Asymptotic Approximation of Fading Mode in Neurooscillator Dynamics

TL;DR

This paper develops an asymptotic approximation for the fading neuron mode in a system of delay differential equations modeling neurooscillator dynamics, revealing how solutions transition from high peaks to small amplitudes over time.

Contribution

It introduces a method to asymptotically construct the fading neuron mode in coupled delay differential equations with large parameters, extending understanding of neurooscillator behavior.

Findings

Constructed asymptotic solutions with high peaks over multiple periods.

Demonstrated the transition from high peaks to asymptotically small solutions.

Provided parameter adjustments to control the number of peaks in solutions.

Abstract

We consider a system consisting of two delay differential equations with a large parameter, modeling the association of a pair of neurooscillators. The unknown functions describe the changes in the normalized membrane potentials of neurons over time, with the large parameter characterizing the speed of electrical processes. The first equation is separated from the system and represents a generalized Hutchinson equation. This equation, as known, possesses periodic solutions with high peaks over the period. The second equation is also based on the generalized Hutchinson equation, but with an additional term, linking it to an oscillator satisfying the first equation. For the second equation, it is possible to asymptotically construct the so-called fading neuron mode, which is as follows: for any natural number $n$, one can adjust the parameters of the problem in such a way that the solution is asymptotically close to a periodic function with high peaks over $n$ periods, and then, after a transient process represented by decreasing peaks, becomes asymptotically small.