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

This article is cited in 7 scientific papers (total in 7 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-а
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:

Document Type: Article
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. Glyzin S.D., Kolesov A.Yu., Rozov N.Kh., “Stable Relaxation Cycle in a Bilocal Neuron Model”, Differ. Equ., 54:10 (2018), 1285–1309  crossref  mathscinet  isi
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