Relaxation Oscillations in a System of Two Pulsed Synaptically Coupled Neurons
https://doi.org/10.18255/1818-1015-2017-1-82-93
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.
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.
About the Authors
Sergei D. Glyzin
P.G. Demidov Yaroslavl State University;
Scientific Center in Chernogolovka RAS
Russian Federation
Russian Federation
Doctor, Professor, 14 Sovetskaya str., Yaroslavl 150003;
9 Lesnaya str., Chernogolovka, Moscow region, 142432
Andrey Yu. Kolesov
P.G. Demidov Yaroslavl State University
Russian Federation
Russian Federation
Doctor, Professor,
14 Sovetskaya str., Yaroslavl 150003
Elena A. Marushkina
P.G. Demidov Yaroslavl State University
Russian Federation
Russian Federation
PhD, Researcher,
14 Sovetskaya str., Yaroslavl 150003
References
1. Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “On a Method for Mathematical Modeling of Chemical Synapses”, Differential Equations, 49:10 (2013), 1193–1210.
2. Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “Relaxation self-oscillations in neuron systems: I”, Differential Equations, 47:7 (2011), 927–941.
3. [Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “Relaxation self-oscillations in neuron systems: II”, Differential Equations, 47:12 (2011), 1697–1713.
4. Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “Relaxation self-oscillations in neuron systems: III”, Differential Equations, 48:2 (2012), 159–175].
5. Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “Discrete autowaves in neural systems”, Computational Mathematics and Mathematical Physics, 52:5 (2012), 702–719].
6. Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “Self-excited relaxation oscillations in networks of impulse neurons”, Russian Math. Surveys, 70:3 (2015), 383–452].
7. Somers D., Kopell N., “Rapid synchronization through fast threshold modulation”, Biol. Cybern., 68 (1993), 393–407.
8. Somers D., Kopell N., “Anti-phase solutions in relaxation oscillators coupled through excitatory interactions”, J. Math. Biol., 33 (1995), 261–280.
9. Mishchenko E. F., Rozov N. Kh., Differentsialnye uravneniya s malym parametrom i relaksatsionnye kolebaniya, Moskva, 1975, 247 pp., (in Russian)].
10. FitzHugh R. A., “Impulses and physiological states in theoretical models of nerve membrane”, Biophysical J., 1 (1961), 445–466.
11. Terman D., “An Introduction to Dynamical Systems and Neuronal Dynamics”, Tutorials in Mathematical Biosciences I, Lecture Notes in Mathematics, 1860 (2005), 21–68.
12. Hutchinson G. E., “Circular causal systems in ecology”, Ann. N. Y. Acad. of Sci., 50 (1948), 221–246.
13. Kolesov A. Yu., Mishchenko E. F., Rozov N. Kh., “A modification of Hutchinson’s equation”, Computational Mathematics and Mathematical Physics, 50:12 (2010), 1990–2002].
14. Glyzin S. D., Kiseleva E. O., “The account of delay in a connecting element between two oscillators”, Model. Anal. Inform. Syst., 17:2 (2010), 133–143 (in Russian)].
15. Glyzin S. D., Soldatova E. A., “The factor of delay in a system of coupled oscillators FitzHugh–Nagumo”, Model. Anal. Inform. Syst., 17:3 (2010), 134–143 (in Russian).
16. Kolesov A. Yu., Mishchenko E. F., Rozov N. Kh., “Relay with delay and its C1-approximation”, Proceedings of the Steklov Institute of Mathematics, 216 (1997), 119–146.
17. Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “Modeling the Bursting Effect in Neuron Systems”, Math. Notes, 93:5 (2013), 676–690 .
18. Glyzin S. D., Marushkina E. A., “Relaxation Cycles in a Generalized Neuron Model with Two Delays”, Model. Anal. Inform. Syst., 20:6 (2013), 179–199 (in Russian).
19. Chay T. R., Rinzel J., “Bursting, beating, and chaos in an excitable membrane model”, Biophys. J., 47:3 (1985.), 357–366.
20. Ermentrout G. B., Kopell N., “Parabolic bursting in an excitable system coupled with a slow oscillation”, SIAM J. Appl. Math., 46:2 (1986), 233–253.
21. Izhikevich E., “Neural excitability, spiking and bursting”, International Journal of Bifurcation and Chaos, 10(6) (2000), 1171–1266.
22. Rabinovich M. I., Varona P., Selverston A. I., and Abarbanel H. D. I., “Dynamical principles in neuroscience”, Rev. Mod. Phys., 78 (2006), 1213–1265.
23. Coombes S., Bressloff P. C., Bursting: the genesis of rhythm in the nervous system., World Scientific Publishing Company, 2005.
Review
For citations:
Glyzin S.D., Kolesov A.Yu., Marushkina E.A. Relaxation Oscillations in a System of Two Pulsed Synaptically Coupled Neurons. Modeling and Analysis of Information Systems. 2017;24(1):82-93. (In Russ.) https://doi.org/10.18255/1818-1015-2017-1-82-93
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