Complex brain networks
Graph theoretical analysis of structural and functional systems
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. We also highlight some of the technical challenges and key questions to be addressed by future developments in this rapidly moving field.
Structural and functional brain networks can be explored using graph theory through the following four steps (see the figure):
• Define the network nodes. These could be defined as electroencephalography or multielectrode array electrodes, or as anatomically defined regions of histological, MRI or diffusion tensor imaging data.
• Estimate a continuous measure of association between nodes. This could be the spectral coherence or Granger causality measures between two magnetoencephalography sensors, or the connection probability between two regions of an individual diffusion tensor imaging data set, or the inter-regional correlations in cortical thickness or volume MRI measurements estimated in groups of subjects.
• Generate an association matrix by compiling all pairwise associations between nodes and (usually) apply a threshold to each element of this matrix to produce a binary adjacency matrix or undirected graph.
• Calculate the network parameters of interest in this graphical model of a brain network and compare them to the equivalent parameters of a population of random networks.
Figure 2 | cellular and whole-brain networks demonstrate consistent topological features. The top panel shows a cellular functional network constructed from multielectrode- array recordings made in the anaesthetized cat; each node (represented by a circle) corresponds approximately to one neuron and the connections represent high functional connectivity between neurons. The different coloured nodes constitute separate clusters or modules. The plots in each circle illustrate cellular responses to stimuli of different orientations, and the circle size corresponds to the degree (number of functional connections) of each node. The bottom panel shows a whole-brain structural network constructed from histological data on the macaque cortex; each node corresponds to a brain area and the connections represent axonal projections between areas. The network has two main modules, shown here with yellow and grey circles corresponding to mostly dorsal and ventral visual regions, respectively. Both networks exhibit the small-world attributes of high clustering and short path length; both have an exponentially truncated power law degree distribution, associated with the existence of high-degree ‘hubs’ (V4 in the anatomical network); and both have a community structure characterized by sparse connectivity between modules (each module is enclosed by stippled lines) and linked by hubs (nodes circled in red). AITv, anterior inferotemporal ventral area; CITd, central inferotemporal dorsal area; CITv, central inferotemporal ventral area; DP, dorsal preluneate area; FEF, frontal eye field; FST, floor of superior temporal area; LIP, lateral intraparietal area; MT, middle temporal area; PIP, posterior intraparietal area; PITd, posterior inferotemporal dorsal area; PITv, posterior inferotemporal ventral area; PO, parieto-occipital area; TF, area TF; TH, area TH ;V1–4, visual cortical areas 1–4; VIP, ventral intraparietal area; VOT, ventral
occipitotemporal area; VP, ventral posterior area.
Figure 3 | Disease-related disorganization of brain anatomical networks derived from structural Mri data. In both parts, the nodes (circles) represent cortical regions and the connections represent high correlation in grey matter density between nodes. The nodes are arranged vertically by degree and are separated horizontally for clarity of representation. The numbers indicate approximate Brodmann area, and the prime symbols (Œ) denote left-sided regions. The clustering coefficient of each node, a measure of its local connectivity, is indicated by its size: nodes with high clustering are larger.
a | The brain anatomical network of the healthy volunteers has a hierarchical organization characterized by low clustering of high-degree nodes.
b | The equivalent network constructed from MRI data on people with schizophrenia shows loss of this hierarchical organization . high-degree nodes are more often highly clustered.
It is clear that certain aspects of the organization of complex brain networks are highly conserved over different scales and types of measurement, across different species and for functional and anatomical networks. The archetypal brain network has a short path length (associated with high global efficiency of information transfer), high clustering (associated with robustness to random error), a degree distribution compatible with the existence of hubs, and a modular community structure. Furthermore, anatomical networks are sparsely connected, especially between nodes in different modules, and the ‘wiring length’ (the physical distance that connections span) is close to minimal. This profile of topological and geometric properties is typical not just of brain networks but also of many other complex networks, including transport
systems and intracellular signalling pathways.
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