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.
A number of very important problems in science and technology involve complex networks of interconnected dynamical systems. Among these problems, synchronization holds a special place. With applications ranging from the genesis of epilepsy to stability of power networks, understanding principles underlying synchronization is of the utmost importance. In this research, the Principal Investigator (PI) studied systems of coupled differential equations on large graphs modeling neuronal networks and multi-agent systems. For coupled systems with different types of dynamics at the individual nodes of the graph, including oscillatory, excitable, and those with multiple time scales, and for a wide range of network architecture, the PI derived sufficient conditions for synchronization, studied transition to synchrony, and its stability to stochastic perturbations. These results cover networks on Caley, random, and small-world graphs, to name a few. Using the combination of the state-of-the-art techniques of the graph theory and analytical methods from the theory of dynamical systems, the PI developed new mathematical approaches for studying synchronization in large networks. The results of this work show how network synchronizability depends on the structure of its connections. The new techniques developed in the course of this research were used to study patterns of electrical activity in Locus Coeruleus network in mammalian brainstem involved in the regulation of cognitive performance and dynamics of electrically coupled pancreatic beta cells responsible for insulin secretion. In addition, these techniques were used to study stochastic stability of consensus protocols for networks of dynamic agents. This project resulted in seven publications in peer-reviewed mathematical journals. The PI organized three minisimposia on the dynamics of networks at international conferences on applied dynamical systems. The PI organized summer research for three undergraduate students and supervised research of one Ph.D. student. The results of this work were integrated in the graduate course on Mathematical Neuroscience that PI teaches at Drexel University.
