Synchronization of oscillators through time-shifted common inputs

Shared upstream dynamical processes are frequently the source of common inputs in various physical and biological systems. However, due to finite signal transmission speeds and differences in the distance to the source, time shifts between otherwise common inputs are unavoidable. Since common inputs can be a source of correlation between the elements of multi-unit dynamical systems, regardless of whether these elements are directly connected with one another or not, it is of importance to understand their impact on synchronization. As a canonical model that is representative for a variety of different dynamical systems, we study limit-cycle oscillators that are driven by stochastic time-shifted common inputs. We show that if the oscillators are coupled, time shifts in stochastic common inputs do not simply shift the distribution of the phase differences, but rather the distribution actually changes as a result. The best synchronization is therefore achieved at a precise intermediate value of the time shift, which is due to a resonance-like effect with the most probable phase difference that is determined by the deterministic dynamics.

Figure 1

(a) Phase difference distribution function of two identical uncoupled oscillators, receiving a fully correlated Gaussian white noise with different time shifts shown above the plots. (b) Same results for nonidentical oscillators with the frequency mismatch $\mathrm{\Delta }\omega =0.5$. Here $D=0.5$, and ${\tau }_s$ is $-2,-1,0,1,2$ from left to right.

Figure 2

(a) Phase difference distribution function of two coupled oscillators in the locked mode, receiving a fully correlated Gaussian white noise with different time shifts shown above each plot. (b) Same results for oscillators in the running mode. The results shown by gray bar plots are calculated by numeric integration of Eq. (3), and the solid lines show the analytical result given by Eq. (6). The vertical dashed lines in (a) show the fixed point of Eq. (3). Other parameters are $D=0.5$ and $\mathrm{\Delta }\omega =0.2$ for the locked case, and $\mathrm{\Delta }\omega =-0.1$ for the running case.

Figure 3

(a) Synchronization index versus input time shift for the case of two identical uncoupled phase oscillators (black dashed line), nonidentical uncoupled phase oscillators (solid black line), coupled phase oscillators in phase-locked mode (solid gray line), and nonidentical coupled phase oscillators in the running mode (gray dashed line). (b) Most probable phase lag which shows the location of the peak of the phase lag distributions in Figs. 1 and 2 is plotted versus input time shift.