Dynamics of Large Networks

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A number of very important problems in science and technology lead to the analysis of large networks of interacting dynamical systems. Our ability to predict epileptic seizures, to effectively control a power grid, or to coordinate a group of robots rely on our understanding of the principles underlying collective behavior in coupled dynamical systems. Many natural and man-made networks around us feature extraordinary richness and complexity of interconnections. Mathematical modeling of such networks poses new challenges for nonlinear science and requires new approaches incorporating combinatorial and probabilistic methods into dynamical analysis of complex systems. Graph Theory holds an extraordinary potential to inspire new powerful techniques for extended dynamical systems and applications to technological, social, economic, and biological networks. In this research, the Principal Investigator (PI) combines state-of-the-art techniques of Graph Theory with analytical methods for Dynamical Systems to develop an effective set of tools for studying coupled dynamical systems and their applications in neuroscience. The results of this research will be integrated into graduate courses in Dynamical Systems and Mathematical Neuroscience. In this project, the PI develops a unified approach for studying dynamical networks as evolution equations on three types of spatial domains: Caley graphs, quasirandom graphs, and graph limits. For problems in each class, analytical and algebraic techniques, which mesh well with the underlying spatial structures, are identified. These techniques are used to study stability of spatial patterns in systems of coupled phase oscillators on Caley graphs, synchronization in systems of coupled chaotic maps, and differential equations on different random graphs including small-world and graphs that exhibit power law behavior, as well as those that exhibit temporally structured stable patterns in a neural network model of learning. The PI seeks systematic ways for describing the role of network connectivity in shaping the dynamics in coupled systems. This work is aimed toward development of a theory for interacting dynamical systems on regular and random graphs.