Topological Analysis of Neural Systems

哺乳动物的大脑由大量的神经元组成，每个神经元都通过树突和轴突与数千个相邻的神经元相连。大脑通过沿着这些导线在神经元之间传递电信号来处理信息。神经元以一种定向的方式自然地相互连接。这些连接形成了一个极其复杂的网络，其结构被认为对其功能至关重要。“活的”或“兴奋的”神经系统是正在经历随时间变化的电化学过程的系统。大脑的结构结构（以及生物、生态、技术和社会网络）通常使用图论进行研究，其中网络被视为由顶点和边缘组成的图，分别模拟神经元和它们之间的连接。人们普遍认为，这些网络的潜在结构塑造了它们的新兴动态，尽管缺乏理解网络结构和功能之间关系的系统方法。神经科学研究通常会产生大量的数据。分析、统计、动力系统和图论的方法在神经科学中得到了应用，并取得了显著的成果。一方面，随着过去10-15年拓扑学应用的增长，另一方面，考虑到神经科学研究中出现的数据自然适合于拓扑分析，令人惊讶的是，到目前为止，拓扑方法才刚刚开始被引入这一学科。这个项目将是用代数拓扑方法解决神经科学问题的主要尝试。该项目的主要数据来源将是在由蓝脑计划（BBP）设计和建造的超级计算机上对一只年轻老鼠的新皮质柱进行数字重建。重建是基于丰富的生物数据与强约束随机过程相结合，并提供一个生物学精确的模型，从中可以提取结构和功能数据在一个前所未有的细节水平。从BBP重构中可以提取可以表示为图的连通性矩阵的数据。更丰富的结构可以通过给图分配适当的权重来表示。这个项目的指导思想是，在神经系统的结构和功能中编码的大部分信息都以高维结构表达出来，人们可以将这种高维结构与这种图联系起来。我们将考虑数据图，为某些应用程序在图上引入权重系统，并通过各种方法将拓扑空间关联到这些图中，这些方法将允许我们从所考虑的对象的数学属性中推断出生物信息。这个项目的挑战是找到一种方法，使神经科学数据产生的拓扑结构，无论来源是BBP还是其他，都能揭示数据中编码的属性和特征。神经科学通常会产生“嘈杂”的数据。然而，任何生物的大脑都能够持续地执行非常复杂的任务。一般来说，神经科学家不断寻找的是数据中的不变属性。拓扑学非常适合于检测几何结构的不变性。因此，这个项目的目的基本上是通过神经系统的结构和功能产生的拓扑结构来发现检测神经系统一致行为的方法。

The mammalian brain is populated by a huge number of neurons, each connected to thousands of its neighbours by dendrites and axons. The brain processes information by sending electrical signals from neuron to neuron along these wires. Neurons are naturally connected to each other in a directed fashion. These connections form an immensely complicated network, whose structure is believed to be of crucial importance to its functionality. A "live" or "excited" neural system is system that is undergoing an electro-chemical process that varies with time. The structural architecture of the brain (as well as of biological, ecological, technological, and social networks) is typically studied using graph theory, where the network is viewed as a graph comprised of vertices and edges that model neurons and connections between them, respectively. It is universally accepted that the underlying structure of these networks shapes their emergent dynamics, even though a systematic approach to understanding the relationship between the structure and function of a networks is lacking. Neuroscience research typically produces immense amounts of data. Methods of analysis, statistics, dynamical systems and graph theory have been used in neuroscience and yielded remarkable results. With the growth of applications of topology in the past 10-15 years on one hand, and considering the fact that data emerging from neuroscience research naturally lends itself to topological analysis on the other hand, it is surprising that so far topological methods are only now starting to be introduced to the subject. This project will be a major attempt to address neuro-scientific questions by the methods of algebraic topology. A primary source of data for this project will be the digital reconstruction of the neocortical column of a young rat on a supercomputer, designed and built by Blue Brain Project (BBP). The reconstruction is based on rich biological data combined with strongly constrained stochastic processes and provide a biologically accurate model from which one can extract structural and functional data at an unprecedented level of detail. From the BBP reconstruction one can extract data that can be expressed as a connectivity matrix of a graph. Richer structures can be expressed by assigning appropriate weights to the graph. The guiding philosophy in this project is that much of the information encoded in the structure and function of a neural system expresses itself in high dimensional structure that one can associate to such graphs. We will consider data graphs, introduce systems of weights on graphs for certain applications, and to those graphs we will associate topological spaces by a variety of methods that will allow us to infer biological information from mathematical properties of the objects under consideration. The challenge in this project is to find ways in which the topology arising from neuro-scientific data, whether the source is the BBP or otherwise, reveals properties and features encoded in the data. Neuroscience typically produces "noisy" data. Yet, the brain of any living being is capable of performing remarkably complicated tasks consistently. It is the invariant properties within the data that neuroscientists in general are constantly searching for. Topology is perfectly suitable for detecting invariant properties in geometric structures. Thus the aim of this project is by and large to discover ways of detecting consistent behaviour of neural systems through the topology their structure and function give rise to.