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，必须首先理解节点之间和节点之间脉冲耦合相互作用的动态后果。每个单独节点（如神经元）的动态都是高度非线性的，潜在的网络拓扑结构绝不是同质的，最相关的网络现象在多个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，开发一个通用的概念框架是至关重要的，该框架能够在复杂的网络拓扑结构上对大规模脉冲耦合网络动态进行粗粒度的统计描述。目前的数学工具，如统计力学的技术，不能很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，平稳性，相互作用网络的同质性等）根本不成立。拟议研究的目的是通过以下方式迈出解决这些概念性问题的第一步：(i)系统地扩展了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；以及由于网络拓扑结构而产生的网络活动波动（ii）为脉冲耦合网络系统寻找熵的适当定义，该定义将作为以下两方面的统一原则：(a)扩展最大熵原理，最大熵原理已成功地简化了与某些理想化脉冲耦合网络系统相关的动力学，以及(b)表征现实网络拓扑结构上网络动力学的全局波动的性质。组织原则的理论框架的发展，可以捕捉复杂拓扑网络动态的统计行为，这对于理解一般网络拓扑的一般信息传输和处理具有潜在的变革性。这里提出的理论方法的成功实施将对我们如何模拟大规模脉冲耦合网络动力学，特别是神经网络动力学，以及我们如何从新的粗粒度角度理论上研究大脑动力学产生强烈影响。这将是解决结构连接如何成为大脑功能和有效连接的基础这一科学挑战的第一步。重要的是要强调，本提案中解决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，例如化学反应级联，遗传网络，交通网络等，特别是系统神经科学。这项工作将为博士后和研究生提供令人兴奋的研究项目，包括应用数学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.