Traveling pulses in Class-I excitable media

We study Class-I excitable $1$-dimensional media showing the appearance of propagating traveling pulses. We consider a general model exhibiting Class-I excitability mediated by two different scenarios: a homoclinic (saddle-loop) and a SNIC (Saddle-Node on the Invariant Circle) bifurcations. The distinct properties of Class-I with respect to Class-II excitability infer unique properties to traveling pulses in Class-I excitable media. We show how the pulse shape inherit the infinite period of the homoclinic and SNIC bifurcations at threshold, exhibiting scaling behaviors in the spatial thickness of the pulses that are equivalent to the scaling behaviors of characteristic times in the temporal case.