Spike Dynamics in the Presence of Noise

Abstract

We study the effect of noise on dynamics of a single spike for the classical Gierer--Meinhardt model on a finite interval. When spatio-temporal noise is introduced in the equation for the activator, we derive a stochastic ODE that describes the motion of a single spike on a slow time scale. The steady state is described by a density distribution for spike positions, obtained via the corresponding Fokker--Planck PDE. For sufficiently small noise level, the spike performs random fluctuations near the center of the domain. As noise level is increased, the spike can deviate from the domain center but remains effectively “trapped” within a certain subinterval that includes the center. For even larger noise levels, the spike starts to undergo large excursions that eventually collide with the domain boundary and temporarily trap the spike there. By reformulating this problem in terms of mean first passage time, we derive the expected time for the spike to collide with the boundary. Several new open problems are also presented.