Noise-induced stabilization of bumps in systems with long-range spatial coupling

The position of a localized region of active neurons (a “bump”) has been proposed to encode information for working memory, the head direction system, and feature selectivity in the visual system. Stationary bumps are ordinarily stable, but including spike frequency adaptation in the neural dynamics causes a stationary bump to become unstable to a moving bump through a supercritical pitchfork bifurcation in bump speed. Adding spatiotemporal noise to the network supporting the bump can cause the average speed of the bump to decrease to almost zero, reversing the effect of the adaptation and “restabilizing” the bump. This restabilizing occurs for noise levels lower than those required to break up the bump. The restabilizing can be understood by examining the effects of noise on the normal form of the pitchfork bifurcation where the variable involved in the bifurcation is bump speed. This noisy normal form can be further simplified to a persistent random walk in which the probability of changing direction is related to the noise level through an Arrhenius-type rate. The probability density function of position for the continuous-time version of this random walk satisfies the telegrapher’s equation, and the closed-form solution of this PDE allows us to find expressions for the mean and variance of the average speed of the particle (the bump) undergoing the random walk. This noise-induced stabilization is a novel example in which moderate amounts of noise have a beneficial effect on a system, specifically, stabilizing a spatiotemporal pattern.