  We study the effects of noise on stationary pulse solutions (bumps) in
spatially extended neural fields. The dynamics of a neural field is described
by an integrodifferential equation whose integral term characterizes synaptic
interactions between neurons in different spatial locations of the network.
Translationally symmetric neural fields support a continuum of stationary bump
solutions, which may be centered at any spatial location. Random fluctuations
are introduced by modeling the system as a spatially extended Langevin equation
whose noise term we take to be multiplicative or additive. For nonzero noise,
these bumps are shown to wander about the domain in a purely diffusive way. We
can approximate the effective diffusion coefficient using a small noise
expansion. Upon breaking the (continuous) translation symmetry of the system
using a spatially heterogeneous inputs or synapses, bumps in the stochastic
neural field can become temporarily pinned to a finite number of locations in
the network. In the case of spatially heterogeneous synaptic weights, as the
modulation frequency of this heterogeneity increases, the effective diffusion
of bumps in the network approaches that of the network with spatially
homogeneous weights.
