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Zh. Vychisl. Mat. Mat. Fiz., 2012, Volume 52, Number 5, Pages 840–858 (Mi zvmmf9713)  

This article is cited in 21 scientific papers (total in 21 papers)

Discrete autowaves in neural systems

S. D. Glyzina, A. Yu. Kolesova, N. Kh. Rozovb

a Faculty of Mathematics, Yaroslavl State University, ul. Sovetskaya 14, Yaroslavl, 150000 Russia
b Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119992 Russia

Abstract: A singularly perturbed scalar nonlinear differential-difference equation with two delays is considered that is a mathematical model of an isolated neuron. It is shown that a one-dimensional chain of diffusively coupled oscillators of this type exhibits the well-known buffer phenomenon. Specifically, as the number of chain links increases consistently with decreasing diffusivity, the number of coexisting stable periodic motions in the chain grows indefinitely.

Key words: differential-difference equations, relaxation cycle, autowaves, stability, buffer phenomenon, bursting effect.

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English version:
Computational Mathematics and Mathematical Physics, 2012, 52:5, 702–719

Bibliographic databases:

UDC: 519.62
Received: 05.12.2011

Citation: S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Discrete autowaves in neural systems”, Zh. Vychisl. Mat. Mat. Fiz., 52:5 (2012), 840–858; Comput. Math. Math. Phys., 52:5 (2012), 702–719

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. 49, no. 10, 2013, 1193–1210  crossref  mathscinet  zmath  isi  elib  scopus
    2. A. O. Tolbei, “Lokalnaya dinamika trekh ostsillyatorov so svyazyu veschatelnogo tipa”, Model. i analiz inform. sistem, 19:3 (2012), 105–112  mathnet
    3. S. D. Glyzin, E. A. Marushkina, “Relaksatsionnye tsikly v obobschennoi neironnoi modeli s dvumya zapazdyvaniyami”, Model. i analiz inform. sistem, 20:6 (2013), 179–199  mathnet
    4. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “On a modification of the FitzHugh–Nagumo neuron model”, Comput. Math. Math. Phys., 54:3 (2014), 443–461  mathnet  crossref  crossref  isi  elib  elib
    5. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Neklassicheskie relaksatsionnye kolebaniya v neirodinamike”, Model. i analiz inform. sistem, 21:2 (2014), 71–89  mathnet
    6. M. M. Preobrazhenskaya, “Primenenie metoda kvazinormalnykh form k matematicheskoi modeli otdelnogo neirona”, Model. i analiz inform. sistem, 21:5 (2014), 38–48  mathnet
    7. Yu. V. Bogomolov, S. D. Glyzin, A. Yu. Kolesov, “O chisle sosuschestvuyuschikh avtovolnovykh reshenii tsepochki diffuzionno svyazannykh ostsillyatorov neironnogo tipa”, Model. i analiz inform. sistem, 21:5 (2014), 162–180  mathnet
    8. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Self-excited relaxation oscillations in networks of impulse neurons”, Russian Math. Surveys, 70:3 (2015), 383–452  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Avtovolnovye protsessy v koltsevoi neironnoi tsepi s odnonapravlennoi svyazyu”, Model. i analiz inform. sistem, 22:3 (2015), 404–419  mathnet  crossref  mathscinet  elib
    10. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Nonclassical relaxation oscillations in neurodynamics”, Autom. Control Comp. Sci., 50:7 (2016), 571–585  crossref  isi  scopus
    11. S. D. Glyzin, A. Yu. Kolesov, E. A. Marushkina, “Relaksatsionnye avtokolebaniya v sisteme iz dvukh sinapticheski svyazannykh impulsnykh neironov”, Model. i analiz inform. sistem, 24:1 (2017), 82–93  mathnet  crossref  mathscinet  elib
    12. M. M. Preobrazhenskaya, “Relaksatsionnye tsikly v modeli sinapticheski vzaimodeistvuyuschikh ostsillyatorov”, Model. i analiz inform. sistem, 24:2 (2017), 186–204  mathnet  crossref  elib
    13. M. M. Preobrazhenskaya, “Impulsno-refrakternyi rezhim v koltsevoi tsepi sinapticheski svyazannykh ostsillyatorov neironnogo tipa”, Model. i analiz inform. sistem, 24:5 (2017), 550–566  mathnet  crossref  elib
    14. S. D. Glyzin, A. Yu. Kolesov, E. A. Marushkina, “Relaxation oscillations in a system of two pulsed synaptically coupled neurons”, Autom. Control Comp. Sci., 51:7 (2017), 658–665  crossref  isi  scopus
    15. M. M. Preobrazhenskaia, “Relaxation cycles in a model of synaptically interacting oscillators”, Autom. Control Comp. Sci., 51:7 (2017), 783–797  crossref  mathscinet  isi  scopus
    16. S. D. Glyzin, A. Yu. Kolesov, M. M. Preobrazhenskaia, “Existence and stability of periodic solutions of quasi-linear Korteweg-de Vries equation”, V International Conference on Problems of Mathematical and Theoretical Physics and Mathematical Modelling, Journal of Physics Conference Series, 788, IOP Publishing Ltd, 2017, UNSP 012016  crossref  isi  scopus
    17. Preobrazhenskaia M.M., “The Impulse-Refractive Mode in a Neural Network With Ring Synaptic Interaction”, Autom. Control Comp. Sci., 52:7 (2018), 777–789  crossref  isi  scopus
    18. Preobrazhenskaia M.M., International Conference on Computer Simulation in Physics and Beyond, Journal of Physics Conference Series, 1163, ed. Shchur L., IOP Publishing Ltd, 2019  crossref  isi
    19. V. E. Goryunov, M. M. Preobrazhenskaya, “Kvaziustoichivost sosuschestvuyuschikh attraktorov neirodinamicheskoi modeli s zapazdyvaniem”, Materialy Voronezhskoi zimnei matematicheskoi shkoly Sovremennye metody teorii funktsii i smezhnye problemy. 28 yanvarya2 fevralya 2019 g. Chast 4, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 173, VINITI RAN, M., 2019, 26–47  mathnet  crossref
    20. M. M. Preobrazhenskaya, “Discrete traveling waves in a relay system of Mackey–Glass equations with two delays”, Theoret. and Math. Phys., 207:3 (2021), 827–840  mathnet  crossref  crossref  isi  elib
    21. A. A. Kaschenko, “Dinamika odnoi modeli s zapazdyvaniem i bolshim parametrom”, Materialy Voronezhskoi vesennei matematicheskoi shkoly Sovremennye metody teorii kraevykh zadach. Pontryaginskie chteniyaXXX. Voronezh, 39 maya 2019 g. Chast 5, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 194, VINITI RAN, M., 2021, 115–123  mathnet  crossref
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