Homological scaffolds of brain functional networks

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.

Figure 1.

Panels (a,b) display an unweighted network and its clique complex, obtained by promoting cliques to simplices. Simplices can be intuitively thought as higher-dimensional interactions between vertices, e.g. as a simplex the clique (b,c,i) corresponds to a filled triangle and not just its sides. The same principle applies to cliques—thus simplices—of higher order. (Online version in colour.)

Figure 2.

Panels (a­–c) display a weighted network (a), its intuitive representation in terms of a stratigraphy in the weight structure according the weight filtration described in the main text (b) and the persistence diagram for H₁ associated with the network shown (c). By promoting cliques to simplices, we identify network connectivity with relations between the vertices defining the simplicial complex. By producing a sequence of networks through the filtration, we can study the emergence and relative significance of specific features along the filtration. In this example, the hole defined by (a,b,c,d) has a longer persistence (vertical solid green bars) implying that the boundary of the cycle are much heavier than the internal links that eventually close it. The other hole instead has a much shorter persistence, surviving only for one step and is therefore considered less important in the description of the network homological properties. Note that the births and deaths are defined along the sequence of descending edge weights in the network, not in time. (Online version in colour.)

Figure 3.

Probability densities for the H₁ generators. Panel (a) reports the (log-)probability density for the placebo group, whereas panel (b) refers to the psilocybin group. The placebo displays a uniform broad distribution of values for the births–deaths of H₁ generators, whereas the plot for the psilocybin condition is very peaked at small values with a fatter tail. These heterogeneities are evident also in the persistence distribution and find explanation in the different functional integration schemes in placebo and drugged brains. (Online version in colour.)

Figure 4.

Comparison of persistence π and birth β distributions. Panel (a) reports the H₁ generators' persistence distributions for the placebo group (blue line) and psilocybin group (red line). Panel (b) reports the distributions of births with the same colour scheme. It is very easy to see that the generators in the psilocybin condition have persistence peaked at shorter values and a wider range of birth times when compared with the placebo condition. (Online version in colour.)

Figure 5.

Statistical features of group homological scaffolds. Panel (a) reports the (log-binned) probability distributions for the edge weights in the persistence homological scaffolds (main plot) and the frequency homological scaffolds (inset). While the weights in the frequency scaffold are not significantly different, the weight distributions for the persistence scaffold display clearly a broader tail. Panel (b) shows instead the scatter plot of the edge frequency versus total persistence. In both cases, there is a clear linear relationship between the two, with a large slope in the psilocybin case. Moreover, the psilocybin scaffold has a larger spread in the frequency and total persistence of individual edges, hinting to a different local functional structure within the functional network of the drugged brains. (Online version in colour.)

Figure 6.

Simplified visualization of the persistence homological scaffolds. The persistence homological scaffolds (a) and (b) are shown for comparison. For ease of visualization, only the links heavier than 80 (the weight at which the distributions in figure 5a bifurcate) are shown. This value is slightly smaller than the bifurcation point of the weights distributions in figure 5a. In both networks, colours represent communities obtained by modularity [49] optimization on the placebo persistence scaffold using the Louvain method [50] and are used to show the departure of the psilocybin connectivity structure from the placebo baseline. The width of the links is proportional to their weight and the size of the nodes is proportional to their strength. Note that the proportion of heavy links between communities is much higher (and very different) in the psilocybin group, suggesting greater integration. A labelled version of the two scaffolds is available as GEXF graph files as the electronic supplementary material. (Online version in colour.)