"Traveling wave'' solutions of Fitzhugh model with cross-diffusion
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Department of Mathematics, Howard University, Washington D.C., 20059
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Department of Mathematical Sciences and Applied Computing, Arizona State University, Glendale, AZ 85306
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National Centre for Biotechnological Information, National Institutes of Health, Bethesda, MD 20894
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Received:
01 May 2007
Accepted:
29 June 2018
Published:
01 March 2008
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MSC :
Primary: 34C20, 34C23; Secondary: 92C99.
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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.
Citation: F. Berezovskaya, Erika Camacho, Stephen Wirkus, Georgy Karev. 'Traveling wave'' solutions of Fitzhugh model with cross-diffusion[J]. Mathematical Biosciences and Engineering, 2008, 5(2): 239-260. doi: 10.3934/mbe.2008.5.239
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Abstract
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.
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