Topological Data Analysis to Improve Mitochondria Measurements in Neurobiology

Sammanfattning

In this project mitochondria from rodent brain cells are investigated using topological data analysis (TDA), for the purpose of analyzing and classifying them and improving measurement tools. The analysis is built on data invariants known as stable ranks, which can be calculated by from data point clouds building distance spaces, after which so-called Vietoris-Rips complexes are found through which the invariants are calculated. By calculating these, a "fingerprint" representing each mitochondrial data can be created. The mitochondria are represented by vertices in three dimensions, from which the distance spaces are built. The data size is however very large, and therefore subsampling, where only a randomly chosen part of the data is used, is necessary. For this reason, statistical convergence analysis is performed to determine the needed subsampling, where the convergence is measured through so-called distances between the stochastically sampled and computed data invariants. After the different convergences of the stable ranks have been investigated, a representation is chosen and used to calculate the data invariants, which are then graphically illustrated. To investigate the accuracy of the data representation, machine learning is performed where the stable ranks are divided into training and test sets. Principal component analysis (PCA) is also used to illustrate the ability of the invariants to represent and illustrate the data. Smaller subsamplings are also investigated and the data is perturbed with randomized stable ranks to investigate its robustness. Afterwards the density of the point clouds is investigated and an experiment taking this into account is conducted. The results are promising and suggest that this method of representing mitochondrial data could yield new ways of investigating the different cell types that can be found in the brain. The project is done in collaboration with Associate Professor Corrado Calì and the University of Turin.