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Mat. Zametki, 2013, Volume 93, Issue 5, Pages 684–701 (Mi mz9293)  

This article is cited in 13 scientific papers (total in 13 papers)

Modeling the Bursting Effect in Neuron Systems

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

a P. G. Demidov Yaroslavl State University
b M. V. Lomonosov Moscow State University

Abstract: We propose a new method for modeling the well-known phenomenon of “bursting behavior” in neuron systems by invoking delay equations. Namely, we consider a singularly perturbed nonlinear difference-differential equation with two delays describing the functioning of an isolated neuron. Under a suitable choice of parameters, we establish the existence of a stable periodic motion with any prescribed number of spikes on a closed time interval equal to the period length.

Keywords: “bursting behavior” in neuron systems, difference-differential equation, relay equation, Cauchy problem, Schauder principle, relaxation cycle, spiking, stability.

DOI: https://doi.org/10.4213/mzm9293

Full text: PDF file (581 kB)
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English version:
Mathematical Notes, 2013, 93:5, 676–690

Bibliographic databases:

UDC: 517.926
Received: 01.12.2011

Citation: S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Modeling the Bursting Effect in Neuron Systems”, Mat. Zametki, 93:5 (2013), 684–701; Math. Notes, 93:5 (2013), 676–690

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. A. S. Bobok, S. D. Glyzin, A. Yu. Kolesov, “Ekstremalnaya dinamika sistemy trekh odnonapravlenno svyazannykh singulyarno vozmuschennykh uravnenii iz neirodinamiki”, Model. i analiz inform. sistem, 20:5 (2013), 158–167  mathnet
    2. 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
    3. M. M. Preobrazhenskaya, “Primenenie metoda kvazinormalnykh form k matematicheskoi modeli otdelnogo neirona”, Model. i analiz inform. sistem, 21:5 (2014), 38–48  mathnet
    4. 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
    5. 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
    6. 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
    7. 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
    8. 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
    9. 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
    10. 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
    11. 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
    12. 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
    13. 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
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