Uncovering Cortical Units of Processing From Multi-Layered Connectomes

Abstract

Modern diffusion and functional magnetic resonance imaging (dMRI/fMRI) provide non-invasive high-resolution images from which multi-layered networks of whole-brain structural and functional connectivity can be derived. Unfortunately, the lack of observed correspondence between the connectivity profiles of the two modalities challenges the understanding of the relationship between the functional and structural connectome. Rather than focusing on correspondence at the level of connections we presently investigate correspondence in terms of modular organization according to shared canonical processing units. We use a stochastic block-model (SBM) as a data-driven approach for clustering high-resolution multi-layer whole-brain connectivity networks and use prediction to quantify the extent to which a given clustering accounts for the connectome within a modality. The employed SBM assumes a single underlying parcellation exists across modalities whilst permitting each modality to possess an independent connectivity structure between parcels thereby imposing concurrent functional and structural units but different structural and functional connectivity profiles. We contrast the joint processing units to their modality specific counterparts and find that even though data-driven structural and functional parcellations exhibit substantial differences, attributed to modality specific biases, the joint model is able to achieve a consensus representation that well accounts for both the functional and structural connectome providing improved representations of functional connectivity compared to using functional data alone. This implies that a representation persists in the consensus model that is shared by the individual modalities. We find additional support for this viewpoint when the anatomical correspondence between modalities is removed from the joint modeling. The resultant drop in predictive performance is in general substantial, confirming that the anatomical correspondence of processing units is indeed present between the two modalities. Our findings illustrate how multi-modal integration admits consensus representations well-characterizing each individual modality despite their biases and points to the importance of multi-layered connectomes as providing supplementary information regarding the brain's canonical processing units.

Keywords: brain parcellation; dMRI; fMRI; multi-layered connectomes; stochastic block model.

Figure 1

Concept of data-driven multi-layer network modeling. (A) Based on functional and structural MRI, graphs of structural and functional connectivity are generated, (B) such that vertices are defined on the same standard surface mesh. (C) From these graphs data-driven parcellations can be inferred and compared, when modeling structure or function either individually or jointly.

Figure 2

Examination of functional connectivity (left) and structural connectivity (right) using the HCP_MMP1.0 atlas parcellation (Glasser et al., 2016). The cortical surface and flatmaps show the link-density within each of the 360 atlas parcels whereas the adjacency matrices outline the whole brain functional and structural connectivity graphs, based on the average of different populations of 50 subjects.

Figure 3

The Stochastic Block Model (SBM) is a generative model capable of discovering a single group-structure from multiple complex networks, i.e., functional connectome A^(f) and structural connectome A^(s). Based on such a shared parcellation z, the model assumes links are independently generated from a Bernoulli distribution such that the probability of observing a link between any two vertices only depends on the modality specific probability of observing a link between the two parcels that the vertices belong to given by the inter parcel (off diagonal elements) and intra parcel (diagonal elements) of the link density matrices η^(f) and η^(s) for the functional and structural connectomes, respectively. Under this assumption, the model hence allows a single parcellation to be inferred from multiple networks while accounting for differences in connectivity profiles. Let Z denote a matrix of the clustering z using a one-hot encoding. Notably, the SBM can be considered a lossy compressed representation of the connectomes such that A^(f) ≈ Z^⊤η^(f)Z and A^(s) ≈ Z^⊤η^(s)Z.

Figure 4

The steps that defines the flow of the investigations. Based on high-resolution functional and structural MRI obtained from publicly released data of the Human Connectome Project, independent networks of structural and functional connectivity are generated. The networks are based on population averages, resulting in a total of five networks for each modality, based on five populations of 50 subjects. The networks are binarized by thresholding at 1% link density. Stochastic Blockmodeling is utilized to infer data-driven parcellations, based on modeling structure or function individually or jointly modeling both modalities for the five populations. The benefits of multimodal integration are hence evaluated by comparing the performance of predicting hold-out population networks using inferred single and multimodal parcellations, contrasted with that of using networks that have been spatially permuted whilst preserving the size of the parcels.

Figure 5

(Top) Flatmap of each vertex in the HCP data color-coded according to the vertex traversal order of the network adjacency matrices. (Middle) Example of parcellation structure learned from the non-predicted modality. (Bottom) The random permutation of the non-predicted modality obtained by re-ordering the vertex traversal according to the learned parcellation structure in which the clusters are ordered in random order. Color-code indicates the original vertex position.

Figure 6

Flatmaps of the inferred parcellations for the five populations of 50 subjects, using the different modalities. Parcels are separated into three groups based on their size: small (<100 vertices), medium (between 100 and 1,000 vertices) and big (more than 1,000 vertices). Also shown, for the per-modality averages of the five parcellations, is the number of parcels that contain nodes from both hemispheres and how evenly these nodes are split across the two hemispheres. Errorbars and ± indicate the standard deviation over the five parcellations.

Figure 7

Averaged Normalized Mutual Information (NMI) for the inferred parcellations between and within modalities, showing the average NMI from 10 comparisons of parcellations between populations and five comparisons within populations. The standard deviations are shown in parenthesis. The flatmaps illustrate one of the five parcellations for a single 50-subject population, showing only the left hemisphere.

Figure 8

AUC and expected predictive log-likelihood when predicting structural and functional hold-out network. The bars show the average score as obtained using the following different parcellations: (1) SBM on the same modality as the test networks, (2) SBM on the other modality, (3) SBM on the permuted networks for the other modality, (4) jointly modeling both modalities, (5) jointly modeling both modalities with the network for the other modality permuted, and (6) using the parcellation defined by the HCP atlas. For each of the five training populations, four evaluations are computed for each modality configuration, when respectively predicting from the training graph to each of the other four graphs of the same modality (providing a total of 20 predictions). The mean value is shown for each bar while the whiskers indicate the standard deviation of the mean correcting by the five independently acquired networks ($\pm std/\sqrt{5}$).