Incoherence-Mediated Remote Synchronization

In previously identified forms of remote synchronization between two nodes, the intermediate portion of the network connecting the two nodes is not synchronized with them but generally exhibits some coherent dynamics. Here we report on a network phenomenon we call incoherence-mediated remote synchronization (IMRS), in which two noncontiguous parts of the network are identically synchronized while the dynamics of the intermediate part is statistically and information-theoretically incoherent. We identify mirror symmetry in the network structure as a mechanism allowing for such behavior, and show that IMRS is robust against dynamical noise as well as against parameter changes. IMRS may underlie neuronal information processing and potentially lead to network solutions for encryption key distribution and secure communication.

Figure 1

Remote synchronization between node groups $\mathsf{A}$ and $\mathsf{C}$ mediated by incoherence in group $\mathsf{B}$. The colors of the nodes schematically represent their states, indicating that nodes 1 and $N$ are identically synchronized, while the dynamics of the nodes in $\mathsf{B}$ are incoherent.

Figure 2

Network exhibiting IMRS. (a) Mirror-symmetric structure of the network, generated with $n_{\mathsf{A}}=6$, $n_{\mathsf{B}}=30$, $n_{\mathsf{B}}^{\prime }=2$, and $p=0.8$. (b) Pairwise cross correlation $C_{i,j}$. (c) Pairwise mutual information $I_{i,j}$. (d)–(f) Phase variable ${\theta }_i$ as a function of $t$. In (d), only the blue curve is clearly visible because the two curves overlap. The calculations in (b)–(f) are based on iterating Eq. (2) with $\beta =1.5$ and $\sigma =1.5$.

Figure 3

Characterizing the network dynamics. (a) Lyapunov exponent ${\lambda }_{\mathrm{iso}}$ of the isolated node dynamics as a function of self-feedback strength $\beta$. (b) Synchronization stability ${\lambda }_{{\mathsf{C}}_1}$ of cluster ${\mathsf{C}}_1$ (and thus between nodes 1 and $N$) as a function of $\beta$ and coupling strength $\sigma$. (c) Lyapunov exponent $\lambda$ measuring the instability parallel to the synchronization manifold as a function of $\beta$ and $\sigma$. The exponents ${\lambda }_{{\mathsf{C}}_1}$ and $\lambda$ are averaged over ten network realizations and ten initial conditions.

Figure 4

Influence of system parameters on IMRS. (a)–(e) The distribution of the correlation $C_{i,j}$ between pairs of nodes in $\mathsf{B}$ is shown as a function of parameters $n_{\mathsf{B}}$, $n_{\mathsf{B}}^{\prime }$, $n_{\mathsf{A}}$, $\beta$, and $\sigma$. Each panel shows the median (solid curve with dots), the range between the minimum and maximum (shaded region), and the 25th and 75th percentiles (dashed curves). These quantities are all averaged over ten network realizations and ten initial conditions. Unless noted otherwise, all parameters are set to the values used in Fig. 2 (indicated by red dots).