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 $(-\infty,0)$.
Highlights
A variety of mathematical models have been developed to describe neuronal dynamics in various levels of detail
The primary goal of this paper is to present the derivation of the eigenvalue problem that will determine the linearized stability of the single spike traveling wave solutions to the theta network model with strong adaptation
It is not surprising that speed plays a crucial role in the stability analysis here, since stability analysis based on spike time perturbations has shown that, in related models, fast waves are stable and slow waves are unstable [2, 3, 6]
Summary
A variety of mathematical models have been developed to describe neuronal dynamics in various levels of detail. The theta model, which is more mathematically tractable, arises in certain limits from a class of biophysically based neural models for cells near an activity threshold [5, 4, 7]. This model applies near a saddle-node bifurcation on a limit cycle, through which the model cell undergoes a transition from being excitable, or intrinsically at rest, to oscillatory, or spontaneously active. If Z is a negative constant, equation (1.1) has stable and unstable fixed points These coalesce in a saddle-node bifu√rcation on a limit cycle at Z = 0, and for Z > 0 the θ-neuron fires with period π/ Z
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