Effective Desynchronization by Nonlinear Delayed Feedback

We show that nonlinear delayed feedback opens up novel means for the control of synchronization. In particular, we propose a demand-controlled method for powerful desynchronization, which does not require any time-consuming calibration. Our technique distinguishes itself by its robustness against variations of system parameters, even in strongly coupled ensembles of oscillators. We suggest our method for mild and effective deep brain stimulation in neurological diseases characterized by pathological cerebral synchronization.

Figure 1

Time course of the $\overline{X}(t)$ component of the mean field of system (1) stimulated with signal (2) in (a) and signal (3) in (d). Coupling ($C$) and stimulation ($K$) are switched on at different times: $C=K=0$ for $t<250$, $C=1$ and $K=0$ for $t\in (250\hspace{0.17em}400)$, and $C=1$ and (a) $K=3$ and (d) $K=150$ for $t>400$. In (a) the mean field is depicted for $N=100$ (blue curve) and $N=900$ (red curve). In (d) the mean field (blue curve) and the corresponding stimulation force $|S(t)|$ (red curve) are shown for $N=100$. In subplots (b) and (e) the graphs are enlarged for $t\in (700,900)$, and in (c) and (f) two trajectories $x_j$ are depicted for $t\in (700,740)$. Delay $\tau =7.5$ in (a) and $\tau =5$ in (d). The mean period of ensemble (1) $T=5$ (mean natural frequency $\overline{\omega }=2\pi /T=1.2566\dots$), the frequency distribution is Gaussian with deviation $\sigma =0.1$; $a_j=1.0$.

Figure 2

Averaged order parameter $⟨R(t)⟩$ of the stimulated ensemble (1) with $N=100$ versus parameters $K$ and $\tau$ with stimulation signal (2) in (a), and signal (3) in (b). The other parameters are the same as in Fig. 1.

Figure 3

Desynchronization mechanism of global stimulation and multistability of desynchronized states. (a) Differences between the individual averaged frequencies ${\overline{\omega }}_j$ and the natural frequencies ${\omega }_j$ for $K=150$ and for $\tau =T/2$ (green curve) and $\tau =T$ (blue and red curves, obtained for two different initial conditions). $\Omega$ is the mean frequency of a desynchronized state. In (b) and (c) the order parameter $R$ (red symbols) and the mean frequency $\Omega$ (blue symbols) obtained by simulation of $N=100$ phase oscillators (4) are shown. The values of $R(t)$ are plotted for time intervals of 40 mean periods $T$ after skipping a transient of 2000 periods. The solid curves are the theoretical prediction of $R$ and $\Omega$ from Eq. (6). In (c) the mean frequencies ${\Omega }_1$ and ${\Omega }_2$ (blue symbols) realized for two different initial conditions of system (4) correspond to red and blue curves in (a), respectively. Parameters $C=1.0$, $T=5.0$, (b) $\tau =2.5$, and (c) $\tau =5.0$.