Heteroclinic connections in a hyperbolic reaction-transport model for excitable media

This work aims at investigating the emergence of continuous and discontinuous heteroclinic connections arising in an excitable media by means of a hyperbolic extension of the activator–inhibitor Barkley model. In particular, the 1D reaction-transport model is considered to qualitatively depict the behaviour of the fast activator variable and the slow inhibitor one by highlighting the role played by inertia. The occurrence of gradual or abrupt phenomena is analysed by studying the formation of continuous and discontinuous orbits between the steady states. Results emphasize the possibility to observe both colonization or degradation phenomena depending on the combination of the inertial effects and the excited migration speed. Finally, numerical investigation are performed to corroborate the theoretical results and get more details on the system responses to any abrupt or gradual change.