Time delay control of symmetry-breaking primary and secondary oscillation death

Zakharova, Anna; Schneider, Isabelle; Kyrychko, Yuliya N.; Blyuss, Konstantin B.; Koseska, Aneta; Fiedler, Bernold; Schöll, Eckehard

doi:10.1209/0295-5075/104/50004

Cited by 63 publications

(63 citation statements)

References 53 publications

(71 reference statements)

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“…The following results, which will be needed later, have been derived in [30] for a system of only two coupled Stuart-Landau oscillators:…”

Section: Modelmentioning

confidence: 99%

“…Due to the coherent nature of the oscillation death patterns, we are able to predict analytically and with high precision the boundaries between different stability regimes. This paper is organized as follows: In section II we introduce our model of nonlocally coupled oscillators and arXiv:1507.01918v2 [nlin.AO] 12 Oct 2015 briefly summarize previous work on two coupled oscillators [30]. Our approach is numerical in the first step, and we observe two different types of oscillation death: transient as well as asymptotically stable.…”

Section: Introductionmentioning

confidence: 97%

“…neural networks [26], genetic oscillators [27], calcium oscillators [28] or stem cell differentiation [29] where it has been proposed as a basic mechanism for morphogenesis and cellular differentiation. Consequently, there has also been theoretical interest in oscillation death and it has been shown to exist for special time-delayed [30] and repulsive [31,32] types of coupling as well as coupling through conjugate or dissimilar variables [33][34][35]. However, research has mostly focussed on small numbers of coupled oscillators in networks with local or global coupling.…”

Section: Introductionmentioning

confidence: 99%

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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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“…The following results, which will be needed later, have been derived in [30] for a system of only two coupled Stuart-Landau oscillators:…”

Section: Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Stable and transient multicluster oscillation death in nonlocally coupled networks

et al. 2015

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“…As an example, we have quenched the oscillations in a Stuart-Landau oscillator. Such an inhomogeneous steady state is known as oscillation death [Koseska et al, 2010[Koseska et al, , 2013aZakharova et al, 2013] and has been found in various systems including tunnel diodes [Heinrich et al, 2010], neuronal networks [Curtu, 2010], and genetic oscillators [Koseska et al, 2009]. This example demonstrates that we have found a very versatile method to construct networks that show a desired dynamical behavior.…”

Section: Resultsmentioning

confidence: 76%

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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“…Time delays are used to account for the fact that in the majority of real-world networks, some processes do not happen instantaneously due to a finite speed of signal propagation, times required for information processing, etc., and their inclusion leads to significant changes in the dynamics of the system [15,16,17,18,19,20].…”

mentioning

confidence: 99%

Dynamics of Neural Systems with Discrete and Distributed Time Delays

Rahman¹,

Blyuss²,

Kyrychko³

2015

SIAM J. Appl. Dyn. Syst.

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Abstract. In real-world systems, interactions between elements do not happen instantaneously, due to the time required for a signal to propagate, reaction times of individual elements, and so forth. Moreover, time delays are normally nonconstant and may vary with time. This means that it is vital to introduce time delays in any realistic model of neural networks. In order to analyze the fundamental properties of neural networks with time-delayed connections, we consider a system of two coupled two-dimensional nonlinear delay differential equations. This model represents a neural network, where one subsystem receives a delayed input from another subsystem. An exciting feature of the model under consideration is the combination of both discrete and distributed delays, where distributed time delays represent the neural feedback between the two subsystems, and the discrete delays describe the neural interaction within each of the two subsystems. Stability properties are investigated for different commonly used distribution kernels, and the results are compared to the corresponding results on stability for networks with no distributed delays. It is shown how approximations of the boundary of the stability region of a trivial equilibrium can be obtained analytically for the cases of delta, uniform, and weak gamma delay distributions. Numerical techniques are used to investigate stability properties of the fully nonlinear system, and they fully confirm all analytical findings.

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Time delay control of symmetry-breaking primary and secondary oscillation death

Cited by 63 publications

References 53 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

Dynamics of Neural Systems with Discrete and Distributed Time Delays

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