On stability of traveling wave solutions in synaptically coupled neuronal networks. (Q1413580)

The author is concerned with the asymptotic stability of traveling wave solutions of integro-differential equations arising from synaptically coupled neuronal networks. By using complex analytic functions, it is proved that there is no nonzero spectrum of some linear operator \(L\) in the region \(Re\lambda \geq 0\) and \(\lambda =0\) which is a simple eigenvalue. By applying a linearized stability criterion, it is shown that the traveling wave solutions are asymptotically stable. Additionally, some explicit analytic functions are found for a scalar integro-differential equation.