The impact of the delay in coupling on the behaviour of dynamical systems

Abstract

The study of the impact of delay in coupling on the behavior of dynamical systems is currently a very important problem [1-4]. In real systems, the presence of a delay is inevitable; therefore, their studies are important from both theoretical and applied points of view. Large values of the delay values (for example, comparable to the period of oscillations of a neuron) are of particular interest, since they describe the interaction of real neurons [4]. The results of such studies can be further applied, for example, in the study of processes occurring in the human brain.
In the framework of the presented work, the FitzHugh-Nagumo oscillator is considered, which is one of the simplest models of neurons and is widely used in numerical simulation. The cases of a system of two coupled FitzHugh-Nagumo oscillators in the excitable mode and a ring of symmetrically and locally coupled FitzHugh-Nagumo oscillators in the presence of a delay in the coupling are studied in detail. The influence of the delay time and the strength of the coupling between oscillators on the process and threshold of ignition (excitation) of neurons and the transmission of impulses between connected oscillators, as well as on the formation of spatiotemporal structures in ensembles, is analyzed in detail. Numerical calculations are carried out for various cases of the initial states of the oscillators.
To compare the results obtained, we also present data on the influence of the delay time on the dynamics of the system of coupled van der Pol oscillators, which in the oscillatory mode exhibit behavior similar to the FitzHugh-Nagumo neuron model and can demonstrate similar effects.
The study was supported by a grant from the Russian Science Foundation (project no. 20-12-00119).

1. Zhou, J., Xiang, L., & Liu, Z. (2007). Global synchronization in general complex delayed dynamical networks and its applications. Physica A: Statistical Mechanics and its Applications, 385(2), 729-742. doi:10.1016/j.physa.2007.07.006
2. Perlikowski, P., Yanchuk, S., Popovych, O. V., & Tass, P. A. (2010). Periodic patterns in a ring of delay-coupled oscillators. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 82(3) doi:10.1103/PhysRevE.82.036208
3. Hauschildt, B., Janson, N. B., Balanov, A., & Schöll, E. (2006). Noise-induced cooperative dynamics and its control in coupled neuron models. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 74(5) doi:10.1103/PhysRevE.74.051906
4. Yamakou, M. E., & Jost, J. (2019). Control of coherence resonance by self-induced stochastic resonance in a multiplex neural network. Physical Review E, 100(2) doi:10.1103/PhysRevE.100.022313

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Bukh Andrei
Saratov State University
Russia

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