Travelling pulses in the Barkley model: A geometric singular perturbation approach

In this work, we investigate travelling pulse solutions to the Barkley model, a prototypical example of excitable system with activator-inhibitor dynamics. Such patterns are numerically observed for a wide range of parameter values and show how coherent structures can be induced by mechanisms different from diffusion-driven instability. The intrinsic multiscale nature of this system allows us to apply Geometric Singular Perturbation Theory (GSPT) to constructively establish the existence of travelling pulses as homoclinic orbits in the corresponding three-dimensional phase-space. The analytical findings are corroborated by a thorough numerical investigation via direct simulation as well as continuation based on the software AUTO.