Reconstruction of stimulus spaces from neural activation sequences and anti-geometric persistence
Date
2023
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Publisher
University of Delaware
Abstract
This dissertation describes two distinct, but related projects. ☐ In the region of the brain known as the hippocampus, neurons known as place cells encode the structure of the animal’s physical environment. Each place cell fires rapidly when the animal is near a corresponding location in a known environment. The region in which the neuron is active is called a place field. When the animal is exploring a known environment and moving from place field to place field, place cell sequences can be observed. Surprisingly, amongst an enormous amount of noise, neural firing patterns that mimic those seen when the animal is moving can be found when the animal is at rest. In this document, we apply tools from data science, both known and novel, to study neural activity patterns such as those appearing in the hippocampus at rest, with the goal of detecting the activity corresponding with a geometric space and if so, to reconstruct the space. ☐ Next, we study a fairly new idea relating to persistent homology, which we call anti-geometric persistence. The input to many persistent homology computation algorithms is a square symmetric dissimilarity matrix, a generalization of a distance matrix, where the rows and columns correspond to points in space and the matrix entries describe the dissimilarity between the points. However, many data analysis pipelines like correlation of time series produce similarity measures, and it is a common experience among TDA practitioners to forget to transform these into dissimilarity measures before computing persistence. If this reverse step is omitted from the workflow, the persistence diagrams resulting from the computations are often wildly different from the expectation. The standard response is to discard such results and “do it right”. However, we have observed that interesting structure arises in these reverse filtrations. We present results concerning the persistence diagrams obtained by reversing the distance matrix of points distributed in euclidean space and explore other questions relating to reverse persistence.
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Keywords
Barcodes, Persistent homology, Topology, Stimulus spaces, Hippocampus