Controlling Synchronization Patterns in Complex Networks

Abstract

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...

Globally attracting synchrony in a network of oscillators with all-to-all inhibitory pulse coupling

“…The master stability function [50] (MSF) is elegant method that is often applied in physics because it separates the network properties from the dynamical properties of individual oscillators. This method also requires one to assume a form for the dynamic system, such as a normalized SNIPER bifurcation, Fitzhugh-Nagumo or Stuart-Landau [51]. Moreover, the derivation of the master stability function assumes linear coupling between the state variables of the oscillators; diffusive coupling is an example of linear coupling.…”

Patterns of Synchrony in Complex Networks of Adaptively Coupled Oscillators

“…This fact allows for the relation between dynamical properties and the Laplacian structure of a network. It has been used, for instance, to introduce the powerful methodology of the master stability function [PEC98,LEH15b] . Linear coupling is in some sense the simplest type of coupling.…”

Analysis and Design of Adaptive Synchronization of a Complex Dynamical Network with Time-Delayed Nodes and Coupling Delays

“…There are a lot of synchronization phenomena in physical sciences and in mathematics. Several books and reviews [6][7][8][9] have also dealt with this topic. Such applications are pervasive and include clock synchronization in complex networks [10][11][12], coordination of unmanned aerial vehicles [13], and fair allocation of network resources [14].…”