Synchronization-desynchronization transitions in complex networks: An interplay of distributed time delay and inhibitory nodes

We investigate the combined effects of distributed delay and the balance between excitatory and inhibitory nodes on the stability of synchronous oscillations in a network of coupled Stuart-Landau oscillators. To this end a symmetric network model is proposed for which the stability can be investigated analytically. It is found that beyond a critical inhibition ratio, synchronization tends to be unstable. However, increasing distributional widths can counteract this trend, leading to multiple resynchronization transitions at relatively high inhibition ratios. The extended applicability of the results is confirmed by numerical studies on asymmetrically perturbed network topologies. All investigations are performed on two distribution types, a uniform distribution and a $\Gamma$ distribution.

Figure 1

Eigenvalue spectra (dots) of the coupling matrix as a function of the inhibition ratio $\eta =m/n$ for two different coupling radii (a) $\kappa =0.25$, (b) $\kappa =0.1$. Network size $N=m+n$, with $n=200$ excitatory nodes and $m=\eta n$ inhibitory nodes. The red (gray) lines show the extremal eigenvalues in the large network limit given by Eqs. (7) and (8).

Figure 2

Amplitude $r$ of the synchronous oscillation for (a) uniformly distributed delay and (b) delay distributed by a ${\Gamma }_2$ distribution $\left(p=2\right)$. Parameters: $\lambda =0.12,\sigma =0.09,\omega =1$. Bottom right white region in (a): forbidden because of causality. Other white regions: no solution to Eqs. (16) or (17).

Figure 3

Master stability function (MSF) in the $(Re\gamma ,\rho )$ plane for uniformly distributed delay of width $\rho$ with different mean delay times $\tau$; $\gamma$ corresponds to the eigenvalues of the coupling matrix. Parameters: $\lambda =0.12,\omega =1,\sigma =0.09$. White region in (a): no solution $\left(r\in \mathbb{R}\right)$ to Eqs. (16) and (17).

Figure 4

Stable synchronization as a function of the inhibition ratio $\eta$ and the delay width $\rho$ for uniformly distributed delay in a network with coupling radius (a) $\kappa =0.25$ and (b) $\kappa =0.1$. Eigenvalue spectra (dots) and master stability function [color code (grayscale)] with (c) coupling radius ratio $\kappa =0.25$ and delay distribution width $\rho =0.5\pi$ and (d) $\kappa =0.1,\rho =0.75\pi$. Network size $N=n+m$ with $n=2000$ excitatory and $m=\eta n$ inhibitory nodes. Other parameters as described in the legend of Fig. 3. Vertical lines in (c) and (d) denote synchronization-desynchronization transitions.

Figure 5

Master stability function for a ${\Gamma }_2$ delay distribution with standard deviation $s$ in the $(Re\gamma ,s)$ plane for two different time offsets (a) ${\tau }_0=3\pi$, (b) ${\tau }_0=4\pi$. Other parameters are as described in the legend of Fig. 3.

Figure 6

Fraction of stable synchronization for asymmetric networks emerging from the rewiring process described in Sec. 6 as a function of the inhibition ratio $\eta$ for increasing numbers of repeated rewirings $R$ from top to bottom. The blue (black) dashed line marks parameters that are considered in Fig. 7. Parameters: 100 realizations, network size $N=n+m$ with $n=200$ excitatory and $m=\eta n$ inhibitory nodes, (a) $\rho =0.7\pi ,\kappa =0.25$, (b) $\rho =0.75\pi ,\kappa =0.1$, other parameters as in Fig. 3.

Figure 7

Stability of synchronization in asymmetrically perturbed networks. Color (gray) shading, master stability function; black dots, eigenvalue spectrum for a network generated by (a) $R=1000$, (b) $R=5000$, and (c) $R=10000$ rewirings. Parameters: $\eta =0.6,\kappa =0.25$ [see Fig. 6 blue (black) dashed line], $n=200,\rho =0.7\pi$, others as in Fig. 3.