Mechanism of Appearing Complex Relaxation Oscillations in a System of Two Synaptically Coupled Neurons
- PDF / 339,001 Bytes
- 17 Pages / 594 x 792 pts Page_size
- 81 Downloads / 162 Views
Journal of Mathematical Sciences, Vol. 249, No. 6, September, 2020
MECHANISM OF APPEARING COMPLEX RELAXATION OSCILLATIONS IN A SYSTEM OF TWO SYNAPTICALLY COUPLED NEURONS S. D. Glyzin ∗ P. G. Demidov Yaroslavl’ State University 14, Sovetskaya St., Yaroslavl’ 150000, Russia [email protected]
M. M. Preobrazhenskaya P. G. Demidov Yaroslavl’ State University 14, Sovetskaya St., Yaroslavl’ 150000, Russia [email protected]
UDC 517.929.8, 517.938
We consider a system of two specially coupled differential-difference equations with delay in the coupling link. We establish that the system has a set of coexisting orbitally asymptotically stable solutions with the total number 2n, n ∈ N, of bursts in the period; moreover, one of the oscillators has m bursts and the other has 2n − m bursts, m = 1, . . . , 2n − 1. From the results obtained it follows that an additional delay leads to the appearance of coexisting attractors in the system with a given number of bursts in the period. Bibliography: 20 titles. Illustrations: 2 figures.
1
Introduction
Among the methods for modeling neural systems and associations, the most popular is the replacement of the modeled system with an electric circuit with similar dynamic properties. Such models are usually called phenomenological since they realize the same qualitative properties as in the original system. Among qualitative (phenomenological) properties of neural systems, a particular important property is the ability of the system to generate oscillations the structure of which corresponds to that observed in living systems. As noted in [1], models appropriate to be used as phenomenological in neurodynamics should not only have impulse solutions, but also provide bursts of impulses for suitable values of the parameters (for the bursting effect we refer to [2, 3]). Furthermore, the coexistence of a number of stable oscillatory solutions to neurodynamic systems is possible, which can be used to model the associative memory. We consider the system of differential-difference equations ∗
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 103, 2020, pp. 71-84. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2496-0894
894
u˙ 1 = (λf (u1 (t − 1)) + bg(u2 (t − h)) ln(u∗ /u1 ))u1 ,
(1.1)
u˙ 2 = (λf (u2 (t − 1)) + bg(u1 (t − h)) ln(u∗ /u2 ))u2
modeling the synaptic coupling of a pair of impulse neurons. Here, u1 (t), u2 (t) > 0 are their normalized membrane potentials. This phenomenological model is based on the idea of fast threshold modulation [4, 5] and is a modification of the system proposed in [6], in contrast to which it contains an additional delay h > 1 in the coupling link. The necessity to take into account additional delays is emphasized in a lot of publications and is caused by delays in the conduction of impulses between nerve cells. In [7]–[9], just such delays were associated with the appearance of the multistability phenomenon in the corresponding systems. We note that each neuron in (1.1) is simulated by the singul
Data Loading...
