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RESEARCH ARTICLE Structure-Function Network Mapping and Its Assessment via Persistent Homology Hualou Liang 1 * , Hongbin Wang 2 1 School of Biomedical Engineering, Science & Health Systems, Drexel University, Philadelphia, PA, United States of America, 2 Center for Biomedical Informatics, Texas A&M University Health Science Center, Houston, TX, United States of America * [email protected] Abstract Understanding the relationship between brain structure and function is a fundamental prob- lem in network neuroscience. This work deals with the general method of structure-function mapping at the whole-brain level. We formulate the problem as a topological mapping of structure-function connectivity via matrix function, and find a stable solution by exploiting a regularization procedure to cope with large matrices. We introduce a novel measure of net- work similarity based on persistent homology for assessing the quality of the network map- ping, which enables a detailed comparison of network topological changes across all possible thresholds, rather than just at a single, arbitrary threshold that may not be optimal. We demonstrate that our approach can uncover the direct and indirect structural paths for predicting functional connectivity, and our network similarity measure outperforms other cur- rently available methods. We systematically validate our approach with (1) a comparison of regularized vs. non-regularized procedures, (2) a null model of the degree-preserving ran- dom rewired structural matrix, (3) different network types (binary vs. weighted matrices), and (4) different brain parcellation schemes (low vs. high resolutions). Finally, we evaluate the scalability of our method with relatively large matrices (2514x2514) of structural and functional connectivity obtained from 12 healthy human subjects measured non-invasively while at rest. Our results reveal a nonlinear structure-function relationship, suggesting that the resting-state functional connectivity depends on direct structural connections, as well as relatively parsimonious indirect connections via polysynaptic pathways. Author Summary One of the major challenges in neuroscience is to understand how brain structure is related to function. In this work, we present a whole-brain method to quantify the struc- ture-function relationship. Our data-driven approach allows the inferred functional con- nectivity matrix to be represented as a weighted sum of the powers of the structural matrix, containing both direct and indirect pathways. We further introduce a novel mea- sure of network similarity based on persistent homology for assessing the goodness of fit for the mapping; such a measure enables the complete comparison of network topological PLOS Computational Biology | DOI:10.1371/journal.pcbi.1005325 January 3, 2017 1 / 19 a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 OPEN ACCESS Citation: Liang H, Wang H (2017) Structure- Function Network Mapping and Its Assessment via Persistent Homology. PLoS Comput Biol 13(1): e1005325. doi:10.1371/journal.pcbi.1005325 Editor: Danielle S. Bassett, University of Pennsylvania, UNITED STATES Received:
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