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.Comment: 28 pages, 8 figure
