Topological Analysis of Neural Systems

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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.