Simplicial and Topological Descriptions of Human Brain Dynamics

Whereas brain imaging tools like functional Magnetic Resonance Imaging (fMRI) afford measurements of whole-brain activity, it remains unclear how best to interpret patterns found amid the data9s apparent self-organization. To clarify how patterns of brain activity support brain function, one might identify metric spaces that optimally distinguish brain states across experimentally defined conditions. Therefore, the present study considers the relative capacities of several metric spaces to disambiguate experimentally defined brain states. One fundamental metric space interprets fMRI data topographically, i.e, as the vector of amplitudes of a multivariate signal, changing with time. Another perspective considers the condition-dependency of the brain9s Functional Connectivity (FC), i.e., the similarity matrix computed across the variables of a multivariate signal. More recently, metric spaces that think of the data topologically, e.g., as an abstract geometric object, have become available. In the abstract, uncertainty prevails regarding the distortions imposed by the mode of measurement upon the object under study. Features that are invariant under continuous deformations, such as rotation and inflation, constitute the features of topological data analysis. While there are strengths and weaknesses of each metric space, we find that metric spaces that track topological features are optimal descriptors of the brain9s experimentally defined states.