Review Article | Published:

Complex brain networks: graph theoretical analysis of structural and functional systems

Nature Reviews Neuroscience volume 10, pages 186198 (2009) | Download Citation

Subjects

  • A Corrigendum to this article was published on 04 March 2009

Abstract

Recent developments in the quantitative analysis of complex networks, based largely on graph theory, have been rapidly translated to studies of brain network organization. The brain's structural and functional systems have features of complex networks — such as small-world topology, highly connected hubs and modularity — both at the whole-brain scale of human neuroimaging and at a cellular scale in non-human animals. In this article, we review studies investigating complex brain networks in diverse experimental modalities (including structural and functional MRI, diffusion tensor imaging, magnetoencephalography and electroencephalography in humans) and provide an accessible introduction to the basic principles of graph theory. We also highlight some of the technical challenges and key questions to be addressed by future developments in this rapidly moving field.

Key points

  • Understanding the network organization of the brain has been a long-standing challenge for neuroscience. In the past decade, developments in graph theory have provided many new methods for topologically analysing complex networks, some of which have already been translated to the characterization of anatomical and functional brain networks.

  • Anatomical networks at whole-brain and cellular scales in several species consistently demonstrate conservation of wiring costs and small-world topology (high clustering and short path length). Human brain anatomical networks, derived from MRI or diffusion tensor imaging data, have high-degree cortical 'hubs' and modular and hierarchical properties.

  • Functional networks also demonstrate small-world properties at whole-brain and cellular spatial scales. Additionally, complex network properties including small-worldness and the existence of hubs are conserved over different frequency scales in functional MRI and electrophysiological data.

  • Convergent experimental and computational data suggest that there is interdependence in the organization of structural and functional networks. The topology, synchronizability and other dynamic properties of functional networks are strongly affected by small-world and other metrics of structural connectivity. Conversely, over a slower timescale the dynamics can modulate structural network topology.

  • Neuropsychiatric disorders can be thought of as dysconnectivity syndromes, and graph theory has already been used to quantify abnormality of structural and functional network properties in schizophrenia, Alzheimer's disease and other disorders. Graph theory can help us to understand the vulnerability of brain networks to lesions and could in future be used to provide markers of genetic risk for disorders or to measure therapeutic effects of drug treatments on functional networks.

  • The network organization of the brain, as it is beginning to be revealed by graph theory, is compatible with the hypothesis that the brain, perhaps in common with other complex networks, has evolved both to maximize the efficiency of information transfer and to minimize connection cost, at all scales of space and time. Key issues for future work include clarifying the relationship between the brain's network properties and its emergent cognitive behaviours in health and disease.

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Acknowledgements

E.B. was supported by a Human Brain Project grant from the National Institute of Mental Health and the National Institute of Biomedical Imaging & Bioengineering. The Behavioural & Clinical Neurosciences Institute in the University of Cambridge is supported by the Wellcome Trust and the Medical Research Council (UK). O.S. was supported by the JS McDonnell Foundation.

Author information

Affiliations

  1. University of Cambridge, Behavioural & Clinical Neurosciences Institute, Department of Psychiatry, Addenbrooke's Hospital, Cambridge, CB2 2QQ, UK.

    • Ed Bullmore
  2. GlaxoSmithKline, Clinical Unit Cambridge, Addenbrooke's Hospital, Cambridge, CB2 2QQ, UK.

    • Ed Bullmore
  3. Department of Psychological and Brain Sciences, Indiana University, Bloomington, Indiana 47405, USA.

    • Olaf Sporns

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Competing interests

Ed Bullmore is employed part-time by GlaxoSmithKline (GSK). He is a shareholder in GSK and the Brain Resource Company.

Corresponding author

Correspondence to Ed Bullmore.

Glossary

Graph theory

A branch of mathematics that deals with the formal description and analysis of graphs. A graph is defined simply as a set of nodes (vertices) linked by connections (edges), and may be directed or undirected. When describing a real-world system, a graph provides an abstract representation of the system's elements and their interactions.

Complex network

An informal description of a network with certain topological features, such as high clustering, small-worldness, the presence of high-degree nodes or hubs, assortativity, modularity or hierarchy, that are not typical of random graphs or regular lattices. Most real-life networks are complex by this definition, and analysis of complex networks therefore forms an important methodological tool for systems biology.

Adjacency matrix

An adjacency matrix indicates the number of edges between each pair of nodes in a graph. For most brain networks, the adjacency matrix is specified as binary — that is, each element is either 1 (if there is an edge between nodes) or 0 (if there is no edge). For undirected graphs the adjacency matrix is symmetrical.

Connectivity

In the brain, connectivity can be described as structural, functional or effective. Structural connectivity denotes a network of anatomical links, functional connectivity denotes the symmetrical statistical association or dependency between elements of the system, and effective connectivity denotes directed or causal relationships between elements.

Microcircuit

A neuronal network composed of specific cell types and synaptic connections, often arranged in a modular architecture and capable of generating functional outputs.

Connectome

The complete description of the structural connections between elements of a nervous system.

Diffusion tensor imaging

(DTI). An MRI technique that takes advantage of the restricted diffusion of water through myelinated nerve fibres in the brain to map the anatomical connectivity between brain areas.

Diffusion spectrum imaging

An MRI technique that is similar to DTI, but with the added capability of resolving multiple directions of diffusion in each voxel of white matter. This allows multiple groups of fibres at each location, including intersecting fibre pathways, to be mapped.

Cortical parcellation

A division of the continuous cortical sheet into discrete areas or regions; Brodmann's division of the cortex into areas defined by their cytoarchitectonic criteria is the most famous but not the only parcellation scheme.

Neuronographic measurements

Recordings of epileptiform electrical activity at specific sites in the cortex following topical application of a pro-convulsive drug to a distant cortical site; rapid propagation of electrical activity from stimulation to recording sites implies that the sites are anatomically connected.

Functional MRI

(fMRI). The detection of changes in regional brain activity through their effects on blood flow and blood oxygenation (which, in turn, affect magnetic susceptibility and tissue contrast in magnetic resonance images).

Electroencephalography

(EEG). A technique used to measure neural activity by monitoring electrical signals from the brain, usually through scalp electrodes. EEG has good temporal resolution but relatively poor spatial resolution.

Magnetoencephalography

(MEG). A method of measuring brain activity by detecting minute perturbations in the extracranial magnetic field that are generated by the electrical activity of neuronal populations.

Multielectrode array

(MEA). A technique for simultaneously measuring the electrical activity of local neuronal populations or single neurons, usually in tissue slices or cell cultures in vitro.

Association matrix

A matrix that represents the strength of the association between each pair of nodes in a graph. Association between nodes can be quantified by many continuously variable metrics, such as correlation or mutual information. Either functional or effective connectivity measures can be used to construct an association matrix.

Blood oxygen level-dependent (BOLD) signals

Changes in magnetic susceptibility and MRI tissue contrast that are indirectly indicative of underlying changes in spontaneous or experimentally controlled brain activation.

Default-mode network

A set of brain regions, including medial frontal and posterior cingulate areas of the cortex, that are consistently deactivated during the performance of diverse cognitive tasks.

Metastable dynamics

Transitions between marginally stable network states; these transitions can occur spontaneously or as a result of weak external perturbations.

Resting state

A cognitive state in which a subject is quietly awake and alert but does not engage in or attend to a specific cognitive or behavioural task.

Assortativity

A measure of the tendency for nodes to be connected to other nodes of the same or similar degree.

Endophenotype

A quantifiable biological marker of the genetic risk for a neuropsychiatric disorder.

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DOI

https://doi.org/10.1038/nrn2575

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