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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