Synchronization in networks of networks: The onset of coherent collective behavior in systems of interacting populations of heterogeneous oscillators

Abstract

The onset of synchronization in networks of networks is investigated. Specifically, we consider networks of interacting phase oscillators in which the set of oscillators is composed of several distinct populations. The oscillators in a given population are heterogeneous in that their natural frequencies are drawn from a given distribution, and each population has its own such distribution. The coupling among the oscillators is global, however, we permit the coupling strengths between the members of different populations to be separately specified. We determine the critical condition for the onset of coherent collective behavior, and develop the illustrative case in which the oscillator frequencies are drawn from a set of (possibly different) Cauchy-Lorentz distributions. One motivation is drawn from neurobiology, in which the collective dynamics of several interacting populations of oscillators (such as excitatory and inhibitory neurons and glia) are of interest.In recent years, there has been considerable interest in networks of interacting systems. Researchers have found that an appropriate description of such systems involves an understanding of both the dynamics of the individual oscillators and the connection topology of the network. Investigators studying the latter have found that many complex networks have a modular structure involving motifs [1], communities [2,3], layers [4], or clusters [5]. For example, recent work has shown that as many kinds of networks (including isotropic homogeneous networks and a class of scale-free networks) transition to full synchronization, they pass through epochs in which well-defined synchronized communities appear and interact with one another [3]. Knowledge of this structure, and the dynamical behavior it supports, informs our understanding of biological [6], social [7], and technological networks [8].Here we consider the onset of coherent collective behavior in similarly structured systems for which the dynamics of the individual oscillators is simple. In seminal work, Kuramoto analyzed a mathematical model that illuminates the mechanisms by which synchronization arises in a large set of globally-coupled phase oscillators [9]. An important feature of Kuramoto's model is that the oscillators are heterogeneous in their frequencies. And, although these mathematical results assume global coupling, they have been fruitfully applied to further our understanding of systems of fireflies, arrays of Josephson junctions, electrochemical oscillators, and many other cases [10]. In this work, we study systems of several interacting Kuramoto systems, i.e., networks of interacting populations of phase oscillators. Our motivation is drawn not only from the applications listed above (e.g., an amusing application might be interacting populations of fireflies, where each population inhabits a separate tree), but also from other biological systems. Rhythms arising from coupled cell populations are seen in many of the body's organs (including the heart, the pancreas, and the kidney...

Solvable Model for Chimera States of Coupled Oscillators

“…The non-trivial fixed points correspond to stationary chimera states, in which the local order parameters ρ 1 (t) ≡ 1 and ρ 2 (t) = r(t) remain constant, as does the phase difference ψ(t) = φ 1 (t) − φ 2 (t), despite the fact that the individual microscopic oscillators in population σ = 2 continue to move in a desynchronized fashion. Figure 3 plots typical phase portraits for (12). Figure 3(a) shows a stable chimera state coexisting with the stable synchronized state; the basin boundary between them is defined by the stable manifold of a saddle chimera.…”

“…4? Do chimeras also exist if the oscillators are non-identical [10,11,12] or arranged in complex networks [13]? It would also be worth looking for experimental examples of chimera states.…”

The Kuramoto model in complex networks

“…Furthermore, in contrast to [166,167] the accuracy of the generalized method no longer depends on the scale of the coupling strength, since the emergence of phase lag between nodes belonging to different communities is guaranteed by the presence of negative couplings [169,170]. It is worth mentioning that many papers analytically investigated the lowdimensional behavior of Kuramoto oscillators coupled in fully connected graphs but organized into subpopulations characterized by different frequency and couplings distributions [171][172][173][174][175][176][177]. Extensions of these studies accounting modular heterogeneous networks are promising topics for future works.…”

Frustration tuning and perfect phase synchronization in the Kuramoto-Sakaguchi model

“…Some variations of the model consider random pinning fields [19] while in others the frustrations are either global or random for every node [20,22], or allocated according to a multi-network structure for interacting populations [23,24]. While such additions introduce additional degrees of disorder to the traditional Kuramoto dynamics, we are interested in whether frustrations may be tuned to enhance synchronisation.…”