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Traveling Wave Solutions to a Coupled System of Spatially Discrete Nagumo Equations
Vateman, Michael D. Van Vleck, Erik S.
Vateman, Michael D.
Van Vleck, Erik S.
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Abstract
We consider a coupled system of discrete Nagumo equations and derive traveling wave solutions to this system using McKean's caricature of the cubic. A certain form of this system is used to model ephaptic coupling between pairs of nerve axons. We study the difference $g(c)=a_1-a_2$ between the detuning parameters $a_i$ that is required to make both waves move at the same speed c. Of particular interest is the effect of a coupling parameter $\alpha$ and an "alignment" parameter A on the function g. Numerical investigation indicates that for fixed A, there exists a time delay value $\beta$ that results in $g=0$, and for large enough wave speeds, multiple such $\beta$ values exist. Also, numerical results indicate that the perturbation of $\alpha$ away from zero will yield additional solutions with positive wave speed when $A={1\over 2}$. We employ both analytical and numerical results to demonstrate our claims.
Description
This is the published version, also available here: http://dx.doi.org/10.1137/050624352.
Date
2006-07-31
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Society for Industrial and Applied Mathematics
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Keywords
discrete Nagumo equations, ephaptic coupling, traveling waves
Citation
Batemen, Michael D. & Van Vleck, Erik."Traveling Wave Solutions to a Coupled System of Spatially Discrete Nagumo Equations." (2006) SIAM J. Appl. Math., 66(3), 945–976. (32 pages). http://dx.doi.org/10.1137/050624352.