# Bulletin KRASEC. Phys. & Math. Sci. 2017. vol. 16, issue. 1. pp. 52-60. ISSN 2313-0156

DOI: 10.18454/2313-0156-2017-16-1-52-60

MATHEMATICAL MODELLING

MSC 34A08

MATHEMATICAL MODEL OF NERVE IMPULSE PROPAGATION WITH REGARD TO HEREDITY

O. D. Lipko

Vitus Bering Kamchatka State University, 683032, Petropavlovsk-Kamchatsky, Pogranichnaya st., 4, Russia
E-mail: lipko__95@list.ru

A mathematical model of FitzHugh-Nagumo nerve impulse propagation is proposed. It takes into account the effect of heredity. This hereditary model is described by an integrodifferential equation with a power kernel, a function of memory. The algorithm for the numerical solution of this model is implemented in a computer program in Maple symbolic mathematics environment. With the help of this program, calculated curves, oscillograms, and phase trajectories were constructed depending on various values of control parameters.

Keywords: heredity, FitzHugh-Nagumo model , finite-difference scheme.

References

1. Volterra V. Sur les ’equations int’egro-diff’erentielles et leurs applications. Acta Mathematica. 1912. vol. 35. no. 1. pp. 295–356.
2. Uchajkin V. V. Metod drobnyh proizvodnyh [Fractional derivative method]. Ul’yanovsk. Artishok. 2008. 512 p.
3. Parovik R. I. Matematicheskoe modelirovanie linejnyh jereditarnyh oscilljatorov [Mathematical modeling of linear heredity oscillators]. Petropavlovsk-Kamchatskij. KamGU im. Vitusa Beringa. 2015. 178 p.
4. Petras I. Fractional-Order Nonlinear Systems. Modeling, Analysis and Simulation. Beijing and Springer-Verlag Berlin Heidelberg. Springer. 2011. 218 p.
5. FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal. 1961. vol. 1. pp. 446–446.
6. Nagumo J., Arimoto S., Yoshizawa S. An active pulse transmission line simulating nerve axon. Proc. IRE. 1962. vol. 50. pp. 2061–2070.
7. Lipko O. D. Jereditarnoe model’noe uravnenie FitcH’ju-Nagumo [Hereditary model FitzHugh-Nagumo equation]. Mezhdunarodnyj studencheskij nauchnyj vestnik. 2017. no. 2. pp. 43–43 https://www.eduherald.ru /ru/article/view?id=16890 (reference date: 22.04.2017)
8. Parovik R. I. Mathematical modeling of nonlocal oscillatory Duffing system with fractal friction. Bulletin KRASEC. Physical and Mathematical Sciences. 2015. vol. 15. no. 1. pp. 16–21.
9. Parovik R. I. Ob issledovanii ustojchivosti jereditarnogo oscilljatora Van der Polja [Investigation of hereditary Van der Pol oscillator]. Fundamental’nye issledovanija. 2016. no. 3(2). pp. 283–287.
10. Parovik R. I. Explicit finite-difference scheme for the numerical solution of the model equation of nonlinear hereditary oscillator with variable order fractional derivatives. Archives of Control Sciences. 2016. vol. 26. no. 3. pp. 429–435.
11. Parovik R. I. Finite-differential schemes for fractal oscillator with variable fractional orders. Bulletin KRASEC. Physical and Mathematical Sciences. 2015. vol. 11. no. 2. pp. 85–92.

For citation: Lipko O. D. Mathematical model of nerve impulse propagation with regard to heredity. Bulletin KRASEC. Physical and Mathematical Sciences 2017, vol. 16, issue 1, 52 — 60. DOI: 10.18454/2313-0156-2017-16-1-52-60.

Original article submitted: 22.03.2017

Lipko Olga Dmitrievna – a student of the 4th year of training «Applied Mathematics and Informatics Vitus Bering Kamchatka State University.