Patterns of Synchrony in Complex Networks of Adaptively Coupled Oscillators (Springer Theses)

The focus of this thesis is the interplay of synchrony and adaptivity in complex networks. Synchronization is a ubiquitous phenomenon observed in different contexts in physics, chemistry, biology, neuroscience, medicine, socioeconomic systems, and engineering. Most prominently, synchronization takes place in the brain, where it is associated with cognitive capacities like learning and memory, but is also a characteristic of neurological diseases like Parkinson and epilepsy. Adaptivity is common in many networks in nature and technology, where the connectivity changes in time, i.e., the strength of the coupling is continuously adjusted depending upon the dynamic state of the system, for instance synaptic neuronal plasticity in the brain. This research contributes to a fundamental understanding of various synchronization patterns, including hierarchical multifrequency clusters, chimeras and other partial synchronization states. After a concise survey of the fundamentals of adaptive and complex dynamical networks and synaptic plasticity, in the first part of the thesis the existence and stability of cluster synchronization in globally coupled adaptive networks is discussed for simple paradigmatic phase oscillators as well as for a more realistic neuronal oscillator model with spike-timing dependent plasticity. In the second part of the thesis the interplay of adaptivity and connectivity is investigated for more complex network structures like nonlocally coupled rings, random networks, and multilayer systems. Besides presenting a plethora of novel, sometimes intriguing patterns of synchrony, the thesis makes a number of pioneering methodological advances, where rigorous mathematical proofs are given in the Appendices. These results are of interest not only from a fundamental point of view, but also with respect to challenging applications in neuroscience and technological systems. Supervisors’ Foreword Abstract Acknowledgements Contents About the Author 1 Introduction 1.1 Dynamics on Complex Networks 1.2 Synchronization and Collective Phenomena 1.3 Dynamics of Complex Networks 1.4 The Role of Phase Oscillator Models for Complex Dynamical Networks 1.5 Outline References 2 Fundamentals of Adaptive and Complex Dynamical Networks 2.1 Complex Networks 2.1.1 Networks, Subnetworks, and Connectivity 2.1.2 Special Network Types 2.1.3 Permutation Symmetries in Networks 2.2 Dynamics 2.2.1 Types of Coupling 2.2.2 Kuramoto–Sakaguchi Type Model 2.2.3 Hodgkin–Huxley Model with Chemical Synapses 2.3 Adaptive Networks in Neuroscience 2.3.1 Spike Timing-Dependent Plasticity 2.3.2 Phase Difference-Dependent Plasticity 2.3.3 A Network of Adaptively Coupled Phase Oscillator 2.4 Summary References Part I Cluster Synchronization in Globally Coupled Adaptive Networks 3 Population of Hodgkin–Huxley Neurons with Spike Timing-Dependent Plasticity 3.1 Coupled Hodgkin–Huxley Neurons on a Network with Spike Timing-Dependent Plasticity 3.2 Numerical Observation of Synchrony and Frequency Clustering 3.2.1 Clustering with Independent Random Input 3.3 Emergence of Two-Cluster States 3.4 Phenomenological Model with Phase Difference-Dependent Plasticity 3.4.1 Properties of the Model 3.4.2 Comparison of the Model and Cluster Dynamics in Hodgkin–Huxley Network 3.4.3 Criteria for the Emergence of Frequency Clusters 3.5 Summary References 4 One-Cluster States in Adaptive Networks of Coupled Phase Oscillators 4.1 Classification of One-Cluster States 4.2 Stability of One-Cluster States 4.3 Adaptation Rate Dependence of One-Cluster Stability 4.4 Double Antipodal States 4.5 Summary References 5 Multicluster States in Adaptive Networks of Coupled Phase Oscillators 5.1 Numerical Observation of Multicluster States 5.1.1 Splay Type Cluster States 5.1.2 Antipodal Type Cluster States 5.1.3 Mixed Type Cluster States 5.2 Splay Type Multicluster States 5.2.1 Conditions for the Emergence of Splay Type Multicluster States 5.2.2 Two-Cluster States of Splay Type 5.2.3 Adaptation Rate Dependence for the Emergence of Two-Cluster States 5.3 Conditions for the Emergence of Multicluster States—A Generalized Approach 5.4 Antipodal Type Multicluster States 5.4.1 Asymptotic Conditions for the Emergence of Antipodal Type Multicluster States 5.4.2 Two-Cluster States of Antipodal Type 5.5 Mixed Type Pseudo-multicluster States 5.5.1 Asymptotic Conditions for the Emergence of Mixed Type Pseudo-multicluster States 5.5.2 Pseudo-two-cluster States of Mixed Type 5.6 Stability of Multicluster States 5.6.1 On the Stability of Multicluster States with Evenly Sized Clusters 5.6.2 An Effective Approach for the Stability of Multicluster States 5.7 Summary References Part II Interplay of Adaptivity and Connectivity 6 Adaptation on Nonlocally Coupled Ring Networks 6.1 Multicluster and Solitary States 6.1.1 One-Cluster States 6.1.2 Multicluster States 6.1.3 Solitary States 6.2 One-Cluster States: Local Versus Global Features 6.2.1 Classification of One-Cluster States 6.2.2 Stability of One-Cluster States 6.3 The Emergence of Solitary States 6.4 Adaptive Networks with Global Base Topology Versus Ring Base Topology: The Differences 6.5 Summary References 7 Synchronization on Adaptive Complex Network Structures 7.1 The Master Stability Function for Adaptive Complex Networks 7.2 Stability Islands in the Presence of Adaptation 7.3 Stability Islands and Implications for the Emergence of Multicluster States 7.4 Summary References 8 Multilayered Adaptive Networks 8.1 Lifted States in Multiplex Networks 8.2 Birth and Robustness of Phase Clusters 8.3 Multiplex Decomposition 8.4 Stabilizing Through Multiplexing 8.5 Applications for the Multiplex Decomposition 8.5.1 The Master Stability Approach for Multiplex Networks 8.5.2 Analytic Treatment of Diffusive Dynamics on Multiplex Networks 8.6 Summary References 9 Conclusion and Outlook References Appendix A Proofs of Results from the Main Text and Supplemental Material A.1 One-Cluster States on Globally Coupled Adaptive Networks A.2 Stability of One-Cluster States on Globally Coupled Networks A.3 Multicluster States of Splay Type A.4 Asymptotic Expansions of Multicluster States A.5 From Local to Global Order Parameter A.6 Stability of One-Cluster States on Nonlocally Coupled Networks A.7 Stability of Lifted One-Cluster States A.8 Example for a Complex Adjacency Matrix References