Delay stabilizes stochastic motion of bumps in layered neural fields

TL;DR

This paper investigates how propagation delays in layered neural fields influence the stability and stochastic motion of bumps, showing delays can reduce variability and stabilize bump positions through asymptotic analysis and delay approximations.

Contribution

It introduces an asymptotic approximation for bump dynamics in delayed, noisy neural fields, revealing how delays stabilize and reduce variability in bump motion.

Findings

Delays cause translating perturbations to decay, stabilizing bumps.

Layer coupling and delays reduce stochastic variability of bump positions.

Asymptotic delay expansion accurately predicts bump diffusion, matching simulations.

Abstract

We study the effects of propagation delays on the stochastic dynamics of bumps in neural fields with multiple layers. In the absence of noise, each layer supports a stationary bump. Using linear stability analysis, we show that delayed coupling between layers causes translating perturbations of the bumps to decay in the noise-free system. Adding noise to the system causes bumps to wander as a random walk. However, coupling between layers can reduce the variability of this stochastic motion by canceling noise that perturbs bumps in opposite directions. Delays in interlaminar coupling can further reduce variability, since they couple bump positions to states from the past. We demonstrate these relationships by deriving an asymptotic approximation for the effective motion of bumps. This yields a stochastic delay-differential equation where each delayed term arises from an interlaminar coupling. The impact of delays is well approximated by using a small delay expansion, which allows us to compute the effective diffusion in bumps' positions, accurately matching results from numerical simulations.