Mathematical analysis of synchronization in complex networks

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Synchronization is an important mode of collective behavior in diverse physical, biological, and technological networks. In many applications, local dynamics is modeled by systems of differential equations and the interaction schemes are defined by weighted graphs. This research is aimed at advancing the mathematical theory of synchronization and pattern formation in coupled systems of differential equations on graphs. Networks with different types of local dynamics, such as those generated by limit cycles, chaotic attractors, or induced by noise, are considered under general assumptions on the network architecture. The principal investigator (PI) develops mathematically rigorous yet practically efficient conditions guaranteeing synchronization, studies robustness of synchrony to noise, and analyzes patterns of electrical activity in gap-junctionally coupled neuronal networks. The graph-theoretic interpretation of the analytical results is emphasized. The PI seeks systematic ways of quantifying the contribution of the network topology to the dynamics of coupled oscillators by integrating combinatorial techniques into dynamical systems analysis.Synchronization is a universal phenomenon with abundant applications across science and technology. Power grid safety, effective communication in information networks, and coordination of unmanned vehicles are just three of many areas of technology where synchronization is crucial. Furthermore, synchronization plays a prominent role in the mechanisms of many vital physiological and cognitive processes such as respiration, sleep, and attention, as well as in the mechanisms of several severe neurodegenerative disorders such as Parkinson's Disease and epilepsy. The PI develops new mathematical tools and uses them to study synchronization in biophysical models including that of the Locus Coeruleus network, a group of neurons in the mammalian brainstem involved in the regulation of cognitive performance and behavior. This study enhances our understanding of how to achieve, control, or destroy synchrony in an important class of models. This investigation fosters research at the interface between theories of dynamical systems, stochastic processes, and algebraic graph theory. The results of this research will be integrated into graduate courses in dynamical systems and mathematical neuroscience that are taught by the PI at Drexel University. This grant supports one graduate student and sponsors summer research for two undergraduate students.