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"Traveling wave'' solutions of Fitzhugh model with cross-diffusion

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  • The FitzHugh-Nagumo 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. In this paper, we utilize a modified version of the FitzHugh-Nagumo equations to model the spatial propagation of neuron firing; we assume that this propagation is (at least, partially) caused by the cross-diffusion connection between the potential and recovery variables. We show that the cross-diffusion version of the model, be- sides giving rise to the typical fast traveling wave solution exhibited in the original ''diffusion'' FitzHugh-Nagumo equations, additionally gives rise to a slow traveling wave solution. We analyze all possible traveling wave solutions of the model and show that there exists a threshold of the cross-diffusion coefficient (for a given speed of propagation), which bounds the area where ''normal'' impulse propagation is possible.
    Mathematics Subject Classification: Primary: 34C20, 34C23; Secondary: 92C99.

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