Zero-Lag Long-Range Synchronization via Dynamical Relaying

We show that isochronous synchronization between two delay-coupled oscillators can be achieved by relaying the dynamics via a third mediating element, which surprisingly lags behind the synchronized outer elements. The zero-lag synchronization thus obtained is robust over a considerable parameter range. We substantiate our claims with experimental and numerical evidence of such synchronization solutions in a chain of three coupled semiconductor lasers with long interelement coupling delays. The generality of the mechanism is validated in a neuronal model with the same coupling architecture. Thus, our results show that zero-lag synchronized chaotic dynamical states can occur over long distances through relaying, without restriction by the amount of delay.

Figure 1

Experimental setup. A central laser (LD2) exchanges information between the other two (LD1 and LD3). The coupling times between the two outer and the central laser agree.

Figure 2

(a), (b), (c) Time series (in pairs) of the output intensity of the lasers, for the case of a central laser with negative detuning $\Omega ={\omega }_2-{\omega }_{1,3}=-4.1\mathrm{GHz}$. (d), (e), (f) Cross-correlation functions of the corresponding time series. The time series of the central laser have been shifted ${\tau }_c$ to allow an easier comparison.

Figure 3

(a), (b), (c) Numerical time series (in pairs) of the output intensity of the lasers, for the case of zero detuning between the three lasers. (d), (e), (f) Cross-correlation functions of the corresponding time series. The time series of the central laser have been shifted by ${\tau }_c$ to facilitate comparison.

Figure 4

Synchronization of three bidirectionally coupled thermoreceptor neurons. The left column shows the time series of the three neurons, and the right column the corresponding cross-correlation functions. The maxima of the cross correlation for panels (e) and (f) is about 20 ms, which roughly corresponds to the coupling time. Following the notation of Ref. [18], the parameters are $g_{\mathrm{Na}}=1.5\mu \mathrm{S}/{\mathrm{cm}}^2$, $g_{\mathrm{K}}=2\mu \mathrm{S}/{\mathrm{cm}}^2$, $g_{\mathrm{sd}}=0.25\mu \mathrm{S}/{\mathrm{cm}}^2$, $g_{\mathrm{sr}}=0.4\mu \mathrm{S}/{\mathrm{cm}}^2$, $g_l=0.1\mu \mathrm{S}/{\mathrm{cm}}^2$, $g_{\mathrm{syn}}=0.15\mu \mathrm{S}/{\mathrm{cm}}^2$, $V_{\mathrm{Na}}=50\mathrm{mV}$, $V_{\mathrm{K}}=-90\mathrm{mV}$, $V_{\mathrm{sd}}=50\mathrm{mV}$, $V_{\mathrm{sr}}=-90\mathrm{mV}$, $V_l=-60\mathrm{mV}$, $V_{\mathrm{syn}}=0\mathrm{mV}$, ${\tau }_{\mathrm{Na}}=0\mathrm{ms}$, ${\tau }_{\mathrm{K}}=2.0\mathrm{ms}$, ${\tau }_{\mathrm{sd}}=10.0\mathrm{ms}$, ${\tau }_{\mathrm{sr}}=20.0\mathrm{ms}$, ${\tau }_{\mathrm{syn}}=5\mathrm{ms}$, ${\tau }_c=15\mathrm{ms}$, $g_i^{\mathrm{syn}}=-0.13{\mathrm{ms}}^{-1}$, $C_m=1\mu \mathrm{F}/{\mathrm{cm}}^2$, $\alpha =0.012\mu \mathrm{A}$, and $\beta =0.17$.