Existence of Traveling Waves in a Neural Model

In 1992 G. B. Ermentrout and J. B. McLeod published a landmark study of traveling wave fronts for a differential-integral equation modeling a neural network. Since then a number of authors have extended the model by adding an additional equation for a "recovery variable", thus allowing the possibility of traveling pulse type solutions. In a recent paper G. Faye gave perhaps the first rigorous proof of the existence (and stability) of a traveling pulse solution for such a model. The excitatory weight function J used in this work allowed the system to be reduced to a set of four coupled ODEs, and a specific firing rate function S, with parameters, was considered. The method of geometric singular perturbation was employed, together with blow-ups. In this paper, while keeping the same J, we extend Faye's work by obtaining both the fast and slow waves and by removing Faye's hypothesis that the wave passes "under the knee". We also consider a more general set of functions S, and we use only standard ode methods.