Existence and Stability of Traveling Pulses in a Continuous Neuronal Network

We examine the existence and stability of traveling pulse solutions of a set of integro-differential equations that describe activity in a spatially extended population of synaptically connected neurons. These equations have been employed extensively to model wave propagation during normal and epileptic brain activity. Compared to previous studies, we make relatively weak assumptions on the pattern of spatial connectivity, namely, that it is positive, homogeneous, and symmetric and that it decays with distance. A Heaviside step function governs the activation of each neuron. We incorporate a relatively slow local recovery variable within each neuron but make no other assumptions about the recovery rate. Our results are guided by the local behavior of individual neurons. When neurons have a single stable state, we demonstrate the existence of two traveling pulse solutions in a connected network. When the neurons are bistable, we demonstrate the existence of a stationary pulse solution and, in some cases, a...

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