Inhibition-induced explosive synchronization in multiplex networks

To date, explosive synchronization (ES) in a network is shown to be originated from considering either degree-frequency correlation, frequency-coupling strength correlation, inertia, or adaptively controlled phase oscillators. Here we show that ES is a generic phenomenon and can occur in any network by multiplexing it with an appropriate layer without even considering any prerequisite for the emergence of ES. We devise a technique which leads to the occurrence of ES with hysteresis loop in a network upon its multiplexing with a negatively coupled (or inhibitory) layer. The impact of various structural properties of positively coupled (or excitatory) and inhibitory layers along with the strength of multiplexing in gaining control over the induced ES transition is discussed. Analytical prediction for the spread of phase distribution of each layer is provided, which is in good agreement with the numerical assessment. This investigation is a step forward in highlighting the importance of multiplex framework not only in bringing phenomena which are not possible in an isolated network but also in providing more structural control over the induced phenomena.

Figure 1

Schematic diagram for a single layer network where nodes are positively coupled, depicting a smooth second order transition to synchrony (top). Multiplex network in which the same nodes are positively and negatively coupled separately in two different layers, called excitatory and inhibitory, respectively (bottom). Parameter $r$ depicts the degree of coherence as a function of coupling strength $\lambda$ as defined in Eq. (3). The positively coupled layer shows ES transition with hysteresis upon multiplexing with a negative layer.

Figure 2

Order parameters illustrating continuous transition as a function of coupling strength in a positively coupled GC network (a) in its isolation and (b) when it is multiplexed with another positively coupled regular layer. Upon multiplexing with a negatively coupled (${\lambda }_{-}=2$) regular layer, (c) it exhibits ES transition with a hysteresis (d) while the inhibitory layer shows the first-order jump to another asynchronous state.

Figure 3

Effective frequencies, order parameters, and $r^{\mathrm{link}}$ of both the excitatory and inhibitory layers as a function of ${\lambda }_{+}$ for different values of inhibitory coupling strength ${\lambda }_{-}$. The number of nodes in each layer $N=200$, $D_x=2$, and $\langle k_2\rangle =10$.

Figure 4

Order parameter $r_1$ of the excitatory layer (for $N=50$ nodes, and $D_x=2$) in forward and backward transitions, for different values of inhibitory coupling strength, demonstrating hysteresis curves.

Figure 5

Order parameter $r$ (square) and $r^{\mathrm{link}}$ (circle) of the multiplex network as a function of ${\lambda }_{+}$ for different values of ${\lambda }_{-}$. Squares and circles in red, green, and blue colors correspond to ${\lambda }_{-}=-0.5,-0.8$, and $-1$, respectively. The results are included for $N=200$ nodes in each layer.

Figure 6

Time series of the phases of the nodes in the excitatory (${\theta }_1$) and inhibitory (${\theta }_2$) layers before [(a) and (b)] and after [(c) and (d)] the onset of ES. Time series are obtained for $N=50$ nodes in each layer interacting under the impression of ${\lambda }_{-}=-0.5$ and $D_x=2$.

Figure 7

Radar plots of the excitatory GC (${\theta }_1$) and the inhibitory regular ring (${\theta }_2$) phases at two different instants $t_1=1$ [(a) and (c)] and $t_2=500$ [(b) and (d)] before [(a) and (b)] and after [(c) and (d)] the triggering of ES. The radar plots are presented for $D_x=2$, ${\lambda }_{-}=-1$, and $N=50$ nodes in each layer with natural frequencies uniformly drawn in the range 0 and 1. All ${\theta }_1$ (${\theta }_2$) rotate with constant frequency maintaining the angular distances with each other over time.

Figure 8

Binned excitatory ${\theta }_1$ (top row panels) and inhibitory ${\theta }_2$ (third row panels) phases with each increase in ${\lambda }_{+}$, associated with inhibitory coupling strength ${\lambda }_{-}=-0.5$ (left panels), $-1$ (middle panels), and $-2$ (right panels). Second and bottom row panels depict distributions of the excitatory and inhibitory phases, respectively, before (${\lambda }_{+}<{\lambda }_c$) and after (${\lambda }_{+}>{\lambda }_c$) the onset of ES. Results are presented for $N=500$ nodes in each layer and $D_x=2$. ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ stand for the gap between peaks for the excitatory and inhibitory layers, respectively.

Figure 9

Separation [Eq. (11)] ${\mathrm{\Delta }}_1$ (blue color) and ${\mathrm{\Delta }}_2$ (red color) between two bimodal peaks in the excitatory, and inhibitory phase distributions, respectively, as a function of ${\lambda }_{+}$. Line denotes analytical prediction, while filled circle denotes numerical estimations. The results are included for $N=200$, $D_x=1$, and ${\lambda }_{-}=-0.5$.

Figure 10

$r_1$ as a function of ${\lambda }_{+}$ for the multiplex network comprised of a positive GC and negative regular networks for different values of (a) interlayer coupling strength $D_x=1$ (circle), 1.5 ($\times$), 2 (square), 3 (triangle up), and 4 (diamond), and (b) inhibitory coupling strength ${\lambda }_{-}=-0.5$ (black circle), $-1$ (red triangle up), $-2$ (green $\times$), $-3$ (yellow circle), $-4$ (blue circle), $-5$ (orange circle), and $-8$ (violet square).

Figure 11

$r_1$ as a function of ${\lambda }_{+}$ for a multiplex network comprised of a positive GC and a negative regular network for different values of the network size.

Figure 12

$r_1$ as a function of ${\lambda }_{+}$ when excitatory layer is fixed as GC network and it is multiplexed with different network architectures for the inhibitory layer, (a) GC, (b) ER, and (c) SF. Here $\langle k_1\rangle =\langle k_2\rangle =10$ and $D_x=2$.

Figure 13

$r_1$ as a function of ${\lambda }_{+}$ for different network architecture of excitatory layer. For all the cases excitatory layer is multiplexed with an inhibitory regular layer, $\langle k_2\rangle =10$. Panels (a) and (b) correspond to SF layer, (c) and (d) correspond to regular layer, and (e) and (f) correspond to the ER layer for $\langle k_1\rangle =10$ and $\langle k_1\rangle =30$, respectively.