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Uspekhi Mat. Nauk, 2015, Volume 70, Issue 3(423), Pages 3–76 (Mi umn9659)  

This article is cited in 16 scientific papers (total in 16 papers)

Self-excited relaxation oscillations in networks of impulse neurons

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

a Yaroslavl State University
b Moscow State University

Abstract: This paper addresses the problem of mathematical modelling of neuron activity. New classes of singularly perturbed differential-difference equations with Volterra-type delay are proposed and used to describe how single neurons and also neural networks function with various kinds of connections (electrical or chemical). Special asymptotic methods are developed which make it possible to analyse questions of the existence and stability of relaxation periodic motions in such systems.
Bibliography: 56 titles.

Keywords: neuron models, differential-difference equations, asymptotic behaviour, relaxation oscillations, stability, buffering, bursting effect.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-01-04066-
Ministry of Education and Science of the Russian Federation 1875 2014/258


DOI: https://doi.org/10.4213/rm9659

Full text: PDF file (1232 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2015, 70:3, 383–452

Bibliographic databases:

UDC: 517.926
MSC: 34C26, 34K26, 92C20
Received: 05.02.2015

Citation: S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Self-excited relaxation oscillations in networks of impulse neurons”, Uspekhi Mat. Nauk, 70:3(423) (2015), 3–76; Russian Math. Surveys, 70:3 (2015), 383–452

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. 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
    2. M. M. Preobrazhenskaya, “Relaksatsionnye tsikly v modeli sinapticheski vzaimodeistvuyuschikh ostsillyatorov”, Model. i analiz inform. sistem, 24:2 (2017), 186–204  mathnet  crossref  elib
    3. 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
    4. L. I. Ivanovsky, “Stable regimes of dynamic systems with impulsive influences”, Lobachevskii J. Math., 38:5, SI (2017), 921–925  crossref  mathscinet  zmath  isi  scopus
    5. E. A. Marushkina, “Periodicheskie i kvaziperiodicheskie resheniya v sisteme trekh uravnenii Khatchinsona s zapazdyvayuschei veschatelnoi svyazyu”, Model. i analiz inform. sistem, 25:1 (2018), 102–111  mathnet  crossref  elib
    6. S. D. Glyzin, E. A. Marushkina, “Neuporyadochennye kolebaniya v neiroseti iz trekh ostsillyatorov s zapazdyvayuschei veschatelnoi svyazyu”, Model. i analiz inform. sistem, 25:5 (2018), 572–583  mathnet  crossref
    7. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Stable relaxation cycle in a bilocal neuron model”, Differ. Equ., 54:10 (2018), 1285–1309  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    8. M. M. Preobrazhenskaia, “The impulse-refractive mode in a neural network with ring synaptic interaction”, Autom. Control Comp. Sci., 52:7 (2018), 777–789  crossref  isi  scopus
    9. S. D. Glyzin, E. A. Marushkina, “Complicated dynamic regimes in a neural network of three oscillators with a delayed broadcast connection”, Autom. Control Comp. Sci., 52:7 (2018), 885–893  crossref  mathscinet  isi  scopus
    10. Aleshin S.V., Glyzin D.S., Glyzin S.D., Goryunov V.E., International Conference on Computer Simulation in Physics and Beyond, Journal of Physics Conference Series, 1163, ed. Shchur L., IOP Publishing Ltd, 2019  crossref  isi
    11. 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
    12. 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
    13. Glyzin S.D., Kolesov A.Yu., Rozov N.Kh., Vii International Conference Problems of Mathematical Physics and Mathematical Modelling, Journal of Physics Conference Series, 1205, IOP Publishing Ltd, 2019  crossref  isi
    14. Glyzin S.D., Preobrazhenskaia M.M., “Multistability and Bursting in a Pair of Delay Coupled Oscillators With a Relay Nonlinearity”, IFAC PAPERSONLINE, 52:18 (2019), 109–114  crossref  isi
    15. M. M. Preobrazhenskaya, “A relay Mackey–Glass model with two delays”, Theoret. and Math. Phys., 203:1 (2020), 524–534  mathnet  crossref  crossref  isi  elib
    16. V. A. Ermolaev, Yu. A. Kropotov, A. Yu. Proskuryakov, “Postroenie modelei sistem obmena informatsiei s diskretnym i raspredelennym zapazdyvaniem i zaderzhannoi obratnoi svyazyu”, Kompyuternaya optika, 44:3 (2020), 454–465  mathnet  crossref
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