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Model. Anal. Inform. Sist., 2017, Volume 24, Number 1, Pages 82–93 (Mi mais550)  

Relaxation oscillations in a system of two pulsed synaptically coupled neurons

S. D. Glyzinab, A. Yu. Kolesova, E. A. Marushkinac

a P. G. Demidov Yaroslavl State University, 14 Sovetskaya str., Yaroslavl 150003, Russia
b Scientific Center in Chernogolovka RAS, 9 Lesnaya str., Chernogolovka, Moscow region, 142432, Russia
c P. G. Demidov Yaroslavl State University, 14 Sovetskaya str., Yaroslavl 150003, Russia

Abstract: We consider a mathematical model of synaptic interaction between two pulse neuron elements. Each of the neurons is modeled by a singularly-perturbed difference-differential equation with delay. Coupling is assumed to be at the threshold, and time delay is taken into consideration. Problems of existence and stability of relaxation periodic movements for obtained systems are considered. It turns out that the ratio between the delay due to internal causes in a single neuron model and the delay in the coupling link between oscillators is crucial. Existence and stability of a uniform cycle of the problem is proved for the case where the delay in the link is less than a period of a single oscillator that depends on the internal delay. As the delay grows, the in-phase regime becomes more complex, particularly, it is shown that by choosing a suitable delay, we can obtain more complex relaxation oscillation and inside a period interval the system can exhibit not one but several high-amplitude splashes. This means that bursting-effect can appear in a system of two synaptic coupled oscillators of neuron type due to a delay in a coupling link.

Keywords: neural models, differential-difference equations, relaxation oscillations, asymptotic behavior, stability, synaptic coupling.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-31-60039_мол_а_дк
The reported study was funded by RFBR, according to the research project No. 16-31-60039 mol_а_dk.


DOI: https://doi.org/10.18255/1818-1015-2017-1-82-93

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Document Type: Article
UDC: 517.9
Received: 25.10.2016

Citation: S. D. Glyzin, A. Yu. Kolesov, E. A. Marushkina, “Relaxation oscillations in a system of two pulsed synaptically coupled neurons”, Model. Anal. Inform. Sist., 24:1 (2017), 82–93

Citation in format AMSBIB
\Bibitem{GlyKolMar17}
\by S.~D.~Glyzin, A.~Yu.~Kolesov, E.~A.~Marushkina
\paper Relaxation oscillations in a system of two pulsed synaptically coupled neurons
\jour Model. Anal. Inform. Sist.
\yr 2017
\vol 24
\issue 1
\pages 82--93
\mathnet{http://mi.mathnet.ru/mais550}
\crossref{https://doi.org/10.18255/1818-1015-2017-1-82-93}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3620402}
\elib{http://elibrary.ru/item.asp?id=28380083}


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