Synchronization and Phase Locking in Networks of Heterogeneous Model Neurons

Abstract

This dissertation examines the effect of two types of system complexity, nonlinearity and heterogeneity, on oscillatory dynamics in networked systems. In particular, we focus on finding conditions for complete synchronization, where the dynamics of multiple systems are identical, phase locking, where the dynamics of multiple systems share critical features, and mixed mode oscillations (MMOs), where the dynamics of a single system demonstrate periodic oscillations with peaks of markedly different sizes. A fascinating application of these conditions is to networks of model neurons and the crucial role of synchronization in brain function.

We establish conditions for synchronization in networks of heterogeneous systems with nonlinear dynamics and diffusive coupling. We leverage a passivity-based Lyapunov approach to find a condition for complete synchronization in networks of identical nonlinear systems in terms of the network structure and the dynamics of individual systems. An application to networked model neurons with biologically relevant parameter values demonstrates improvement over alternative methods. Cluster synchronization is an extension of complete synchronization where the network can be partitioned into distinct subgroups of systems that are synchronized. We find conditions for cluster synchronization in networks of non-identical systems with nonlinear dynamics and diffusive coupling using a passivity-based Lyapunov approach and a contraction based approach.

We examine a system of two model neurons where the first neuron receives a constant external input and the second neuron receives input from the first through diffusive coupling. Large networks that are cluster synchronized can be represented by simpler systems; in particular, the dynamics of a network synchronized in two clusters can be represented by a system of two coupled model neurons. We use techniques from dynamical systems theory to characterize parameter regimes where each model neuron is resting, firing, or sustaining MMOs. The system of two model neurons and its extensions represent a foundation for investigating how network structure and external stimuli interact to influence the dynamics in networks of neurons.

Characterization of the conditions for when synchronization may arise in networks of heterogeneous nonlinear systems is a crucial step toward understanding complex networked systems.