Numerical Simulation of Traveling Waves of the FitzHugh-Nagumo System
DOI:
https://doi.org/10.17268/sel.mat.2018.02.06Keywords:
Traveling wave, Stable solution, PDEs, Mobile coordinatesAbstract
The FitzHugh-Nagumo system has a special type of solution called traveling wave, which has the form u(x, t) = (x − μt) and w(x, t) = (x − μt), which is a stable solution over time. Our interest is to numerically characterize the profile of a traveling wave (, ) and its propagation speed μ(t). With a
change of variables, we transform the problem of finding the solutions in original coordinates to a problem of finding the equilibria in a new coordinate system called mobile coordinates or non-local coordinate
system. aa With numerical examples we will demonstrate that the solutions of the system of EDPs in non-local coordinates converge to a traveling wave of the original problem. The non-local coordinate system also allows to calculate the exact propagation speed.
References
Fitzhugh, R. Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961) 445-466.
Glass, L. Time series analysis of complex dynamics in phisiology and medicine, Natural Sciences and Engineering Research Council of Canada (1992).
Nagumo, J.S. et al. An active pulse transmission line simulating nerve axon, Proc.IRE, 50 (1962) 2061-2071.
Panfilov, A. V., Pertsov, A.M. Vortex ring in three-dimensional active medium in reaction-diffusion system, Doklady AN SSSR, 274 (1984) 1500-1503.
Gao, W. y Wang, J. Existence of waves fronts and impulses to FitzHugh-Nagumo equations. Nonlinear Analysis: Theory, Methods and Applications, 57(5): 667-676. 2004.
Yanagida, E. Stability of travelling front solutions of the FitzHugh-Nagumo equations. Mathematical and Computer Modelling, 12(3): 289-301. 1989.
Hecht, F., New development in FreeFem++, J. Numer. Math. 20(2012), no. 3-4, 251-265.
FreeFem++ Free Manual. http://www.freefem.org. [accessed 10-December-2018].
Arrieta, J. M. Lopez-Fernandez, M. y Zuazua, E. Approximating travelling waves by equilibria of non local equations. Asymptotic Analysis, (78): 145-186. 2012.
Beyn, W., Otter, D. y Rottmann-Matthes, J. Stability and computation of dynamic patterns in PDEs, Current Challenges in Stability Issues for Numerical DE, 89-172, 2011.
Feng, H., Numerical simulation of the FHN model with strong reaction, Master thesis, Texas A and M International University, 2012.
Paton, K. M., A study of wave propagation in the FHN system, Master thesis, University of British Columbia, 2011.
Verão, G. B. Aproximando ondas viajantes por equilíbrio de uma equação não local, Tese de doutorado, USP, 2016.
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