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Abstract
We consider a scalar nonlinear differential-difference equation with two delays, which models the behavior of a single neuron. Under some additional suppositions for this equation it is applied a well-known method of quasi-normal forms. Its essence lies in the formal normalization of the Poincare – Dulac, the production of a quasi-normal form and the subsequent application of the conformity theorems. In this case, the result of the application of quasi-normal forms is a countable system of differential-difference equations, which manages to turn into a boundary value problem of the Korteweg – de Vries equation. The investigation of this boundary value problem allows to make the conclusion about the behavior of the original equation. Namely, for a suitable choice of parameters in the framework of this equation it is implemented the buffer phenomenon consisting in the presence of the bifurcation mechanism for the birth of an arbitrarily large number of stable cycles.
Highlights
Общая постановка задачиЭто уравнение впервые было введено в [1, 2] на основе подхода, описанного в книге [3], для моделирования электрической активности отдельного нейрона
which models the behavior of a single neuron
Summary
Это уравнение впервые было введено в [1, 2] на основе подхода, описанного в книге [3], для моделирования электрической активности отдельного нейрона. Как и в статье [4], предположим, что функции f (u), g(u) удовлетворяют следующим трём условиям. Что f (u) < 0, g (u) > 0 ∀u ∈ R+ и, кроме того, выполняются равенства f (0) = 1, g(0) = 0, lim f (u) = 0, lim g(u) = 1. Монотонность функций f (u) и g(u) обеспечивает существование и единственность решения u = u0(a, b) > 0 уравнения (a + 1)f (u) − a − bg(u) = 0. Что параметр b меняется в окрестности критического значения b = b∗, о котором говорится в условии 2
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