  We study the effects of additive noise on traveling pulse solutions in
spatially extended neural fields with linear adaptation. Neural fields are
evolution equations with an integral term characterizing synaptic interactions
between neurons at different spatial locations of the network. We introduce an
auxiliary variable to model the effects of local negative feedback and consider
random fluctuations by modeling the system as a set of spatially extended
Langevin equations whose noise term is a $Q$-Wiener process. Due to the
translation invariance of the network, neural fields can support a continuum of
spatially localized bump solutions that can be destabilized by increasing the
strength of the adaptation, giving rise to traveling pulse solutions. Near this
criticality, we derive a stochastic amplitude equation describing the dynamics
of these bifurcating pulses when the noise and the deterministic instability
are of comparable magnitude. Away from this bifurcation, we investigate the
effects of additive noise on the propagation of traveling pulses and
demonstrate that noise induces wandering of traveling pulses. Our results are
complemented with numerical simulations.
