Abstract
In this paper, we consider a mathematical model of synaptic interaction between two pulse neuron elements. Each of the neurons is modeled by a singularly perturbed difference-differential equation with delay. Coupling is assumed to be at the threshold with the time delay being taken into account. The problems of existence and stability of relaxation periodic movements for the systems derived are considered. It turns out that the critical parameter is the ratio between the delay caused by internal factors in the single-neuron model and the delay in the coupling link between the oscillators. The existence and stability of a uniform cycle for the problem are proved in the case where the delay in the link is less than the period of a single oscillator, which depends on the internal delay. As the delay grows, the in-phase regime becomes more complex; specifically, it is shown that, by choosing an adequate delay, we can obtain more complex relaxation oscillations and, during a period, the system can exhibit more than one high-amplitude splash. This means that the bursting effect can appear in a system of two synaptically coupled neuron-type oscillators due to the delay in the coupling link.
Highlights
We consider a mathematical model of synaptic interaction between two pulse neuron elements
Each of the neurons is modeled by a singularly-perturbed difference-differential equation with delay
Coupling is assumed to be at the threshold
Summary
Рассматривается новый подход к моделированию химических синапсов, сформулированный в статье [1]. Характерная особенность этого способа состоит в том, что правые части соответствующих дифференциальных уравнений меняются скачкообразно при переходе некоторых управляющих переменных через свои критические значения. Предположим, что напряжение u = u(t) и сила тока v = v(t) в отдельной нейронной клетке удовлетворяют системе дифференциальных уравнений εu = f (u, v), v = g(u, v). А именно, будем считать, что 0 при x < 0, sj(uj) = H(uj − u∗∗), H(x) = 1 при x > 0,. U 2 = λf (u2(t − 1))u2 + b s1(u1(t − h))(u∗ − u2), где функции s1, s2 заданы равенствами (3), а положительный параметр h отвечает за запаздывание в цепи связи между осцилляторами. А отсюда, в свою очередь, следует, что при ε → 0 система (12) переходит в релейную систему x 1 = R(x1(t − 1)) + b (c − x1)H(x2(t − h)),.
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