contribution

It introduces a stochastic amplitude equation framework to analyze bump extinction times in neural fields near bifurcation points, incorporating noise effects.

Findings

Extinction time increases as system approaches bifurcation

Derived quadratic amplitude equation describes bump dynamics near criticality

Noise accelerates bump extinction, with timescales predictable by the model

Abstract

We study the effects of additive noise on stationary bump solutions to spatially extended neural fields near a saddle-node bifurcation. The integral terms of these evolution equations have a weight kernel describing synaptic interactions between neurons at different locations of the network. Excited regions of the neural field correspond to parts of the domain whose fraction of active neurons exceeds a sharp threshold of a firing rate nonlinearity. For sufficiently low firing threshold, a stable bump coexists with an unstable bump and a homogeneous quiescent state. As the threshold is increased, the stable and unstable branch of bump solutions annihilate in a saddle node bifurcation. Near this criticality, we derive a quadratic amplitude equation that describes the slow evolution of the even mode (bump contractions) as it depends on the distance from the bifurcation. Beyond the bifurcation, bumps eventually become extinct, and the time it takes for this to occur increases for systems nearer the bifurcation. When noise is incorporated, a stochastic amplitude equation for the even mode can be derived, which can be analyzed to reveal bump extinction time both below and above the saddle-node.