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Regul. Chaotic Dyn., 2018, том 23, выпуск 2, страницы 152–160 (Mi rcd315)  

Эта публикация цитируется в 10 научных статьях (всего в 10 статьях)

Asymptotic and Exact Solutions of the FitzHugh–Nagumo Model

Nikolay A. Kudryashov

Department of Applied Mathematics, National Research Nuclear University MEPhI, Kashirskoe sh. 31, Moscow, 115409 Russia

Аннотация: The standard FitzHugh–Nagumo model for description of impulse from one neuron to another is considered. The system of equations is transformed to a nonlinear second-order ordinary differential equation. It is shown that the differential equation does not pass the Painlevé test in the general case and the general solution of this equation does not exist. The simplest solutions of the system of equations are found. The second-order differential equation is transformed to another asymptotic equation with the general solution expressed via the Jacobi elliptic function. This transformation allows us to obtain the asymptotic solutions of the FitzHugh–Nagumo model. The perturbed FitzHugh–Nagumo model is studied as well. Taking into account the simplest equation method, the exact solutions of the perturbed system of equations are found. The asymptotic solutions of the perturbed model are presented too. The application of the exact solutions for construction of the neural networks is discussed.

Ключевые слова: neuron, FitzHugh–Nagumo model, system of equations, Painelevé test, exact solution

Финансовая поддержка Номер гранта
Российский научный фонд 17-71-20111
This work was supported by the Research Science Foundation grant 17-71-20111 “Study and justification of mechanisms for spiking neural networks learning based on synaptic plasticity in order to create biologically inspired nonlinear information models capable of solving practical tasks”.


DOI: https://doi.org/10.1134/S1560354718020028

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Тип публикации: Статья
MSC: 34M05, 34E10
Поступила в редакцию: 01.11.2017
Принята в печать:22.12.2017
Язык публикации: английский

Образец цитирования: Nikolay A. Kudryashov, “Asymptotic and Exact Solutions of the FitzHugh–Nagumo Model”, Regul. Chaotic Dyn., 23:2 (2018), 152–160

Цитирование в формате AMSBIB
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\by Nikolay A. Kudryashov
\paper Asymptotic and Exact Solutions of the FitzHugh–Nagumo Model
\jour Regul. Chaotic Dyn.
\yr 2018
\vol 23
\issue 2
\pages 152--160
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  • http://mi.mathnet.ru/rcd315
  • http://mi.mathnet.ru/rus/rcd/v23/i2/p152

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    Эта публикация цитируется в следующих статьяx:
    1. Nikolay A. Kudryashov, “Exact Solutions and Integrability of the Duffing–Van der Pol Equation”, Regul. Chaotic Dyn., 23:4 (2018), 471–479  mathnet  crossref  mathscinet
    2. A. I. Zemlyanukhin, A. V. Bochkarev, “Analytical Properties and Solutions of the FitzHugh – Rinzel Model”, Нелинейная динам., 15:1 (2019), 3–12  mathnet  crossref  elib
    3. N. A. Kudryashov, “On Integrability of the FitzHugh – Rinzel Model”, Нелинейная динам., 15:1 (2019), 13–19  mathnet  crossref  elib
    4. Nikolay A. Kudryashov, Dariya V. Safonova, Anjan Biswas, “Painlevé Analysis and a Solution to the Traveling Wave Reduction of the Radhakrishnan – Kundu – Lakshmanan Equation”, Regul. Chaotic Dyn., 24:6 (2019), 607–614  mathnet  crossref
    5. E. G. Fedorov, I A. Popov , I. Y. Popov, “Metric graph version of the FitzHugh-Nagumo model”, Nanosyst.-Phys. Chem. Math., 10:6 (2019), 623–626  crossref  isi
    6. N. A. Kudryashov, “Exact solutions of the equation for surface waves in a convecting fluid”, Appl. Math. Comput., 344 (2019), 97–106  crossref  mathscinet  zmath  isi  scopus
    7. A. Sboev, A. Serenko, R. Rybka, D. Vlasov, “Influence of input encoding on solving a classification task by spiking neural network with STDP”, International Conference on Numerical Analysis and Applied Mathematics (ICNAAM-2018), AIP Conf. Proc., 2116, eds. T. Simos, C. Tsitouras, Amer. Inst. Phys., 2019, 270007  crossref  isi  scopus
    8. N. A. Kudryashov, “Remarks on the Fuchs indices and the first integrals for nonlinear ordinary differential equations”, VII International Conference Problems of Mathematical Physics and Mathematical Modelling, Journal of Physics Conference Series, 1205, IOP Publishing Ltd, 2019, 012031  crossref  mathscinet  isi  scopus
    9. S. F. Lavrova, N. A. Kudryashov, D. I. Sinelshchikov, “On some properties of the coupled FitzHugh-Nagumo equations”, VII International Conference Problems of Mathematical Physics and Mathematical Modelling, Journal of Physics Conference Series, 1205, IOP Publishing Ltd, 2019, 012035  crossref  isi  scopus
    10. A. I. Zemlyanukhin, A. V. Bochkarev, “Analytical properties and exact solution of the Aliev-Panfilov model”, VII International Conference Problems of Mathematical Physics and Mathematical Modelling, Journal of Physics Conference Series, 1205, IOP Publishing Ltd, 2019, 012060  crossref  mathscinet  isi  scopus
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