Traveling Waves Impulses of FitzHugh Model with Diffusion and Cross-Diffusion

Abstract

The FitzHugh equations have been used as a caricature of the Hodgkin–Huxley equations of neuron firing and to capture, qualitatively, the general properties of an excitable membrane. The spatial propagation of neuron firing due to diffusion of the current potential was described by the FitzHugh–Nagumo model. Assuming that the spatial propagation of neuron firing is caused by not diffusion but cross-diffusion connection between the potential and recovery variables the cross-diffusion version of the FitzHugh model gives rise to the typical fast traveling wave solutions characteristic to the FitzHugh model, and additionally gives rise to the slow traveling wave solutions exhibited in the diffusion FitzHugh–Nagumo equations (Berezovskaya et al., Math Biosci Eng 5:239–260, 2008).