Birth and Stabilization of Phase Clusters by Multiplexing of Adaptive Networks

Abstract

We propose a concept to generate and stabilize diverse partial synchronization patterns (phase clusters) in adaptive networks which are widespread in neuro-and social sciences, as well as biology, engineering, and other disciplines. We show by theoretical analysis and computer simulations that multiplexing in a multi-layer network with symmetry can induce various stable phase cluster states in a situation where they are not stable or do not even exist in the single layer. Further, we develop a method for the analysis of Laplacian matrices of multiplex networks which allows for insight into the spectral structure of these networks enabling a reduction to the stability problem of single layers. We employ the multiplex decomposition to provide analytic results for the stability of the multilayer patterns. As local dynamics we use the paradigmatic Kuramoto phase oscillator, which is a simple generic model and has been successfully applied in the modeling of synchronization phenomena in a wide range of natural and technological systems.Complex networks are an ubiquitous paradigm in nature and technology, with a wide field of applications ranging from physics, chemistry, biology, neuroscience, to engineering and socio-economic systems. Of particular interest are adaptive networks, where the connectivity changes in time, for instance, the synaptic connections between neurons are adapted depending on the relative timing of neuronal spiking [1][2][3][4][5]. Similarly, chemical systems have been reported [6], where the reaction rates adapt dynamically depending on the variables of the system. Activity-dependent plasticity is also common in epidemics [7] and in biological or social systems [8]. Synchronization is an important feature of the dynamics in networks of coupled nonlinear oscillators [9][10][11][12][13]. Various synchronization patterns are known, like cluster synchronization where the network splits into groups of synchronous elements [14], or partial synchronization patterns like chimera states where the system splits into coexisting domains of coherent (synchronized) and incoherent (desynchronized) states [15][16][17]. These patterns were also explored in adaptive networks [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. Furthermore, adapting the network topology has also successfully been used to control cluster synchronization in delay-coupled networks [34].Another focus of recent research in network science are multilayer networks, which are systems interconnected through different types of links [35][36][37][38]. A prominent example are social networks which can be described as groups of people with different patterns of contacts or interactions between them [39][40][41]. Other applications are communication, supply, and transportation networks, for instance power grids, subway networks, or airtraffic networks [42]. In neuroscience, multilayer networks represent for instance neurons in different areas of the brain, neurons connected either by a chemical link or by an electrical synapsis, or...

Optimal Self-Induced Stochastic Resonance in Multiplex Neural Networks: Electrical vs. Chemical Synapses

“…Several studies have shown that multiplex networks can generate patterns with significant differences from those observed in single-layer networks (Kouvaris et al, 2015 ; Majhi et al, 2016 , 2017 ; Berner et al, 2020 ). Their use in the optimization and control of dynamical behaviors have therefore attracted much attention recently.…”

Effect of topology upon relay synchronization in triplex neuronal networks

“…Multiplex networks can be associated with social and technological systems, 20 for instance, with dynamical processes where layers correspond to the states of the system at different times, 21 or with different types of links. Multilayer structures allow for full 22 or partial 23 synchronization between the layers, and different synchronization scenarios can be observed, such as explosive synchronization. 24…”

Desynchronization Transitions in Adaptive Networks

“…It drastically simplifies the problem by reducing the dimension and unifying the synchronization study for different networks. Since its introduction, the master stability approach has been extended and refined for multilayer [39], multiplex [40,41] and hypernetworks [42,43]; to account for single and distributed delays [44][45][46][47][48][49]; and to describe the stability of clustered states [50][51][52][53]. The master stability function has been used to understand effects in temporal [54,55] as well as adaptive networks [56] within a static formalism.…”

“…Moreover, adaptive networks have been reported for chemical [67,68], epidemic [69], biological [70], transport [71], and social systems [72,73]. A paradigmatic example of adaptively coupled phase oscillators has recently attracted much attention [12,41,[74][75][76][77][78][79][80][81], and it appears to be useful for predicting and describing phenomena in more realistic and detailed models [82][83][84][85]. Systems of phase oscillators are important for understanding synchronization phenomena in a wide range of applications [86][87][88].…”

“…We consider the following general class of N adaptively coupled systems [12,41,[74][75][76][77][78][79][80]89] x…”