Relaxation Cycles in a Model of Synaptically Interacting Oscillators
https://doi.org/10.18255/1818-1015-2017-2-186-204
Abstract
In this paper the mathematical model of a neural network with a ring synaptic interaction elements is considered. The model is a system of scalar nonlinear differential-difference equations, the right parts of which depend on a large parameter. The unknown functions included in the system characterize the membrane potentials of the neurons. The search of relaxation cycles within the system of equations is interested. To this end solutions of the task are finded in the form of discrete traveling waves. It allows to research a scalar nonlinear differential-difference equations with two delays instead of system. Further, a limit a object that represents a relay equation with two delays is defined by large parameter tends to infinity. There are six cases of restrictions on the parameters. In every case exist alone periodic solution of relay equation started from initial function from suitable function class. It is structurally proved by using the step method. Next, the existence of a relaxation periodic solutions of a singularly perturbed equation with two delays is proved by using Poincare operator and Schauder principle. The asymptotics of this solution is constructed, and then it is proved that the solution is close to decision of the relay equation. Because of the exponential estimate Frechet derivative of the Poincare operator implies the uniqueness and stability of solutions of differential-difference equation with two delays.
About the Author
Margarita M. Preobrazhenskaia
P.G. Demidov Yaroslavl State University
Scientific Center in Chernogolovka RAS
Russian Federation
Russian Federation
14 Sovetskaya str., Yaroslavl 150003, Russia
9 Lesnaya str., Chernogolovka, Moscow region 142432, Russia
References
1. Глызин С.Д., Колесов А.Ю, Розов Н.Х., “Об одном способе математического моделирования химических синапсов”, Дифференциальные уравнения, 49:10 (2013), 1227– 1244; [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. Somers D., Kopell N., “Rapid synchronization through fast threshold modulation”, Biol. Cybern., 68 (1993), 393–407.
3. Somers D., Kopell N., “Anti-phase solutions in relaxation oscillators coupled through excitatory interactions”, J. Math. Biol., 33 (1995), 261–280.
4. Izhikevich E. M., Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, MIT Press, 2010.
5. FitzHugh R. A., “Impulses and physiological states in theoretical models of nerve membrane”, Biophysical J., 1 (1961), 445–466.
6. Terman D., “An Introduction to Dynamical Systems and Neuronal Dynamics”, Tutorials in Mathematical Biosciences I, Lecture Notes in Mathematics, 1860 (2005), 21–68.
7. Hutchinson G. E., “Circular causal systems in ecology”, Ann. N. Y. Acad. of Sci., 50 (1948), 221–246.
8. Колесов А.Ю, Мищенко Е.Ф., Розов Н.Х., “Реле с запаздыванием и его C1-аппроксимация”, Тр. Мат. ин-та им. В.А. Стеклова РАН, 216 (1997), 126–153; [Kolesov A. Yu., Mishchenko E. F., Rozov N. Kh., “Relay with delay and its C1-approxi- mation”, Proceedings of the Steklov Institute of Mathematics, 216 (1997), 119–146].
9. Глызин С.Д., Колесов А.Ю, Розов Н.Х., “Релаксационные автоколебания в нейронных системах. I”, Дифференциальные уравнения, 47:7 (2011), 919–932; [Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “Relaxation self-oscillations in neuron systems: I”, Differential Equations, 47:7 (2011), 927–941].
10. Глызин С.Д., Колесов А.Ю, Розов Н.Х., “Релаксационные автоколебания в нейронных системах. II”, Дифференциальные уравнения, 47:12 (2011), 1675–1692; [Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “Relaxation self-oscillations in neuron systems: II”, Differential Equations, 47:12 (2011), 1697–1713].
11. Глызин С.Д., Колесов А.Ю, Розов Н.Х., “Релаксационные автоколебания в нейронных системах. III”, Дифференц. уравнения, 48:2 (2012), 155–170; [Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “Relaxation self-oscillations in neuron systems: III”, Differential Equations, 48:2 (2012), 159–175].
12. Глызин С.Д., Колесов А.Ю, Розов Н.Х., “Дискретные автоволны в нейронных системах”, ЖВМ и МФ, 52:5 (2012), 840–858; [Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “Discrete autowaves in neural systems”, Computational Mathematics and Mathematical Physics, 52:5 (2012), 702–719].
13. Колесов А.Ю, Мищенко Е.Ф., Розов Н.Х., “Об одной модификации уравнения Хатчинсона”, ЖВМ и МФ, 50:12 (2010), 2099–2112; [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. Преображенская М.М., “Существование и устойчивость релаксационных циклов в нейродинамической модели с двумя запаздываниями”, Вестник НИЯУ МИФИ, 5:4 (2016), 351–366; [Preobrazhenskaia M. M., “Existence and stability of relaxation cycles in a neurodynamic model with two delays”, Vestnik NIYaU MIFI, 5:4 (2016), 351–366].
15. Глызин С.Д., Колесов А.Ю, Розов Н.Х., “Релаксационные автоколебания в сетях импульсных нейронов”, УМН, 70:3(423) (2015), 3–76; [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].
16. Глызин С.Д., Колесов А.Ю, Розов Н.Х., “Релаксационные автоколебания в сетях Хопфилда с запаздыванием”, Изв. РАН. Сер. матем., 77:2 (2013), 53–96; [Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “Relaxation self-oscillations in Hopfield networks with delay”, Izvestiya: Mathematics, 77:2 (2013), 271–312].
Review
For citations:
Preobrazhenskaia M.M. Relaxation Cycles in a Model of Synaptically Interacting Oscillators. Modeling and Analysis of Information Systems. 2017;24(2):186-204. (In Russ.) https://doi.org/10.18255/1818-1015-2017-2-186-204
Views:
1092
Email this article 