Curvature-based Methods for Brain Network Analysis

The human brain forms functional networks on all spatial scales. Modern fMRI scanners allow for resolving functional brain data in high resolution, enabling the study of large-scale networks that relate to cognitive processes. The analysis of such networks forms a cornerstone of experimental neuroscience. Due to the immense size and complexity of the underlying data sets, efficient evaluation and visualization pose challenges for data analysis. In this study, we combine recent advances in experimental neuroscience and applied mathematics to perform a mathematical characterization of complex networks constructed from fMRI data. We use task-related edge densities [Lohmann et al., 2016] for constructing networks whose nodes represent voxels in the fMRI data and whose edges represent the task-related changes in synchronization between them. This construction captures the dynamic formation of patterns of neuronal activity and therefore efficiently represents the connectivity structure between brain regions. Using geometric methods that utilize Forman-Ricci curvature as an edge-based network characteristic [Weber et al., 2017], we perform a mathematical analysis of the resulting complex networks. We motivate the use of edge-based characteristics to evaluate the network structure with geometric methods. Our results identify important structural network features including long-range connections of high curvature acting as bridges between major network components. The geometric features link curvature to higher order network organization that could aid in understanding the connectivity and interplay of brain regions in cognitive processes.