Bifurcation structure of traveling pulses in Type-I excitable media

TL;DR

This paper analyzes the bifurcation structure and stability of traveling pulses in a Type-I excitable medium, linking bifurcations to local excitability scenarios and identifying drift instabilities outside the stability region.

Contribution

It provides a comprehensive bifurcation analysis of traveling pulses in Type-I excitable media, connecting bifurcation scenarios with local excitability mechanisms.

Findings

Identified stability regions for traveling pulses.

Connected bifurcations to homoclinic and saddle-node on invariant circle scenarios.

Linked pulse existence outside stability region to drift pitchfork instability.

Abstract

We have studied the existence of traveling pulses in a general Type-I excitable 1-dimensional medium. We have obtained the stability region and characterized the different bifurcations behind either the destruction or loss of stability of the pulses. In particular, some of the bifurcations delimiting the stability region have been connected, using singular limits, with the two different scenarios that mediated the Type-I local excitability, i.e. homoclinic (saddle-loop) and Saddle-Node on the Invariant Circle bifurcations. The existence of the traveling pulses has been linked, outside the stability region, to a drift pitchfork instability of localized steady structures.