  We analyze the effects of spatiotemporal noise on stationary pulse solutions
(bumps) in neural field equations on planar domains. Neural fields are
integrodifferential equations whose integral kernel describes the strength and
polarity of synaptic interactions between neurons at different spatial
locations of the network. Fluctuations in neural activity are incorporated by
modeling the system as a Langevin equation evolving on a planar domain. Noise
causes bumps to wander about the domain in a purely diffusive way. Utilizing a
small noise expansion along with a solvability condition, we can derive an
effective stochastic equation describing the bump dynamics as two-dimensional
Brownian motion. The diffusion coefficient can then be computed explicitly. We
also show that weak external inputs can pin the bump so it no longer wanders
diffusively. Inputs reshape the effective potential that guides the dynamics of
the bump position, so it tends to lie near attractors which can be single
points or contours in the plane. Perturbative analysis shows the bump position
evolves as a multivariate Ornstein-Uhlenbeck process whose relaxation constants
are determined by the shape of the input. Our analytical approximations all
compare well to statistics of bump motion in numerical simulations.
