Curvature-based Methods for Brain Network Analysis

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.

[1]  Andreas Bartels,et al.  Task-Related Edge Density (TED)—A New Method for Revealing Dynamic Network Formation in fMRI Data of the Human Brain , 2015, PloS one.

[2]  Abraham Z. Snyder,et al.  Function in the human connectome: Task-fMRI and individual differences in behavior , 2013, NeuroImage.

[3]  Y. Ollivier.  Ricci curvature of Markov chains on metric spaces , 2007, math/0701886.

[4]  T. Georgiou,et al.  Market Fragility, Systemic Risk, and Ricci Curvature , 2015, 1505.05182.

[5]  Essa Yacoub,et al.  The WU-Minn Human Connectome Project: An overview , 2013, NeuroImage.

[6]  D. Schacter,et al.  The Brain's Default Network , 2008, Annals of the New York Academy of Sciences.

[7]  J. Jost,et al.  Forman curvature for complex networks , 2016, 1603.00386.

[8]  Y. Ollivier.  A survey of Ricci curvature for metric spaces and Markov chains , 2010 .

[9]  D. Louis Collins,et al.  Unbiased average age-appropriate atlases for pediatric studies , 2011, NeuroImage.

[10]  Emil Saucan,et al.  Forman-Ricci Flow for Change Detection in Large Dynamic Data Sets , 2016, Axioms.

[11]  Daniel S. Margulies,et al.  A software tool for interactive exploration of intrinsic functional connectivity opens new perspectives for brain surgery , 2011, Acta Neurochirurgica.

[12]  Emil Saucan,et al.  Discrete curvatures and network analysis , 2017 .

[13]  Alessandro Vespignani,et al.  Detecting global bridges in networks , 2015, J. Complex Networks.

[14]  Robin Forman,et al.  Bochner's Method for Cell Complexes and Combinatorial Ricci Curvature , 2003, Discret. Comput. Geom..

[15]  Mark Jenkinson,et al.  The minimal preprocessing pipelines for the Human Connectome Project , 2013, NeuroImage.

[16]  Emil Saucan,et al.  Can one see the shape of a network? , 2016, ArXiv.

[17]  Yi Pan,et al.  Identification of Essential Proteins Based on Edge Clustering Coefficient , 2012, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[18]  Emil Saucan,et al.  Characterizing complex networks with Forman-Ricci curvature and associated geometric flows , 2016, J. Complex Networks.

[19]  Jie Gao,et al.  Discrete Ricci Flow for Geometric Routing , 2016, Encyclopedia of Algorithms.