Computational Methods for Studying Heterogeneous Pulse-Coupled Network Dynamics

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。这个建议的目标是开发分析和计算工具，能够解决与异构脉冲耦合动力系统相关的特定架构的动力学。这项建议有两个主要组成部分。 首先，一个图形子网扩展的发展将提供一个系统的方法来分析动态观测（如活动率或相关性）的长期动态与任何脉冲耦合网络。由于网络动力学的脉冲耦合，这个图解展开中的每个项对应于跨越原始网络的子网络的事件的因果序列。 第二，延迟微分种群动力学方程的数值算法的发展将为计算子网络扩展中的项以及求解更一般的延迟微分方程提供一种自然的方法。脉冲耦合网络是一种非常普遍和重要的动力系统类型，经常在物理科学中进行研究。事实上，任何可以用通过瞬时信息突发相互作用的连接节点的网络来表示的系统都可以正式地描述为脉冲耦合网络。例如，互联网、大脑中的神经元网络以及计算机科学中使用的许多神经网络都可以被认为是脉冲耦合网络。与研究任何脉冲耦合网络相关的主要问题之一是“它做什么？如何理解复杂的脉冲耦合网络的动力学？在本提案中开发的技术将提供一个用于分析脉冲耦合网络动态的框架，详细描述任何给定脉冲耦合网络的结构（即，网络是如何构建的）以及该网络如何运行。拟议的研究将有助于理解许多重要的脉冲耦合网络的功能，这些网络在各个领域进行了研究，从图像处理和模式检测到机器人技术和电路设计，再到神经科学。

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.