标题： 分析线虫神经网络的特性：数学建模生物系统 显示英文标题

标题： Analysing properties of the C. Elegans neural network: mathematically modeling a biological system

摘要： 大脑是生物世界中研究最多、最复杂的系统之一。 它是所有脊椎动物和大多数无脊椎动物生命的信息中心，因此已成为当前研究的主要焦点。 虽然许多研究集中在直接研究大脑上，但我们的重点是大脑本身的结构：其核心是一个由节点（神经元）相互连接的网络。 对大脑结构方面的更好理解应该能阐明其一些功能特性。 在本文中，我们分析了线虫秀丽隐杆线虫的大脑。 由仅302个神经元组成，它是更被理解的神经网络之一。 使用该网络的279个神经元“巨大组件”的拉普拉斯矩阵，我们使用特征值计数函数来寻找类似分形的自相似性。 这种矩阵表示还用于绘制（在特征函数坐标中）神经网络的二维和三维可视化。 进一步的分析检查了系统的小世界特性，包括平均路径长度和聚类系数。 然后，我们使用图能和空间方差来测试特征函数的局域化。 为了更好地理解这些结果，所有这些计算也在随机网络、分支树和已知分形以及已经被“重新布线”以具有小世界特性的分形上进行。 这种分析是神经网络研究中的许多踏脚石之一。 虽然大脑内的许多结构和功能已经为人所知，但理解它们之间的相互作用也很重要。 对神经网络结构特性的更深入理解是这一过程中的关键一步。 显示英文摘要

摘要： The brain is one of the most studied and highly complex systems in the biological world. It is the information center behind all vertebrate and most invertebrate life, and thus has become a major focus in current research. While many of these studies have concentrated on studying the brain directly, our focus is the structure of the brain itself: at its core an interconnected network of nodes (neurons). A better understanding of the structural aspects of the brain should elucidate some of its functional properties. In this paper we analyze the brain of the nematode Caenorhabditis elegans. Consisting of only 302 neurons, it is one of the better-understood neural networks. Using a Laplacian matrix of the 279-neuron "giant component" of the network, we use an eigenvalue counting function to look for fractal-like self similarity. This matrix representation is also used to plot (in eigenfunction coordinates) both 2 and 3 dimensional visualizations of the neural network. Further analysis examines the small-world properties of the system, including average path length and clustering coefficient. We then test for localization of eigenfunctions, using graph energy and spacial variance. To better understand these results, all of these calculations are also performed on random networks, branching trees, and known fractals, as well as fractals which have been "rewired" to have small-world properties. This analysis is one of many stepping-stones in the research of neural networks. While many of the structures and functions within the brain are known, understanding how the two interact is also important. A firmer grasp on the structural properties of the neural network is a key step in this process