# American Institute of Mathematical Sciences

2008, 5(2): 239-260. doi: 10.3934/mbe.2008.5.239

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

 1 Department of Mathematics, Howard University, Washington D.C., 20059 2 Department of Mathematical Sciences and Applied Computing, Arizona State University, Glendale, AZ 85306, United States, United States 3 National Centre for Biotechnological Information, National Institutes of Health, Bethesda, MD 20894, United States

Received  May 2007 Revised  January 2008 Published  March 2008

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.
Citation: F. Berezovskaya, Erika Camacho, Stephen Wirkus, Georgy Karev. "Traveling wave'' solutions of Fitzhugh model with cross-diffusion. Mathematical Biosciences & Engineering, 2008, 5 (2) : 239-260. doi: 10.3934/mbe.2008.5.239
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