Partial synchronization and partial amplitude death in mesoscale network motifs

Poel, Winnie; Zakharova, Anna; Schöll, Eckehard

doi:10.1103/physreve.91.022915

Cited by 47 publications

(45 citation statements)

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“…Coupled nonlinear systems exhibit a plethora of different collective behavior such as partial or cluster synchronization [1][2][3][4][5][6], spatio-temporal patterns under delayed feedback control [7][8][9], and chimera states which consist of spatially coexisting domains of synchronized and unsynchronized dynamics [10][11][12]. Recently, special attention has been paid to the different types of oscillation quenching, amplitude death and oscillation death [13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Stable and transient multicluster oscillation death in nonlocally coupled networks

et al. 2015

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In a network of nonlocally coupled Stuart-Landau oscillators with symmetry-breaking coupling, we study numerically, and explain analytically, a family of inhomogeneous steady states (oscillation death). They exhibit multi-cluster patterns, depending on the cluster distribution prescribed by the initial conditions. Besides stable oscillation death, we also find a regime of long transients asymptotically approaching synchronized oscillations. To explain these phenomena analytically in dependence on the coupling range and the coupling strength, we first use a mean-field approximation which works well for large coupling ranges but fails for coupling ranges which are small compared to the cluster size. Going beyond standard mean-field theory, we predict the boundaries of the different stability regimes as well as the transient times analytically in excellent agreement with numerical results.

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Section: Introductionmentioning

confidence: 99%

Stable and transient multicluster oscillation death in nonlocally coupled networks

et al. 2015

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“…Recently, more complex synchronization patterns, including cluster and group synchronization, have received growing interest both in theory [Sorrentino and Ott, 2007;Kestler et al, 2007;Ashwin et al, 2007;Kestler et al, 2008;Kori and Kuramoto, 2001;Lücken and Yanchuk, 2012;Dahms et al, 2012;Kanter et al, 2011b,a;Golubitsky and Stewart, 2002;Sorrentino, 2014;Pecora et al, 2014;Poel et al, 2015] and in experiments [Illing et al, 2011;Aviad et al, 2012;Blaha et al, 2013;Rosin et al, 2013;Williams et al, 2012Williams et al, , 2013aRosin, 2015]. These scenarios appear in many biological systems, examples include dynamics of nephrons [Mosekilde et al, 2002], central pattern generation in animal locomotion [Ijspeert, 2008], or population dynamics [Blasius et al, 1999].…”

Section: Cluster and Group Synchrony: The Theorymentioning

confidence: 99%

“…Besides zero-lag synchronization, a state where all nodes undergo the same dynamics without a phase shift, group and cluster synchronization have received growing interest both in theory [Sorrentino and Ott, 2007;Kestler et al, 2007Kestler et al, , 2008Dahms et al, 2012;Kanter et al, 2011b,a;Golubitsky and Stewart, 2002;Lücken and Yanchuk, 2012;Sorrentino, 2014;Pecora et al, 2014;Poel et al, 2015] and in experiments [Illing et al, 2011;Aviad et al, 2012;Blaha et al, 2013;Williams et al, 2012Williams et al, , 2013aRosin, 2015]. In the case of group synchrony, the network consists of several groups where the nodes within one group are in zero-lag synchrony Dahms et al, 2012].…”

Section: Dynamics On Networkmentioning

confidence: 99%

Controlling Synchronization Patterns in Complex Networks

Lehnert¹

2016

Springer Theses

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In this thesis, I consider the control of synchronization in delay-coupled complex networks. As one main focus, several applications to neural networks will be discussed. In the first part, I focus on the stability of synchronization in complex networks meaning that the control is realized by considering the stability of synchrony in dependence on the parameters. In the second part, adaptive control of synchronization is studied. To this end, adaptive control algorithms are developed that tune the system parameters such that the desired control goal is reached.Besides zero-lag synchrony -a state where all nodes follow the same dynamics without a phase lag -groups and cluster states are considered, i.e., states where the network consists of several groups where the nodes within one group are in zero-lag synchrony and, in the case of cluster synchrony, with a constant phase lag between the clusters.The stability of synchronization can be accessed via the master stability function [Pecora and Carroll, 1998]. This convenient tool allows for treating the node dynamics and the network topology in two separate steps, and, thus, allows for a quite general treatment of different network topologies. In this thesis, I will discuss the generalization of the master stability function to group and cluster states and to non-smooth systems.The master stability function can be used to investigate synchrony in neural networks. Neurons are excitable systems where type-I and type-II excitability can be distinguished. Here, the stability of synchrony for both types in complex networks with excitatory and inhibitory links is investigated on two generic models, namely the normal form of the saddle-node infinite period bifurcation and the FitzHugh-Nagumo system. Furthermore, synchronization in systems with heterogeneous delays or node parameters is studied.In situations where parameters are unknown or drift, adaptive control methods are useful since they allow for an automatically realized adaption of the control parameters. A convenient adaptive method is the speed-gradient method that minimizes a predefined goal function [Fradkov, 1979[Fradkov, , 2007. I first apply this method in the control of an unstable focus and an unstable periodic orbit embedded in the Rössler attractor. Furthermore, I show that clusters states in networks of delayed coupled Stuart-Landau oscillators can be controlled by adaptively tuning the phase of the complex coupling strength or by adapting the topology. The first method is particularly simple because only one parameter has to be adapted, while the second method is more reliable in the sense that its success is widely independent of the choice of the control parameters and the initial conditions. ZUSAMMENFASSUNGIn dieser Arbeit wird die Kontrolle von Synchronisation in komplexen Netzwerken mit retardierten Kopplungen untersucht. Dabei werden verschiedene Anwendungen auf neuronale Netzwerke diskutiert. Der erste Teil der Arbeit beschäftigt sich mit der Stabilität von Synchronisation in komplexen Ne...

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“…Examples of synchrony range from genetic oscillators [2] and population dynamics [3] via data mining [4] and power grid networks [5,6] to opinion formation [7]. While early research focused on in-phase (or zero-lag) synchronization, recently more complex synchronization patterns such as group or cluster [8][9][10][11][12][13][14][15][16][17][18][19][20], or partial synchronization [21][22][23][24] have moved towards the center of scientific interest -in theoretical studies as well as in experiments. Partial synchronization describes a state where a part of the network is in synchrony -this can be in-phase, cluster, or group synchronization -while other parts exhibit oscillation quenching, i.e., amplitude death or oscillation death [25][26][27], or oscillate incoherently.…”

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confidence: 99%

“…A first step in this direction has been made by Do et al establishing that certain mesoscale subgraphs are of crucial importance for the global dynamics of the network [65]. An analytical method for studying partial synchronization states in mesoscale motifs has been presented in [24]. Here, we show how, in hierarchical networks, we can analytically predict the global dynamics from the topology of the small motifs by extending the eigensolution concept suggested in [24].As a model, we consider coupled Stuart-Landau oscillators, a paradigmatic normal form that naturally arises in an expansion of oscillator systems close to a Hopf bifurcation.…”

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confidence: 99%

Synchronization patterns: from network motifs to hierarchical networks

Krishnagopal

Lehnert

Poel

et al. 2017

Phil. Trans. R. Soc. A.

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One contribution of 15 to a theme issue 'Horizons of cybernetical physics' . We investigate complex synchronization patterns such as cluster synchronization and partial amplitude death in networks of coupled Stuart-Landau oscillators with fractal connectivities. The study of fractal or self-similar topology is motivated by the network of neurons in the brain. This fractal property is well represented in hierarchical networks, for which we present three different models. In addition, we introduce an analytical eigensolution method and provide a comprehensive picture of the interplay of network topology and the corresponding network dynamics, thus allowing us to predict the dynamics of arbitrarily large hierarchical networks simply by analysing small network motifs. We also show that oscillation death can be induced in these networks, even if the coupling is symmetric, contrary to previous understanding of oscillation death. Our results show that there is a direct correlation between topology and dynamics: hierarchical networks exhibit the corresponding hierarchical dynamics. This helps bridge the gap between mesoscale motifs and macroscopic networks. Subject Areas: complexity KeywordsThis article is part of the themed issue 'Horizons of cybernetical physics'.

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Partial synchronization and partial amplitude death in mesoscale network motifs

Cited by 47 publications

References 65 publications

Stable and transient multicluster oscillation death in nonlocally coupled networks

Stable and transient multicluster oscillation death in nonlocally coupled networks

Controlling Synchronization Patterns in Complex Networks

Synchronization patterns: from network motifs to hierarchical networks

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