Dynamical Mechanism of Anticipating Synchronization in Excitable Systems

We analyze the phenomenon of anticipating synchronization of two excitable systems with unidirectional delayed coupling which are subject to the same external forcing. We demonstrate for different paradigms of excitable system that, due to the coupling, the excitability threshold for the slave system is always lower than that for the master. As a consequence the two systems respond to a common external forcing with different response times. This allows us to explain in a simple way the mechanism behind the phenomenon of anticipating synchronization in excitable systems.

Figure 1

Time series of the master system $x$ (solid line) and slave system $y$ (dashed line) subjected to white Gaussian noise of zero-mean and correlations $\langle \xi (t)\xi (t^{\prime })\rangle =D\delta (t-t^{\prime })$, obtained by numerical simulation of Eqs. (4, 5). Other parameters are: $\mu =0.95$, $K=0.01$, and $\tau =1$. The noise intensity is $D=0.017$.

Figure 2

(a) Response time $t_r$ versus $\mu$ for the Adler system $x$ perturbed by $p_0\delta (t)$ with $p_0=2$ from Eq. (6). (b) Time series for $x(t)$ for $\mu =0.95$ (solid line) and $\mu =0.97$ (dashed line). Both systems have been perturbed at $t_0=10$ by a pulse of constant amplitude $p=1.7$ and duration $\Delta t=0.4$.

Figure 3

Response of the master (solid line) and slave (dashed line) for three different amplitudes of the singular perturbation of duration $\Delta t=0.4$ at time $t_0=10$: (a) $p=1.7$, (b) $p=1.65$, and (c) $p=1.61$. Other parameters are $\mu =0.95$, $\tau =5$, and $K=0.01$.

Figure 4

The ratio between the slave and the master excitability threshold as a function of $K$ for ${\tau }_1=0.05$, ${\tau }_2=0.2$, ${\tau }_3=0.35$, and ${\tau }_4=0.5$. Considered systems have parameter $\mu =0.95$. Perturbation is applied at time $t_0=10$ with magnitude $p=1.635$ and duration $\Delta t=0.4$. The dashed line corresponds to the constant excitability threshold of the master.

Figure 5

Two coupled systems (master and slave) with a coupling parameter $K=0.01$ and delay time (a) $\tau =1$, (b) $\tau =5$, and (c) $\tau =50$. Both systems have $\mu =0.95$ and are perturbed at time $t_0=60$ with a pulse of magnitude $p=1.7$ and duration $\Delta t=0.4$.

Figure 6

Time series for the variable $x_1$ of the FitzHugh-Nagumo system for $a=-1.01$ (dashed line) and $a=-1.08$ (solid line). In both cases it is $ϵ=0.09$. As indicated by the vertical dotted line, the system is perturbed at a time $t_0=200$ by a pulse of amplitude $p=0.4$ and duration $\Delta t=1$.

Figure 7

Response of the master ($x_1$, solid line) and slave ($y_1$, dashed line) for two coupled FitzHugh-Nagumo systems with $a=-1.01$, $ϵ=0.09$, $\tau =4$, and $K=0.1$, after perturbation at $t_0=200$ by a pulse of amplitude $p$ and duration $\Delta t=1$. (a) For large amplitude $p=0.4$ both systems pulse, whereas (b) for the smaller amplitude $p=0.3$ there is only pulse in the slave variable.