Blinking coupling enhances network synchronization

This paper studies the synchronization of a network with linear diffusive coupling, which blinks between the variables periodically. The synchronization of the blinking network in the case of sufficiently fast blinking is analyzed by showing that the stability of the synchronous solution depends only on the averaged coupling and not on the instantaneous coupling. To illustrate the effect of the blinking period on the network synchronization, the Hindmarsh-Rose model is used as the dynamics of nodes. The synchronization is investigated by considering constant single-variable coupling, averaged coupling, and blinking coupling through a linear stability analysis. It is observed that by decreasing the blinking period, the required coupling strength for synchrony is reduced. It equals that of the averaged coupling model times the number of variables. However, in the averaged coupling, all variables participate in the coupling, while in the blinking model only one variable is coupled at any time. Therefore, the blinking coupling leads to an enhanced synchronization in comparison with the single-variable coupling. Numerical simulations of the average synchronization error of the network confirm the results obtained from the linear stability analysis.

Figure 1

(a) Time series of the membrane potential $x_1$ of the single Hindmarsh-Rose neuron model [Eq. (11)] that exhibits chaotic bursting. The parameters are set to $I_{\mathrm{ext}}=3.2, r=0.006, s=4$. (b) Synchronous time series of two neurons coupled through $x_1$ variables and coupling strength $\sigma =1$. The blue and red colors represent the time series of the membrane potential of the first and second neurons $x_{11}$ and $x_{21}$, respectively. The parameters are as in (a).

Figure 2

The largest Lyapunov exponent $\mathrm{\Lambda }$ of the variational Eq. (5) for $\mu =-2$ with constant coupling function ($H$) in dependence on the coupling strength $\sigma$. (a) Coupling in $x_1$ variables, (b) coupling in $x_2$ variables, (c) coupling in $x_3$ variables, (d) averaged network. Parameters as in Fig. 1.

Figure 3

The largest Lyapunov exponent ($\mathrm{\Lambda }(\mu =-2)$) of Eq. (3) vs the coupling strength ($\sigma$) for different blinking periods $\tau$. (Blue) $\tau =T=30$, (red) $\tau =T/5=6$, (orange) $\tau =T/10=3$, (purple) $\tau =T/100=0.3$, (green) $\tau =T/1000=0.03$. For faster blinking, the largest Lyapunov exponent is the same as the green line with threshold $\sigma =0.021$. Other parameters as in Fig. 1.

Figure 4

The synchronization error vs the coupling strength $\sigma$ with a constant coupling in (a) $x_1$ variables, (b) $x_2$ variables, (c) $x_3$ variables. Parameters as in Fig. 1.

Figure 5

The synchronization error vs coupling strength $\sigma$ with blinking coupling for different blinking periods $\tau$. (a) $\tau =T=30$, (b) $\tau =T/5=6$, (c) $\tau =T/10=3$, (d) $\tau =T/100=0.3$, (e) $\tau =T/1000=0.03$, (f) $\tau =T/10000=0.003$. Other parameters as in Fig. 1.

Figure 6

The region of stable synchronization in the parameter plane of coupling strength $\sigma$ and blinking period $\tau$. (a) The largest Lyapunov exponent of the linearized system [Eq. (5)] for $\mu =-2$; the asynchronous and synchronous states' regions are separated by the black curve ($\mathrm{\Lambda }=0$). (b) Numerically calculated synchronization error. Parameters as in Fig. 1.