  The human brain forms functional networks on all spatial scales. Modern fMRI
scanners allow to resolve functional brain data in high resolutions, allowing
to study 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 remain a challenge 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 of task-related changes in synchronization. This
construction captures the dynamic formation of patterns of neuronal activity
and therefore represents efficiently 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. The geometric features could aid in understanding the connectivity and
interplay of brain regions in cognitive processes.
