Patterns of Synchrony in Complex Networks of Adaptively Coupled Oscillators

Abstract

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.

Modeling Tumor Disease and Sepsis by Networks of Adaptively Coupled Phase Oscillators

“…Synchronization is an important feature of the dynamics in networks of coupled nonlinear oscillators Pikovsky et al (2001), Strogatz (2001), Albert and Barabási (2002), Newman (2003), Boccaletti et al (2018), Berner (2021). Various synchronization patterns are known such as cluster synchronization where the network splits into groups of synchronous elements Dahms et al (2012), or partial synchronization patterns such as chimera states where the system splits into coexisting domains of coherent (synchronized) and incoherent (desynchronized) states Kuramoto and Battogtokh (2002), Abrams and Strogatz (2004), Panaggio and Abrams (2015), Sawicki (2019), Schöll (2020), Schöll et al (2020).…”

Collective Activity Bursting in a Population of Excitable Units Adaptively Coupled to a Pool of Resources

“…Of particular interest are adaptive networks in which connectivity changes are related to intrinsic nodal dynamics (Gross and Blasius, 2008;Berner, 2021). For example, these types of networks can model synaptic neuronal plasticity (Meisel and Gross, 2009;Markram et al, 2011), chemical (Jain and Krishna, 2001;Kuehn, 2019), epidemic (Gross et al, 2006), biological, and social systems (Horstmeyer and Kuehn, 2020).…”

Synchronization scenarios in three-layer networks with a hub

“…Other relevant applications are communication, supply, and transportation networks, for instance power grids, subway networks, or airtraffic networks 54 . Moreover, multilayer networks are also known to generate and stabilize diverse partial synchronization patterns in adaptive networks 55,56 , where the connectivity changes in time 57 .…”