Abstract
Networks, as efficient representations of complex systems, have appealed to scientists for a long time and now permeate many areas of science, including neuroimaging (Bullmore and Sporns 2009 Nat. Rev. Neurosci. 10, 186–198. (doi:10.1038/nrn2618)). Traditionally, the structure of complex networks has been studied through their statistical properties and metrics concerned with node and link properties, e.g. degree-distribution, node centrality and modularity. Here, we study the characteristics of functional brain networks at the mesoscopic level from a novel perspective that highlights the role of inhomogeneities in the fabric of functional connections. This can be done by focusing on the features of a set of topological objects—homological cycles—associated with the weighted functional network. We leverage the detected topological information to define the homological scaffolds, a new set of objects designed to represent compactly the homological features of the correlation network and simultaneously make their homological properties amenable to networks theoretical methods. As a proof of principle, we apply these tools to compare resting-state functional brain activity in 15 healthy volunteers after intravenous infusion of placebo and psilocybin—the main psychoactive component of magic mushrooms. The results show that the homological structure of the brain's functional patterns undergoes a dramatic change post-psilocybin, characterized by the appearance of many transient structures of low stability and of a small number of persistent ones that are not observed in the case of placebo.
Highlights
We introduced two new objects, the homological scaffolds, to go beyond the picture given by persistent homology to represent and summarize information about individual links
The homological scaffolds represent a new measure of topological importance of edges in the original system in terms of how frequently they are part of the generators of the persistent homology groups and how persistent are the generators to which they belong to
We applied this method to an functional magnetic resonance imaging (fMRI) dataset comprising a group of subjects injected with a placebo and another injected with psilocybin
Summary
The understanding of global brain organization and its large-scale integration remains a challenge for modern neurosciences. In the case of brain functional activity, this often implies the use of similarity measures such as (partial) correlations or coherence [6,7,8], which generally yield fully connected, weighted and possibly signed adjacency matrices. Definition a graph G 1⁄4 (V, E) is a representation of a set V of nodes i interconnected by edges or links eij [ E; this interaction can be weighted, directional and signed a completely connected subgraph C 1⁄4 (V’, E’) contained in an undirected and unweighted graph G 1⁄4 (V, E) (V0 , V, eij [ E08i, j [ E0). An element of the generating set of Hk, a subset of Hk such that all elements can be expressed as combination of generators a weighted graph constructed from the persistent homology generators of H1 of a simplicial complex K between information completeness and clarity. A summary of all the keywords and concepts introduced in this paper can be found in table 1
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