Abstract
For the system of FitzHugh-Nagumo equations and certain values of parameters, there exists a relaxation wave solution: a moving spatially periodic structure for which one of the dependent variables has intervals of slow and fast changes within its period. An asymptotic expansion for the solution is constructed (as a power series in the small parameter ε) consisting of regular terms and transition layers. Along with the asymptotic expansion, the velocity of the structure, its period, and the distance between “jumps” of one of the functions are determined.
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References
Feroe, J. A.: Existence and stability of multiple impulse solutions of a nerve equation. SIAM J. Appl. Math. 42, 235–246 (1982)
Fife, P. C.: Mathematical Aspects of Reacting and Diffusing System. (Lect. Notes Biomath., vol. 28) Berlin Heidelberg New York: Springer 1979
Fife, P. C., McLeod, J. B.: The approach of solutions of nonlinear diffusion equations to travelling wave solutions. Arch. Ration. Mech. Anal. 65, 335–361 (1977)
FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961)
Hodgkin, A. L., Huxley, A. F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (London) 117, 500–544 (1952)
Keener, J. P.: Waves in excitable media. SIAM J. Appl. Math. 39, 528–548 (1980)
Kevorkian, J.: Partial Differential Equations: Analytical Solution Techniques. Pacific Grove, CA: Wadsworth & Brooks/Cole 1990
Kevorkian, J., Cole, J. D.: Perturbation Methods in Applied Mathematics. Berlin Heidelberg New York: Springer 1981
McKean, H. P.: Nagumo's equation. Adv. Math. 4, 209–233 (1970)
Murray, J. D.: Mathematical Biology. Berlin Heidelberg New York: Springer 1989
Nagumo, J. S., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2071 (1962)
Namias, V.: Simple derivation of the roots of a cubic equation. Am. J. Phys. 53, 775 (1985)
Nishiura, Y., Mimura, M.: Layer oscillations in reaction-diffusion systems. SIAM J. Appl. Math. 49, 481–514 (1989)
O'Malley, R. E.: Introduction to Singular Perturbations. New York: Academic Press 1974
O'Malley, R. E.: Singular Perturbations Methods for Ordinary Differential Equations. Berlin Heidelberg New York: Springer 1991
McAllister, R. E., Noble, D., Tsien, R. W.: Reconstruction of the electric activity of cardiac Purkinje fibres. J. Physiol. 251, 1–59 (1975)
Rinzel, J.: Models in neurobiology. In: Enus, R. H., Jones, B. L., Miura, R. M., Rangnekar, S. S. (eds.) Nonlinear Phenomena in Physics and Biology, pp. 345–367. New York: Plenum Press (1981)
Shibata, J., Bureš, J.: Optimum topographical conditions for reverberating cortical spreading depression in rats. J. Neurobiol. 5, 107–118 (1974)
Tsujikawa, T., Nagai, T., Mimura, M., Kobayashi, R., Ikeda, H.: Stability properties of travelling pulse solutions of the higher dimensional FitzHugh-Nagumo equations. Japan J. Appl. Math. 6, 341–366 (1989)
Tyson, J. J., Keener, J. P.: Singular perturbation theory of travelling waves in excitable media (a review). Physica D 32, 327–361 (1988)
Vasil'eva, A. B., Butuzov, V. F.: Asymptotic Expansions of the Solutions of Singularly Perturbed Equations (in Russian). Moscow: Nauka 1973
Vasil'eva, A. B., Butuzov, V. F.: Singularly Perturbed Equations in the Critical Case. MRC-TSR 2039, Math. Res. Center, University of Wisconsin, Madison, WI, USA 1980
Vasil'eva, A. B., Butuzov, V. F.: Asymptotic methods in the theory of singular perturbations (in Russian). Moscow: Vischaja Shkola 1990
Winfree, A. T.: Sudden cardiac death: a problem in topology. Sci. Am. 248(5), 144–161 (1983)
Zykov, V. S.: Modelling of Wave Processes in Excitable Media. Manchester: Manchester University Press 1988
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Visiting from the Moscow State University, Russia.
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Kalachev, L.V. A relaxation wave solution of the FitzHugh-Nagumo equations. J. Math. Biol. 31, 133–147 (1993). https://doi.org/10.1007/BF00171222
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DOI: https://doi.org/10.1007/BF00171222