Nonequilibrium Statistical Physics Description of Pulse-Coupled Dynamics on Complex Network Topologies

New York University, New York NY

Investigators

Abstract

The last few decades of accumulated experimental and theoretical evidence indicate that, in order to understand network dynamics arising from the study of complex information processing systems, such as the brain, one must first understand dynamical consequences of the pulse-coupled interactions between and amongst nodes. The dynamics of each individual node, such as a neuron, are highly nonlinear, the underlying network topology is anything but homogeneous, and the most relevant network phenomena exhibit structural features over a multitude of spatiotemporal scales. To advance the understanding of pulsed-coupled networks arising from the study of complex information-processing systems, it is critical to develop a general conceptual framework capable of formulating coarse-grained, statistical descriptions of large-scale pulse-coupled network dynamics over complex network topologies. Current mathematical tools, such as techniques from statistical mechanics, cannot be readily applied to these pulsed-coupled systems, since most of the assumptions upon which these techniques are based (e.g., stationarity, homogeneity of the interaction network, etc.) simply do not hold. The aim of the proposed research is to take a first step towards tackling these conceptual issues via: (i) The systematic extension of kinetic theory formulations of uncorrelated homogeneous pulsed-coupled network dynamics to incorporate pairwise correlations between pulse-coupled nodes within the network, as well as fluctuations in network activity that arise as a consequence of network topology (ii) The search for an appropriate definition of entropy for pulse-coupled network systems that will serve as a unifying principle to both (a) extend the maximum entropy principle that has been successful in simplifying the dynamics associated with certain idealized pulsed-coupled network systems, and (b) characterize the nature of global fluctuations of network dynamics over realistic network topology. A development of a theoretical framework of organizing principles that can capture statistical behaviors of network dynamics over complex topologies is potentially transformative in understanding of general information transmission and processing over general network topologies. A successful implementation of theoretical methodologies proposed here will have strong impact on how we model large-scale, pulse-coupled network dynamics, in particular, neuronal network dynamics, and on how we theoretically investigate brain dynamics from a new coarse-grained perspective. This would be a first step towards undertaking the scientific challenge of addressing how structural connectivity underlies functional and effective connectivity in the brain. It is important to emphasize that the general theoretical issues addressed in this proposal will have ramifications in assisting the analysis and understanding of many other dynamics on complicated networks, such as chemical reaction cascades, genetic networks, traffic networks etc, in particular to systems neuroscience. The proposed work will provide postdocs and graduate students with exciting research projects in applied mathematics as well as in theoretical problems arising from systems neuroscience.