Synchronization in a temporal multiplex neuronal hypernetwork

Synchronization in time-varying complex dynamical systems has been explored in a variety of different coupling topologies during the last two decades. In most of the previous cases, the basic time-varying coupling topologies were considered to be a single interaction between the nodes or in a monolayer configuration, although in many real situations the types of interactions are more than one or in the form of multilayer. In this work, we study the synchronization in multiplex neuronal network, which evolves with time, and each layer consists of more than one interaction function. Specifically, we consider a neuronal hypernetwork at each layer in which neurons are communicated with each other via electrical and chemical synapses simultaneously and independently. The network corresponding to the electrical gap junctional coupling form a small-world network while the connection associated with the chemical synaptic interaction forms a unidirectional random network. Then intralayer connections are allowed to switch stochastically over time with a characteristic rewiring frequency, whereas interlayer connections via electrical synapses are time invariant. We explore the intralayer and interlayer neuronal synchrony in such network and analytically derive the necessary stability conditions for synchrony using master stability function approach, and excellently match with our numerical findings. Interestingly, we find that the higher frequency of switching links in the intralayer enhances both intralayer and interlayer synchrony and conferring larger windows of synchrony. We also analyze the robustness of these synchronization states with respect to initial conditions using the basin stability framework. Furthermore we find that rapidly changing networks take much less time to reach synchronization state. Lastly, we inspect the dynamical robustness of interlayer synchronization against stochastic demultiplexing of each replica, with analytical justification.

Figure 1

Schematic representation of time-varying connections in a hypernetwork at two different time instants: (a) $t=t_1$ and (b) $t=t_2$. Each node is denoted by a gray solid circle. The red dashed lines denote unidirectional interactions, and forms a random network with constant in-degree 1, while the green solid lines represent bidirectional coupling, forming a small-world network of average degree 4.

Figure 2

Different types of neuronal interactions through electrical and chemical synapses. (a) Chemical transmission occurring unidirectionally through chemical synapses. (b) Bidirectional electrical communication intercedes by the electrical gap junction. (c) Mixed synaptic interaction: coexistence of both electric and chemical synaptic interactions eventuate between two neurons. (d) Heterosynaptic interaction: one neuron simultaneously coupled with two other neurons, one through the chemical synapse and the other through gap junction channel.

Figure 3

Left and right panels represent the intralayer and interlayer synchronization error in ($\epsilon ,\eta$) parameter space with different values of $f=0.001$ (top row), $f=0.1$ (middle row), $f=10.0$ (bottom row). Other parameters: $g_c=\frac{\epsilon }{2},k_e=6,k_c=5$, and $p=0.125$.

Figure 4

Basin stability of intralayer (left) and interlayer (right) synchronization in ($\epsilon ,\eta$) parameter space with different values of $f=0.001$ (top), $f=0.1$ (middle), $f=10.0$ (bottom). Other parameters are same as in Fig. 3.

Figure 5

Left and right panels represent the synchronization error of intralayer and interlayer synchronization in ($f,\epsilon$) parameter space with $g_c=\frac{\epsilon }{2}$ and $\eta =\frac{\epsilon }{3}$. Here $k_c=5$ and $p=0.125$ ($k_e=4$ for top panel, $k_e=6$ for bottom panel).

Figure 6

The BS measure of intralayer (left) and interlayer (right) synchronization in ($f,\epsilon$) parameter space for $k_e=4$ (top) and $k_e=6$ (bottom). The other parameters are same as in Fig. 5.

Figure 7

(Left) intralayer and (right) interlayer synchronization error in ($f,p$) parameter plane. For $k_e=4,\epsilon =1.8$ (top row), $k_e=6,\epsilon =1.8$ (middle row), $k_e=6,\epsilon =2.7$ (bottom row). Other parameters: $k_c=5,g_c=\frac{\epsilon }{2}$, and $\eta =\frac{\epsilon }{3}$.

Figure 8

BS for intralayer (left) and interlayer (right) synchronization in ($f,p$) parameter space corresponding to Fig. 7.

Figure 9

(a) Intralayer and (b) interlayer synchronization time with respect to $\epsilon$ for different rewiring frequencies. Other parameters: $g_c=\frac{\epsilon }{2},\eta =\frac{\epsilon }{3},k_e=6,k_c=5,p=0.125$.

Figure 10

Variation of maximum Lyapunov exponent corresponding to (a) intralayer synchronization for parameter space $(\epsilon ,\eta )$ and (b) interlayer synchronization with respect to the parameter $\eta$ where $\epsilon =1.8$. Other parameters: $g_c=\frac{\epsilon }{2},k_e=6,k_c=5$, and $p=0.125$.

Figure 11

The variation of the interlayer synchronization error $E_{\mathrm{inter}}$ with respect to the demultiplexed probability $p_{dm}$ (a) with fixed $\epsilon =2.0, g_c=\frac{\epsilon }{2}$ and different intralayer rewiring frequencies $f$, (b) with fixed $f=1.0$ and varied intralayer coupling strength $\epsilon ,g_c$. Other parameters: $\eta =0.8,N=200$, and $p=0.125$.