Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

A nonlocal eigenvalue problem for the stability of a traveling wave in a neuronal medium

Pages: 925 - 940, Volume 10, Issue 4, June 2004      doi:10.3934/dcds.2004.10.925

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Jonathan E. Rubin - Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, United States (email)

Abstract: Past work on stability analysis of traveling waves in neuronal media has mostly focused on linearization around perturbations of spike times and has been done in the context of a restricted class of models. In theory, stability of such solutions could be affected by more general forms of perturbations. In the main result of this paper, linearization about more general perturbations is used to derive an eigenvalue problem for the stability of a traveling wave solution in the biophysically derived theta model, for which stability of waves has not previously been considered. The resulting eigenvalue problem is a nonlocal equation. This can be integrated to yield a reduced integral equation relating eigenvalues and wave speed, which is itself related to the Evans function for the nonlocal eigenvalue problem. I show that one solution to the nonlocal equation is the derivative of the wave, corresponding to translation invariance. Further, I establish that there is no unstable essential spectrum for this problem, that the magnitude of eigenvalues is bounded, and that for a special but commonly assumed form of coupling, any possible eigenfunctions for real, positive eigenvalues are nonmonotone on $(-\infty,0)$.

Keywords:  Neuronal dynamics, traveling wave, stability, eigenvalues, integrodifferential equations, nonlocal equations.
Mathematics Subject Classification:  34L05, 37B55, 45J05, 92C20.

Received: September 2002;      Revised: October 2003;      Available Online: March 2004.