Topological Analysis of Neural Systems

神经系统的拓扑分析

• 批准号: EP/P025072/1
• 负责人: Ran Levi
• 金额: $ 90.63万
• 依托单位: University of Aberdeen
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FP025072%2F1
• 关键词: Topological; Analysis; Neural; Systems

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.

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。大脑通过这些电线将电信号从神经元发送到神经元，从而处理信息。神经元自然地以定向的方式相互连接。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”神经系统是指正在经历一个随时间变化的电化学过程的系统。大脑的结构架构(以及生物、生态、技术和社会网络)通常使用图论来研究，其中网络被视为由分别建模神经元和它们之间的连接的顶点和边组成的图。人们普遍认为，这些网络的基本结构决定了它们的涌现动态，尽管缺乏系统的方法来理解网络结构和功能之间的关系。神经科学研究通常会产生海量数据。分析、统计、动力系统和图论等方法在神经科学中得到了广泛的应用，并取得了显著的成果。一方面，随着拓扑学在过去10-15年中应用的增长，另一方面，考虑到神经科学研究中产生的数据自然有助于拓扑分析，令人惊讶的是，到目前为止，拓扑学方法才刚刚开始引入这一学科。这个项目将是用代数拓扑学的方法解决神经科学问题的一次重大尝试。该项目的主要数据来源将是由蓝脑项目(BBP)设计和建造的超级计算机上对一只年轻大鼠的新皮质柱进行的数字重建。重建是基于丰富的生物数据和强约束的随机过程，并提供了一个生物准确的模型，从中人们可以提取前所未有的详细程度的结构和功能数据。从BBP重构中，可以提取可以表示为图的连通性矩阵的数据。通过为图形赋予适当的权重，可以表达更丰富的结构。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许多信息以高维结构表达自己，人们可以将其与这样的图表联系在一起。我们将考虑数据图，为某些应用引入图上的权重系统，并通过各种方法将拓扑空间与这些图相关联，这些方法将允许我们从所考虑对象的数学属性中推断生物信息。这个项目的挑战是找到一种方法，让神经科学数据产生的拓扑图揭示数据中编码的属性和特征，无论来源是BBP还是其他。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始终如一地执行非常复杂的任务。这是神经科学家们一直在寻找的数据中的不变特性。拓扑学非常适合于检测几何结构中的不变性。因此，这个项目的目的大体上是通过神经系统的结构和功能引起的拓扑结构来发现检测神经系统一致行为的方法。

项目成果

How many simplices are needed to triangulate a Grassmannian?

对格拉斯曼函数进行三角剖分需要多少个单纯形？

• DOI: 10.12775/tmna.2020.027
• 发表时间: 2020
• 期刊: Topological Methods in Nonlinear Analysis
• 影响因子: 0.7
• 作者: Govc D

An application of neighbourhoods in digraphs to the classification of binary dynamics.

• DOI: 10.1162/netn_a_00228
• 发表时间: 2022-06
• 期刊: NETWORK NEUROSCIENCE
• 影响因子: 4.7
• 作者: Conceicao, Pedro;Govc, Dejan;Lazovskis, Janis;Levi, Ran;Riihimaki, Henri;Smith, Jason P.
• 通讯作者: Smith, Jason P.

Asymptotic behaviour of the containment of certain mesh patterns

• DOI: 10.1016/j.disc.2022.112813
• 发表时间: 2020-11
• 期刊: Discret. Math.
• 作者: Dejan Govc;Jason P. Smith

Persistent magnitude

持续震级

• DOI: 10.1016/j.jpaa.2020.106517
• 发表时间: 2021
• 期刊: Journal of Pure and Applied Algebra
• 影响因子: 0.8
• 作者: Govc D

Moduli spaces of morse functions for persistence

持久性莫尔斯函数的模空间

• DOI: 10.1007/s41468-020-00055-x
• 发表时间: 2020
• 期刊: Journal of Applied and Computational Topology
• 作者: Catanzaro, Michael J.;Curry, Justin M.;Fasy, Brittany Terese;Lazovskis, Jānis;Malen, Greg;Riess, Hans;Wang, Bei;Zabka, Matthew
• 通讯作者: Zabka, Matthew