Bifurcation structure of traveling pulses in Type-I excitable media

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.