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Hierarchical modularity in human brain functional networks
D. Meunier, R. Lambiotte, A. Fornito, K. D. Ersche, E. T. Bullmore
TL;DR
The paper addresses limitations in studying hierarchical modularity in brain networks, including computational demands and single-level analyses. It applies an efficient algorithm to individual fMRI networks and compares their community structures using mutual information. The resulting methods provide computationally feasible, high-resolution hierarchical decompositions with plausible and reasonably consistent group-level descriptions.
Problem
Previous neuroimaging studies were limited by computational time, network size, single-level community analyses, and difficulty quantifying similarity between modular decompositions.
Method
The study applies a computationally efficient algorithm to derive hierarchical modular decompositions of individual fMRI brain networks and compares subjects using mutual information.
Results
The methods generated plausible and reasonably consistent descriptions of brain functional-network community structure in a group.
Takeaways & Limitations
Computationally feasible methods are available for rapid, high-resolution hierarchical modular decomposition of individual brain functional networks.
Takeaways & Limitations
The algorithm’s validation remained arguable because it had not yet been studied in detail, while prior methods addressed only relatively low-resolution networks.
Abstract
from arXiv · showhide
The idea that complex systems have a hierarchical modular organization originates in the early 1960s and has recently attracted fresh support from quantitative studies of large scale, real-life networks. Here we investigate the hierarchical modular (or "modules-within-modules") decomposition of human brain functional networks, measured using functional magnetic resonance imaging (fMRI) in 18 healthy volunteers under no-task or resting conditions. We used a customized template to extract networks with more than 1800 regional nodes, and we applied a fast algorithm to identify nested modular structure at several hierarchical levels. We used mutual information, 0 < I < 1, to estimate the similarity of community structure of networks in different subjects, and to identify the individual network that is most representative of the group. Results show that human brain functional networks have a hierarchical modular organization with a fair degree of similarity between subjects, I=0.63. The largest 5 modules at the highest level of the hierarchy were medial occipital, lateral occipital, central, parieto-frontal and fronto-temporal systems; occipital modules demonstrated less sub-modular organization than modules comprising regions of multimodal association cortex. Connector nodes and hubs, with a key role in inter-modular connectivity, were also concentrated in association cortical areas. We conclude that methods are available for hierarchical modular decomposition of large numbers of high resolution brain functional networks using computationally expedient algorithms. This could enable future investigations of Simon's original hypothesis that hierarchy or near-decomposability of physical symbol systems is a critical design feature for their fast adaptivity to changing environmental conditions.
3 Institute for Mathematical Sciences, Imperial College, London, UK
The listed affiliation is the Institute for Mathematical Sciences at Imperial College, London, UK.
- The affiliation is the Institute for Mathematical Sciences, Imperial College, London, UK.
5 GSK Clinical Unit Cambridge, Addenbrooke’s Hospital, Cambridge, UK
The listed affiliation is associated with GSK Clinical Unit Cambridge at Addenbrooke’s Hospital, Cambridge, UK.
- The affiliation is GSK Clinical Unit Cambridge, Addenbrooke’s Hospital, Cambridge, UK.
- The listed paper title is “Hierarchical modular human brain networks.”
Keywords
The keywords identify the paper’s focus on graph theory, brain networks, modularity, hierarchy, near-decomposability, and information.
- The keywords cover graph theory, brain networks, modularity, hierarchy, near-decomposability, and information.
1 Introduction
The introduction motivates hierarchical modular analysis of brain networks, addressing limits in computational scale, resolution, multilevel structure, and subject comparison. The study applies an efficient algorithm and mutual-information approach to individual fMRI networks.
- Network modularity: Modularity provides a merit function for finding network partitions that reveal community structure.
- Motivation: Hierarchical modularity describes complex systems whose components form nested communities across multiple levels.Simon’s near-decomposability links dense intra-module connectivity with sparse inter-module connectivity.
- Prior limitations: Prior brain-network studies were constrained by computational cost, low spatial resolution, single-level analyses, and difficulty comparing subjects’ decompositions.
- Study approach: 56 healthy-volunteer fMRI networks were analyzed with a computationally efficient algorithm for hierarchical modular decomposition.The approach targeted whole-brain networks with thousands of equally sized nodes, improving spatial or anatomical resolution over earlier studies.
- Study approach: The method avoided biases from arbitrary anatomical templates while enabling rapid, high-resolution decomposition of individual fMRI networks.
- Study approach: Mutual information was used to compare modular community structures across subjects and identify the most representative network.
2.1 Experimental data
The study analyzed resting-state fMRI from 18 healthy volunteers using a high-resolution parcellation with 1808 regional nodes. Functional connectivity networks were constructed from wavelet correlations, thresholded sparsely, and analyzed with a multilevel community-detection procedure.
- Participants and acquisition: 18 healthy volunteers underwent resting-state fMRI while lying quietly with eyes closed.Each participant contributed 300 images; the first four were discarded and the first 256 remaining images were used for wavelet-correlation estimation.
- Participants and acquisition: The imaging protocol used whole-brain EPI acquisition at 3 Tesla with 32 slices and 3.0 mm in-plane voxels.The sequence used TR = 2000 ms, TE = 30 ms, and a 78-degree flip angle.
- Hierarchical analysis: The Louvain method iteratively optimized modularity through greedy node reassignment and construction of community-level meta-networks.Repeated passes generated partitions at multiple hierarchical levels until modularity could no longer improve.
- Connectivity construction: Networks were thresholded at 0.5% connection density, yielding approximately 8,000 edges and disconnecting up to 10% of nodes.An edge was retained when the absolute wavelet correlation exceeded the threshold τ.
2.2 Graph theoretical analysis
The analysis characterized network communities by modularity, hierarchical partitions, node roles, and similarity between subjects. It used the computationally efficient Louvain method to obtain multilevel community structure and normalized mutual information to compare partitions.
- Modularity and hierarchical partitions: Modularity evaluates whether a partition concentrates within-community edges beyond the expectation for an equivalent random graph.The analysis seeks the partition with the largest modularity value Q.
- Modularity and hierarchical partitions: The method was selected because it is computationally expedient and yields modularity values close to those from slower optimization methods.Intermediate partitions were also reported to correspond to meaningful resolutions.
- Modularity and hierarchical partitions: The Louvain method produces a complete hierarchy by repeating greedy optimization and meta-network construction across successive passes.The final pass provides the optimal partition, while intermediate partitions represent communities at intermediate resolutions.
- Node roles: Node roles were assigned using normalized within-module degree and participation coefficient, distinguishing hubs, non-hubs, connectors, and kinless nodes.A node with zi > 2.5 was classified as a hub; participation-coefficient ranges then separated topological roles within hub and non-hub categories.
- Node roles: The participation coefficient captures how broadly a node’s links are distributed across modules, whereas within-module degree captures its connectivity within its own module.Large zi indicates many intra-modular connections, while larger Pi indicates greater inter-modular distribution.
- Similarity measure: Normalized mutual information compared community partitions across hierarchical levels and subjects, ranging from 0 for independent partitions to 1 for identical partitions.The measure used the number of nodes shared between each pair of communities in two partitions.
3 Results
Across 18 subjects, brain functional networks showed hierarchical modular structure with substantial community similarity, organized into anatomically concentrated modules and submodules. Inter-modular connector roles were concentrated in association-cortex regions, while the decomposition remained distinct from randomized networks.
- Network similarity and modularity: 0.604 mean modularity at the highest hierarchical level exceeded 0.303 for equivalent random networks.The brain-network value had SD = 0.097, compared with SD = 0.003 for random networks.
- Network similarity and modularity: 0.63 average pairwise similarity at the lowest non-trivial level exceeded 0.57 at the highest level.Similarity was calculated between subjects at each hierarchical level; similarities across levels were highly correlated.
- Hierarchical modular organization: Occipital modules commonly contained a dominant submodule plus smaller peripheral submodules, whereas the fronto-temporal module decomposed more evenly into multiple submodules.The paper interprets the number of submodules as a module’s span of control.
- Node roles: 95% of nodes were ultra-peripheral or peripheral, while 5% had hub and/or connector roles.Most nodes had no inter-modular connections, and 4% had a high proportion of inter-modular connections qualifying for connector status.
- Node roles: Inter-modular connections and the connector nodes and hubs mediating them were most numerous in posterior association-cortex modules.The fronto-temporal module contained many such nodes, whereas the medial occipital module had relatively few connector nodes.
4 Discussion
The study demonstrates hierarchical modularity in resting-state human brain functional networks and finds reasonably consistent community structures across individuals. It also identifies anatomical patterns in modular organization and highlights challenges for interpreting and comparing these networks.
- Methodological contribution: The computationally efficient approach enabled hierarchical decompositions of many high-resolution networks containing thousands of nodes, addressing limitations of earlier low-resolution analyses.Earlier work was constrained by computational expense, while this study combined network decomposition with information-based similarity to identify a representative subject.
- Hierarchical modularity: The analysis found clear evidence for hierarchical modularity, with community structure reasonably similar across subjects at all hierarchical levels.Similarity was approximately I ∼0.6, supporting modularity as a potentially replicable phenomenon.
- Hierarchical modularity: Major modules comprised functionally or anatomically related cortical regions, with sub-modular separation demonstrated within the central somatosensorimotor module.Its lower-level decomposition separated medial regions from lateral regions, including precentral and postcentral areas.
- Anatomical organization: A symmetrical posterior-to-anterior progression of cortical modules was observed, including medial occipital, central, and parieto-frontal systems.The progression was clearest on the medial cerebral surfaces.
- Inter-modular connectivity: Connector nodes and hubs involved in inter-modular communication were concentrated in posterior association cortex, whereas the fronto-temporal module was sparsely connected to other modules.The authors suggest that time-varying functional connectivity could test whether non-stationary interactions explain these differences.
- Limitations and future directions: Interpretation remains empirical: the identified neuroanatomical systems require further validation, and statistical comparison of modularity across groups presents technical challenges.The authors also note that averaged group networks can neglect between-subject variability and be influenced by outlying functional-connectivity values.
5 Conclusion
The paper presents graph-theoretical tools for analyzing hierarchical modularity in human brain functional networks derived from fMRI. The techniques are computationally feasible and produce plausible, reasonably consistent descriptions of community structure in normal volunteers.
- The study describes graph-theoretical tools for hierarchical modularity analysis in human brain functional networks derived from fMRI.
- The techniques are computationally feasible and generate plausible, reasonably consistent descriptions of brain functional network community structure in normal volunteers.
- The analysis is theoretically connected to Simon’s theory of hierarchy and decomposability in information-processing systems.