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Mathematics
Volume 14
Issue 1
10.3390/math14010008
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Article

An Analytical Study on the Synchronization of a Two-Cell Inhibitory Neural Dynamics

by
Julia V. Chaparova
1,
Dimitar R. Chaparov
2 and
Teodor G. Georgiev
1,*
1
Department of Mathematics, Faculty of Natural Sciences and Education, “Angel Kanchev” University of Ruse, 8 Studentska Str., 7017 Ruse, Bulgaria
2
Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria
*
Author to whom correspondence should be addressed.
Mathematics 2026, 14(1), 8; https://doi.org/10.3390/math14010008
Submission received: 30 October 2025 / Revised: 30 November 2025 / Accepted: 9 December 2025 / Published: 19 December 2025
(This article belongs to the Special Issue Advances in Mathematical Biology and Applications)
Versions Notes

Abstract

Biophysical observations in different brain regions have displayed synchronous firing in inhibitory neural networks. Although inhibition reduces the postsynaptic neuron’s activity, coordinated synchronous rhythms could be predicted provided that inhibition delay is incorporated into the biophysical models. In this article, we study the local dynamics of two mutually coupled neurons connected via inhibitory synapses. The key assumption in the model is that the neurotransmitter release activates some secondary synaptic processes that give additional time to the postsynaptic neuron before it feels the inhibition. Stability conditions for synchrony are derived for this simple inhibitory network. Numerical experiments are presented that justify theoretical conclusions. The geometric singular perturbation theory is used as well as a variational argument.
Keywords: local neural networks; mutually coupling; inhibition; synchronous oscillations; stability conditions local neural networks; mutually coupling; inhibition; synchronous oscillations; stability conditions

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MDPI and ACS Style

Chaparova, J.V.; Chaparov, D.R.; Georgiev, T.G. An Analytical Study on the Synchronization of a Two-Cell Inhibitory Neural Dynamics. Mathematics 2026, 14, 8. https://doi.org/10.3390/math14010008

AMA Style

Chaparova JV, Chaparov DR, Georgiev TG. An Analytical Study on the Synchronization of a Two-Cell Inhibitory Neural Dynamics. Mathematics. 2026; 14(1):8. https://doi.org/10.3390/math14010008

Chicago/Turabian Style

Chaparova, Julia V., Dimitar R. Chaparov, and Teodor G. Georgiev. 2026. "An Analytical Study on the Synchronization of a Two-Cell Inhibitory Neural Dynamics" Mathematics 14, no. 1: 8. https://doi.org/10.3390/math14010008

APA Style

Chaparova, J. V., Chaparov, D. R., & Georgiev, T. G. (2026). An Analytical Study on the Synchronization of a Two-Cell Inhibitory Neural Dynamics. Mathematics, 14(1), 8. https://doi.org/10.3390/math14010008

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