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
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 $(-\infty,0)$.