A note on traveling fronts and pulses in a firing rate model of a neuronal network

We study the activity of a one-dimensional synaptically coupled neural network by means of a firing rate model developed by Coombes et al. [Physica D 178 (2003)]. Their approach incorporates the biologically motivated finite conduction velocity of action potentials into a neural field equation of Wilson and Cowan type [Kybernetik 13 (1973)]. The resulting integro-differential equation with a space depending delay term under the convolution may exhibit a variety of traveling and stationary patterns. In this paper we construct traveling wave solutions for the case of a firing rate given by the Heaviside step function, exponential synaptic kernel, and exponential synaptic footprint. In contrast to Coombes et al., where the model equation is first reduced to an equivalent system of partial differential equations, we make the traveling pattern ansatz into the initial integro-differential equation. We analyse two types of traveling patterns: fronts and pulses, for which we derive shape and speed. We further determine necessary conditions for the linear stability of the traveling waves.