This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The goal of this proposal is to develop analytical and computational tools capable of addressing the architecture-specific dynamics associated with heterogeneous pulse-coupled dynamical systems. There are two major components to this proposal. First, the development of a diagrammatic subnetwork expansion will provide a systematic way of analyzing dynamic observables (such as activity rates or correlations) for the long-time dynamics associated with any pulse-coupled network. Due to the pulse-coupling of the network dynamics, each term within this diagrammatic expansion corresponds to a causal sequence of events spanning a subnetwork of the original network. Second, the development of numerical algorithms for solving delay-differential population-dynamics equations will provide a natural method for computing the terms within the subnetwork expansion, as well as for solving more general delay-differential equations.
Pulse-coupled networks are a very general, and important, type of dynamical system which are often studied within the physical sciences. Indeed, any system which can be represented using a network of connected nodes which interact through instantaneous bursts of information can be formally described as a pulse-coupled network. For example, the internet, networks of neurons within the brain, and many neural networks used in computer science can all be thought of as pulse-coupled networks. One of the major questions associated with the study of any pulse-coupled network is "what does it do?" How can a complicated pulse-coupled network's dynamics be understood? The techniques to be developed within this proposal will provide a framework for analyzing the dynamics of pulse-coupled networks, casting a detailed picture of the relationship between any given pulse-coupled network's structure (i.e., how the network is built) and how that network behaves. The proposed research will aid in understanding the function of many important pulse-coupled networks studied in a variety of fields, ranging from image-processing and pattern detection to robotics and circuit design to neuroscience.
