Structure-Function Network Mapping and Its Assessment via Persistent Homology

Understanding the relationship between brain structure and function is a fundamental problem 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 network similarity based on persistent homology for assessing the quality of the network mapping, 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 currently 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 random 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 structure-function relationship. Our data-driven approach allows the inferred functional connectivity 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 measure 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 changes across all possible thresholds, and thus effectively circumvents the problem of selecting the arbitrary threshold for the resulting functional networks. Our results show that our approach is able to uncover both direct and indirect structural paths for predicting functional connectivity in all three connectivity datasets, suggesting that the resting-state functional connectivity is at least in part mediated by indirect pathways, in addition to direct structural connections. The finding of a nonlinear relationship between brain structure and function is conceptually new, thus advances our understanding of how structural networks shape functional networks. This work demonstrates the potential utility of our approach in a rapidly growing field of network neuroscience.