Geodesic
Mathematical Model of the Hereditary FitzHugh--Nagumo Oscillator
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, Tome 154 (2018), pp. 72-80

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In this paper, we propose a new mathematical FitzHugh–Nagumo model with memory, which describes the propagation of nerve impulses in membranes. This model is an integro-differential equation with initial conditions (the Cauchy problem). The difference kernel (memory function) of the model equation is a power function; this allows one to rewrite it in terms of fractional derivatives. For the Cauchy problem, an explicit finite-difference scheme was constructed and examined by computer experiments on stability and convergence. The finite-difference scheme was implemented in the Maple software; simulation results were visualized, oscillograms and phase trajectories were obtained.
Keywords: heredity, FitzHugh–Nagumo model, fractional derivative, finite-difference scheme. 
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     author = {O. D. Lipko},
     title = {Mathematical {Model} of the {Hereditary} {FitzHugh--Nagumo} {Oscillator}},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {72--80},
     publisher = {mathdoc},
     volume = {154},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2018_154_a8/}
}
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O. D. Lipko. Mathematical Model of the Hereditary FitzHugh--Nagumo Oscillator. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, Tome 154 (2018), pp. 72-80. http://geodesic.mathdoc.fr/item/INTO_2018_154_a8/