Synchronization and amplitude death in hypernetworks

The impact of inner-coupling and time delay on synchronization: From single-layer network to hypernetwork

Though synchronization of complex dynamical systems has been widely studied in the past few decades, few studies pay attention to the impact of network parameters on synchronization in hypernetworks. In this paper, we focus on a specific hypernetwork model consisting of coupled Rössler oscillators and investigate the impact of inner-coupling and time delay on the synchronized region (SR). For the sake of simplicity, the inner-coupling matrix is chosen from three typical forms, which result in classical bounded, unbounded, and empty SR in a single-layer network, respectively. The impact of inner-couplings or time delays on unbounded SR is the most interesting one among the three types of SR. Once the SR of one subnetwork is unbounded, the SR of the whole hypernetwork is also unbounded with a different inner-coupling matrix. In a hypernetwork with unbounded SR, the time delays change not only the size but also the type of SR. In a hypernetwork with bounded or empty SR, the time delays have almost no effect on the type of SR.

Emergent hypernetworks in weakly coupled oscillators

Networks of weakly coupled oscillators had a profound impact on our understanding of complex systems. Studies on model reconstruction from data have shown prevalent contributions from hypernetworks with triplet and higher interactions among oscillators, in spite that such models were originally defined as oscillator networks with pairwise interactions. Here, we show that hypernetworks can spontaneously emerge even in the presence of pairwise albeit nonlinear coupling given certain triplet frequency resonance conditions. The results are demonstrated in experiments with electrochemical oscillators and in simulations with integrate-and-fire neurons. By developing a comprehensive theory, we uncover the mechanism for emergent hypernetworks by identifying appearing and forbidden frequency resonant conditions. Furthermore, it is shown that microscopic linear (difference) coupling among units results in coupled mean fields, which have sufficient nonlinearity to facilitate hypernetworks. Our findings shed light on the apparent abundance of hypernetworks and provide a constructive way to predict and engineer their emergence.

Networks with higher-order interactions are relevant to variety of real-world applications, they can be good description of data even if the system has only pairwise interactions. The authors uncover the hypernetwork emergence in coupled nonlinear oscillators and electrochemical experiments.

Synchronization in networks with multiple interaction layers

The structure of many real-world systems is best captured by networks consisting of several interaction layers. Understanding how a multilayered structure of connections affects the synchronization properties of dynamical systems evolving on top of it is a highly relevant endeavor in mathematics and physics and has potential applications in several socially relevant topics, such as power grid engineering and neural dynamics. We propose a general framework to assess the stability of the synchronized state in networks with multiple interaction layers, deriving a necessary condition that generalizes the master stability function approach. We validate our method by applying it to a network of Rössler oscillators with a double layer of interactions and show that highly rich phenomenology emerges from this. This includes cases where the stability of synchronization can be induced even if both layers would have individually induced unstable synchrony, an effect genuinely arising from the true multilayer structure of the interactions among the units in the network.

Synchronization properties of coupled chaotic neurons: The role of random shared input

Spike-time correlations of neighbouring neurons depend on their intrinsic firing properties as well as on the inputs they share. Studies have shown that periodically firing neurons, when subjected to random shared input, exhibit asynchronicity. Here, we study the effect of random shared input on the synchronization of weakly coupled chaotic neurons. The cases of so-called electrical and chemical coupling are both considered, and we observe a wide range of synchronization behaviour. When subjected to identical shared random input, there is a decrease in the threshold coupling strength needed for chaotic neurons to synchronize in-phase. The system also supports lag-synchronous states, and for these, we find that shared input can cause desynchronization. We carry out a master stability function analysis for a network of such neurons and show agreement with the numerical simulations. The contrasting role of shared random input for complete and lag synchronized neurons is useful in understanding spike-time correlations observed in many areas of the brain.

Finite-size-induced transitions to synchrony in oscillator ensembles with nonlinear global coupling

We report on finite-sized-induced transitions to synchrony in a population of phase oscillators coupled via a nonlinear mean field, which microscopically is equivalent to a hypernetwork organization of interactions. Using a self-consistent approach and direct numerical simulations, we argue that a transition to synchrony occurs only for finite-size ensembles and disappears in the thermodynamic limit. For all considered setups, which include purely deterministic oscillators with or without heterogeneity in natural oscillatory frequencies, and an ensemble of noise-driven identical oscillators, we establish scaling relations describing the order parameter as a function of the coupling constant and the system size.