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Interplay of time-delayed feedback control and temporally correlated noise in excitable systems. (English) Zbl 1197.93081

The subject of the paper is the FitzHugh-Nagumo model as well as a system of two coupled FitzHugh-Nagumo models with a delayed feedback control
\[ \begin{aligned} \varepsilon_1 \frac{dx_1(t)}{dt}&= x_1(t)-\frac{x_1^3(t)}{3}-y_1(t) + C(x_2(t)-x_1(t)),\\ \frac{dy_1(t)}{dt}&= x_1(t)+a+\eta(t) + K(y_1(t-\tau)-y_1(t)),\\ \varepsilon_2 \frac{dx_2(t)}{dt}&= x_1(t)-\frac{x_2^3(t)}{3}-y_2(t) + C(x_1(t)-x_2(t)),\\ \frac{dy_2(t)}{dt}&= x_2(t)+a + D_2 \xi(t), \end{aligned} \]
where \(\varepsilon_1\) and \(\varepsilon_2\) are small parameters, \(C\) is the coupling constant, \(\xi(t)\) and \(\eta(t)\) are white and colored noise, respectively, \(x_1,y_1,x_2,y_2\in \mathbb R\) are variables.
The main purpose of the paper is study the combined effects of the delayed feedback and the noise on the dynamics. In order to achieve this goal, a systematic numeric analysis of the influence of the noise correlation time, noise intensity, time-delay value \(\tau\) and the feedback strength \(C\) on the dynamical behavior of the system has been performed. The main qualitative indicators, which have been monitored are as follows: an index, which measures the synchronization between two subsystems, average values and fluctuations of inter-spike intervals, correlation time.

MSC:

93B52 Feedback control
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34K50 Stochastic functional-differential equations
34K23 Complex (chaotic) behavior of solutions to functional-differential equations
34F05 Ordinary differential equations and systems with randomness
34K35 Control problems for functional-differential equations
92C20 Neural biology
93C15 Control/observation systems governed by ordinary differential equations
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