Basin stability for burst synchronization in small-world networks of chaotic slow-fast oscillators

The impact of connectivity and individual dynamics on the basin stability of the burst synchronization regime in small-world networks consisting of chaotic slow-fast oscillators is studied. It is shown that there are rewiring probabilities corresponding to the largest basin stabilities, which uncovers a reason for finding small-world topologies in real neuronal networks. The impact of coupling density and strength as well as the nodal parameters of relaxation or excitability are studied. Dynamic mechanisms are uncovered that most strongly influence basin stability of the burst synchronization regime.

Figure 1

(a) Chaotic attractor of the single map (1) for $a=0.1, \beta =0.3, d=0.45, \varepsilon =0.001, J=0.1$, and $I=0$. (b) Corresponding spike-burst oscillations.

Figure 2

Properties of chaotic spike-burst oscillations generated by Eq. (1): (a, b) their average period, (c, d) average duration of the active phase, and (e, f) average number of spikes per burst, depending on (a, c, e) $\varepsilon$ (for $J=0.1$) and (b, d, f) $J$ (for $\varepsilon =0.001$). The other parameters are $a=0.1, \beta =0.3, d=0.45$, and $I=0$.

Figure 3

Burst oscillations in small-world networks of $N=50$ nodes for different rewiring probabilities: (a) $P_{\text{rew}}=0$; (b) $P_{\text{rew}}=0.3$. The other parameters are $c=1, k=11, J_0=0.1$, and $\delta J=0.015$.

Figure 4

Distribution of $\sigma$ values for different $\varepsilon$ values. The parameters are $N=50, P_{\text{rew}}=0.3, k=11, c=1, J_0=0.1$, and $\delta J=0.01$.

Figure 5

Basin stability versus rewiring probability $P_{\text{rew}}$ for different values of $\varepsilon$. The parameters are $N=50, k=11, c=1, J_0=0.1$, and $\delta J=0.01$.

Figure 6

Basin stability versus (a) degree $k$ (for $c=1$) and (b) coupling strength $c$ (for $k=11$) for different values of $\varepsilon$. The parameters are $N=50, P_{\mathrm{rew}}=0.3, J_0=0.1$, and $\delta J=0.01$.

Figure 7

Basin stability versus (a) $N$ and $k$ (for $c=1$) and (b) $N$ and $c$ (for $k=0.2\times N$). The parameters are $J_0=0.1, \delta J=0.01, \varepsilon ={10}^{-3}, P_{\mathrm{rew}}=0.3$.

Figure 8

Probability distribution ${\rho }_{x,y}$ in phase plane $(x,y)$ of the trajectories $(x_{\mathrm{mean}},y_{\mathrm{mean}})$ averaged over the network for (a) $P_{\mathrm{rew}}=0$ and (b) $P_{\mathrm{rew}}=0.3$ (for $N=50, k=11, c=1, J_0=0.1$, and $\delta J=0.01$). Projections of these distributions on line (c) $x=0.2$, (d) $x=0.3$, and (e) $x=0.4$.

Figure 9

Basin stability versus $\varepsilon$. The influence of ${\sigma }_{th}$ and network size $N$ are shown. For the case of $N=50, k=11$; for the case of $N=100, k=22$. In all the cases $P_{\mathrm{rew}}=0.3, c=1, J_0=0.1$, and $\delta J=0.01$.

Figure 10

Basin stability versus (a) $J_0$ (for $\delta J=0.01$) and (b) $\delta J$ (for $J_0=0.1$) for different values of $\varepsilon$. The parameters are $N=50, P_{\mathrm{rew}}=0.3, k=11, c=1$.

Figure 11

(a, b) Basin stability, (c, d) average number of spikes per burst, and (e, f) average duration of the active phase versus $J_0$ for different values of $\varepsilon$: $\varepsilon =0.004$, left, and $\varepsilon =0.005$, right. The parameters are $N=50, P_{\mathrm{rew}}=0.3, k=11, c=1$, and $\delta J=0.01$.

Figure 12

Basin stability versus $\varepsilon$ and $J_0$. The parameters are $N=50, P_{\mathrm{rew}}=0.3, k=11, c=1$, and $\delta J=0.01$. The solid curves divide the plane into areas with different numbers of spikes per bursts indicated on the plot by $1\cdots 10$.