Traveling Waves and Synchrony in an Excitable Large-Scale Neuronal Network with Asymmetric Connections

We study (i) traveling wave solutions, (ii) the formation and spatial spread of synchronous oscillations, and (iii) the effects of variations of threshold in a system of integro-differential equations which describe the activity of large-scale networks of excitatory neurons on spatially extended domains. The independent variables are the activity level u of a population of excitatory neurons which have long range connections, and a recovery variable v. In the integral component of the equation for u the firing rate function is the Heaviside function, and the coupling function w is positive. Thus, there is no inhibition in the system. There is a critical value of the parameter $\beta$ ($\beta_{*} > 0$) that appears in the equation for v, at which the eigenvalues $\mu^{\pm}$ of the linearization of the system around the rest state $(u,v) = (0,0)$ change from real to complex. We focus on the range $\beta > \beta_{*}$, where $\mu^{\pm}$ are complex, and analyze properties of wave fronts and 1-pulse and 2-puls...

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