Eigenvalue Functions in Excitatory-inhibitory Neuronal Networks

Abstract

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We study the exponential stability of traveling wave solutions of non-linear systems of integral differential equations arising from nonlinear, nonlocal, synaptically coupled, excitatory-inhibitory neuronal networks. We have proved that exponential stability of traveling waves is equivalent to linear stability. Moreover, if the real parts of nonzero spectrum of an associated linear differential operator have a uniform negative upper bound, namely, max{Re\u03bb: \u03bb \u2208 \u03c3(L), \u03bb \u2260 0} \u2264 -D, for some positive constant D, and \u03bb = 0 is an algebraically simple eigenvalue of L, then the linear stability follows, where L is the linear differential operator obtained by linearizing the nonlinear system about its traveling wave and \u03c3(L) denotes the spectrum of L. The main aim of this paper is to construct complex analytic functions (also called eigenvalue or Evans functions) for exploring eigenvalues of linear differential operators to study the exponential stability of traveling waves. The zeros of the eigenvalue functions coincide with the eigenvalues of L. When studying multipulse solutions, some components of the traveling waves cross their thresholds for many times. These crossings cause great difficulty in the construction of the eigenvalue functions. In particular, we have to solve an over-determined system to construct the eigenvalue functions. By investigating asymptotic behaviors as z \u2192 -\u221e of candidates for eigenfunctions, we find a way to construct the eigenvalue functions. By analyzing the zeros of the eigenvalue functions, we can establish the exponential stability of traveling waves arising from neuronal networks.<\/p>"