\[fig:isi\_tau\]. Other parameters: $a=1.05$, $\epsilon_1=0.005$, $\epsilon_2=0.1$, $\tau=0$, and $D_2=0.09$.](fig04b "fig:"){width="0.5\linewidth"}

Control of synchronization by time-delayed feedback
===================================================

In this Section, we consider the control of global cooperative dynamics by local application of a stimulus to a single system. This stimulus is realized by time-delayed feedback, which was initially introduced by Pyragas in order to stabilize periodic orbits in deterministic systems [@PYR92]:

$$\begin{aligned}
    \label{eq:2FHN_1}
    \epsilon_1 \dot{x}_1 &=& x_1 -\frac{x_1^3}{3} -y_1 + C [x_2(t) -x_1(t)]\\
    \dot{y}_1 &=& x_1 + a + K \left[y_1(t-\tau_K) - y_1(t) \right] + D_1 \xi_1(t) \nonumber\\
    \label{eq:2FHN_2}
    \epsilon_2 \dot{x}_2 &=& x_2 -\frac{x_2^3}{3} -y_2 + C [x_1(t) -x_2(t)]\\
    \dot{y}_2 &=& x_2 + a + D_2 \xi_2(t). \nonumber \end{aligned}$$

The parameters of the time-delayed feedback scheme are the feedback gain $K$ and the time delay $\tau_K$. With this method, a control force is constructed from the differences of the states of the system which are one time unit $\tau_K$ apart. One could also consider application of the feedback scheme to both subsystems and effects of different values of the control parameters for each subsystem, but these investigations are beyond of the scope of this work. Previously, time-delayed feedback has also been used to influence noise-induced oscillations of a single excitable 