Exploring persistent homology as a method for capturing functional connectivity differences in Parkinson’s Disease.

Abstract: Parkinson’s Disease (PD) is the fastest growing neurodegenerative disease, currently affecting two to three percent of the population over 65. Studying functional connectivity (FC) in PD patients may provide new insights into how the disease alters brain organization in different subjects. We explored persistent homology (PH) as a method for studying FC based on the functional magnetic resonance imaging (fMRI) recordings of 63 subjects, of which 56 were diagnosed with PD.  We used PH to translate each set of fMRI recordings into a stable rank. Stable ranks are homological invariants that are amenable for statistical analysis. The pipeline has multiple parameters, and we explored the effect of these parameters on the shape of the stable ranks. Moreover, we fitted functions to reduce the stable ranks to points in two or three dimensions. We clustered the stable ranks based on the fitted parameter values and based on the integral distance between them. For some of the parameter combinations, not all clusters were located in the space covered by controls. These clusters correspond to patients with a topologically distinct connectivity structure, which may be clinically relevant. However, we found no relation between the clusters and the medication status or cognitive ability of the patients. It should be noted that this study was an exploration of applying persistent homology to PD data, and that statistical testing was not performed. Consequently, the presented results should be considered with care. Furthermore, we did not explore the full parameter space, as time was limited and the data set was small. In a follow-up study, a measurable desired outcome of the pipeline should be defined and the data set should be expanded to allow for optimizing over the full parameter space.