Topological Detection in Spatially and Directionally Tuned Neural Network Activity
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Persistent homology has become the main tool in topological data analysis, using methods from algebraic topology to describe the underlying space of data sets. In this thesis, persistent homology is used to detect topological characteristics of the dynamics of head direction, grid and place cell network activity. We simulate the neural activity (neuron firing rate) of the networks (chiefly) based on simple continuous attractor network models. This activity is used to generate a poisson spike train, from which a continuous time series obtained by means of gaussian smoothing. We construct Vietoris-Rips and flag complexes based on the n-dimensional points sampled at discrete times from the time series, where n denotes the number of neurons considered, and apply persistent homology, resulting in what is known as persistence diagrams. These may reveal the topology of the underlying space of the point cloud - in this case describe the activity of the neural networks (and thus the networks themselves). The method is shown to produce interesting results, providing a way to assess the feasibility of a neural network model and to understand its properties.