Exploring Relationships Between Structure, Dynamics And The Effects Of Local Perturbations In Networks
In recent years, an abundance of studies in complex systems research have focused on deciphering the properties and behaviors of networks. Within this arena, much effort has been devoted to characterizing the structural organization of networks with complicated topologies. However, network elements are often dynamical entities, and in this case, it becomes relevant to understand what collective dynamics arise from particular architectures, as those activity patterns may in turn shape system function. Moreover, a network’s dynamical behaviors may be significantly affected if the underlying interactions can themselves evolve over time, or if external influences also act upon the system. In this thesis, we numerically study such scenarios using a generic model of interacting oscillators, and in the context of network models more closely inspired by neural population activity. Beginning with a system of canonical coupled phase-oscillators, we ask whether global synchronization can be enhanced in networks whose connectivity co-evolves with their dynamics. This work presents a simple adaptive strategy which, although it relies on only local information, can reorganize initially unstructured networks towards topologies that better support collective behavior. We next turn our attention to models inspired specifically by mesoscale and macroscale brain network dynamics. In particular, we start by studying small circuits of interacting oscillatory neuronal populations, wherein multistability can enable distinct network activity patterns to arise from a single anatomical backbone. For both deterministic and stochastic networks, we then show how different types of local perturbations can be harnessed to modulate these collective states in functionally meaningful ways, bypassing the need to rewire circuit structure. We then build a model of whole-brain network dynamics by coupling oscillatory neural masses according to empirically-derived anatomical connectivity, and we investigate the impacts of focal stimulation on the system’s dynamics. Our results suggest that network responses can depend not only on the perturbed site within the structural scaffold, but also on the nature of the system’s ongoing rhythmic activity at baseline. As a whole, this work aims to elucidate various interplays between structure, perturbations, and collective dynamics in model systems of interacting elements, including generic coupled oscillator networks and biologically-inspired brain networks.