Adaptation controls synchrony and cluster states of coupled threshold-model neurons

Ladenbauer, Josef; Lehnert, Judith; Rankoohi, Hadi; Dahms, Thomas; Schöll, Eckehard; Obermayer, Klaus

doi:10.1103/physreve.88.042713

Cited by 32 publications

(29 citation statements)

References 59 publications

(77 reference statements)

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“…Indeed the techniques for constructing the relevant saltation matrices for handling synaptic dynamics with pulsatile forcing as well as discontinuous reset in non-linear integrate-and-fire systems have recently been described by Ladenbauer et al [33]. The combination of this with pwl caricatures of Izhikevich style integrate-and-fire models [46], such as described in [47], opens the door for a very thorough examination of network states in biologically realistic spiking networks.…”

Section: Discussionmentioning

confidence: 99%

“…The synchronous state of the system of coupled oscillators is stable if the MSF is negative at α l = λ l , where λ l ranges over the eigenvalues of the matrix G (excluding λ 1 = 0). For a further discussion about the use of the MSF formalism in the analysis of synchronisation of oscillators on complex networks, we refer the reader to [3,32], and for the use of this formalism in spiking neural networks of non-linear integrate-and-fire type see [11,33]. This approach has recently been extended to cover the case of cluster states by making extensive use of tools from computational group theory to determine admissible patterns of synchrony [34] in unweighted networks.…”

Section: Extending the Master Stability Functionmentioning

confidence: 99%

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Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function

Coombes

Thul

2016

Eur. J. Appl. Math

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The master stability function is a powerful tool for determining synchrony in high-dimensional networks of coupled limit cycle oscillators. In part, this approach relies on the analysis of a low-dimensional variational equation around a periodic orbit. For smooth dynamical systems, this orbit is not generically available in closed form. However, many models in physics, engineering and biology admit to non-smooth piece-wise linear caricatures, for which it is possible to construct periodic orbits without recourse to numerical evolution of trajectories. A classic example is the McKean model of an excitable system that has been extensively studied in the mathematical neuroscience community. Understandably, the master stability function cannot be immediately applied to networks of such non-smooth elements. Here, we show how to extend the master stability function to non-smooth planar piece-wise linear systems, and in the process demonstrate that considerable insight into network dynamics can be obtained. In illustration, we highlight an inverse period-doubling route to synchrony, under variation in coupling strength, in globally linearly coupled networks for which the node dynamics is poised near a homoclinic bifurcation. Moreover, for a star graph, we establish a mechanism for achieving so-called remote synchronisation (where the hub oscillator does not synchronise with the rest of the network), even when all the oscillators are identical. We contrast this with node dynamics close to a non-smooth Andronov-Hopf bifurcation and also a saddle node bifurcation of limit cycles, for which no such bifurcation of synchrony occurs.

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Section: Discussionmentioning

confidence: 99%

Section: Extending the Master Stability Functionmentioning

confidence: 99%

Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function

Coombes

Thul

2016

Eur. J. Appl. Math

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“…This gives rise to a discontinuity in the time series. In [Ladenbauer et al, 2013], we derived an MSF for these non-smooth systems.…”

Section: Dynamics On Networkmentioning

confidence: 99%

“…In [Pecora et al, 2014], a very general answer to the question which kinds of topologies exhibit states of group synchrony and a discussion of the stability of these states via a MSF is given. In [Ladenbauer et al, 2013], group synchrony in non-smooth systems was considered.…”

Section: Dynamics On Networkmentioning

confidence: 99%

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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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Patterns of Synchrony in Complex Networks of Adaptively Coupled Oscillators

Berner¹

2021

Springer Theses

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Collective phenomena in systems of interconnected dynamical units are omnipresent in nature. The swarm behavior in flocks of birds or schools of fish, the synchronous flashing of fireflies or even coherent spiking of neurons in the human brain are just a few examples of collective motion. Elucidating the mechanisms that give rise to synchronization is crucial in order to understand biological self-organization. For this sake, the theory of dynamical networks has been successfully applied over the last decades to boil down the complex dynamics from natural systems to their essentials. In addition, dynamical networks with adaptive couplings and hence non-constant network structure appear naturally in real-world systems such as power grid networks, social networks as well as neuronal networks. In this thesis, we study adaptive networks and their properties which give rise to the emergence of a variety of synchronization patterns, including complete and cluster synchronization as well as solitary and chimera-like states. One of the main fields of application that is investigated in this thesis concerns neuroscience. However, the methods and approaches developed in this work are not restricted to a specific field of application.aus gekoppelten Phasenoszillatoren an und zeigen, wie das subtile Zusammenspiel zwischen Adaptivität und Netzwerkstruktur die Entstehung komplexer partieller Synchronisationsmuster hervorruft.Trotz des regen Interesses an adaptiven Netzwerken ist über das Zusammenspiel adaptiv gekoppelter Netzwerkgruppen wenig bekannt. Solche adaptiven Mehrschicht-oder Multiplex-Netzwerke kommen in neuronalen Netzwerken ganz natürlich vor. Wir schlagen ein Konzept zur Erzeugung und Stabilisierung diverser partieller Synchronisationsmuster (Phasen-Cluster) in adaptiven Netzwerken vor. Wir zeigen, dass das "Multiplexen" verschiedene stabile Phasen-Cluster-Zustände induzieren kann, selbst wenn diese in den Einschicht-Systemen nicht stabil sind oder gar nicht existieren. Weiterhin entwickeln wir eine Methode zur Analyse von Laplace-Matrizen von Multiplex-Netzwerken, die einen Einblick in die spektrale Struktur dieser Netzwerke ermöglicht und damit eine Reduktion des Stabilitätsproblems auf die von Einschicht-Netzwerken ermöglicht. Wir setzen die neue Methode ein, um analytische Ergebnisse für die Stabilität der Zustände im Mehrschicht-System zu erhalten.

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Adaptation controls synchrony and cluster states of coupled threshold-model neurons

Cited by 32 publications

References 59 publications

Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function

Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function

Controlling Synchronization Patterns in Complex Networks

Patterns of Synchrony in Complex Networks of Adaptively Coupled Oscillators

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