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

过去几十年积累的实验和理论证据表明，为了理解复杂信息处理系统(如大脑)研究中产生的网络动力学，必须首先了解节点之间和节点之间脉冲耦合相互作用的动力学后果。每个单独节点(如神经元)的动力学是高度非线性的，底层网络拓扑绝不是同质的，最相关的网络现象在许多时空尺度上表现出结构特征。为了促进对复杂信息处理系统研究中产生的脉冲耦合网络的理解，关键是开发一个通用的概念框架，该框架能够对复杂网络拓扑上的大规模脉冲耦合网络动力学进行粗粒度的统计描述。目前的数学工具，例如来自统计力学的技术，不能容易地应用于这些脉冲耦合系统，因为这些技术所基于的大多数假设(例如，相互作用网络的平稳性、均质性等)。根本就是拿不住。所提议的研究的目的是为解决这些概念性问题迈出第一步，方法是：(I)系统地扩展不相关的均匀脉冲耦合网络动力学的动力学理论公式，以纳入网络内脉冲耦合节点之间的成对相关性，以及作为网络拓扑的结果而引起的网络活动的波动；(Ii)为脉冲耦合网络系统寻找将用作统一原则的适当的熵定义，以(A)扩展已成功简化与某些理想化的脉冲耦合网络系统相关联的动力学的最大熵原理，以及(B)描述现实网络拓扑上网络动态的全局波动的性质。建立能够捕获复杂拓扑上的网络动态的统计行为的组织原则的理论框架的发展在理解一般网络拓扑上的一般信息传输和处理方面具有潜在的变革。本文提出的理论方法的成功实施将对我们如何建模大规模的脉冲耦合网络动力学，特别是神经元网络动力学，以及如何从一个新的粗粒度角度从理论上研究大脑动力学产生重大影响。这将是朝着迎接科学挑战迈出的第一步，即解决结构连接如何支撑大脑功能和有效连接的问题。必须强调的是，本提案中涉及的一般理论问题将有助于分析和理解复杂网络上的许多其他动力学，如化学反应级联、遗传网络、交通网络等，特别是对系统神经科学。拟议的工作将为博士后和研究生提供令人兴奋的应用数学研究项目，以及系统神经科学产生的理论问题。

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.