Complex network measures of brain connectivity: Uses and interpretations
Introduction
Modern brain mapping techniques—such as diffusion MRI, functional MRI, EEG, and MEG—produce increasingly large datasets of anatomical or functional connection patterns. Concurrent technological advancements are generating similarly large connection datasets in biological, technological, social, and other scientific fields. Attempts to characterize these datasets have, over the last decade, led to the emergence of a new, multidisciplinary approach to the study of complex systems (Strogatz, 2001, Newman, 2003, Boccaletti et al., 2006). This approach, known as complex network analysis, describes important properties of complex systems by quantifying topologies of their respective network representations. Complex network analysis has its origins in the mathematical study of networks, known as graph theory. However, unlike classical graph theory, the analysis primarily deals with real-life networks that are large and complex—neither uniformly random nor ordered.
Brain connectivity datasets comprise networks of brain regions connected by anatomical tracts or by functional associations. Brain networks are invariably complex, share a number of common features with networks from other biological and physical systems, and may hence be characterized using complex network methods. Network characterization of structural and functional connectivity data is increasing (Bassett and Bullmore, 2006, Bassett and Bullmore, 2009, Stam and Reijneveld, 2007, Bullmore and Sporns, 2009) and rests on several important motivations. First, complex network analysis promises to reliably (Deuker et al., 2009) quantify brain networks with a small number of neurobiologically meaningful and easily computable measures (Sporns and Zwi, 2004, Achard et al., 2006, Bassett et al., 2006, He et al., 2007, Hagmann et al., 2008). Second, by explicitly defining anatomical and functional connections on the same map of brain regions, network analysis may be a useful setting for exploring structural–functional connectivity relationships (Zhou et al., 2006, Honey et al., 2007, Honey et al., 2009). Third, comparisons of structural or functional network topologies between subject populations appear to reveal presumed connectivity abnormalities in neurological and psychiatric disorders (Stam et al., 2007, Stam et al., 2009, Bassett et al., 2008, Leistedt et al., 2009, Ponten et al., 2009, Wang et al., 2009b).
In this article, we provide a non-technical introduction to complex network analysis of brain connectivity and outline important conceptual issues associated with its use. We begin by discussing the construction of structural and functional brain connectivity networks. We then describe the most commonly used measures of local and global connectivity, as well as their neurobiological interpretations. We focus on recently developed network measures (Boccaletti et al., 2006, Costa et al., 2007b) and provide a freely available Matlab toolbox, containing these measures, as well as their weighted and directed variants (Table A1). Finally, we discuss some of the issues associated with comparing structural and functional connectivity in the same subject and comparing connectivity patterns between subjects.
Note that while we concentrate on the analysis of large-scale connectivity, our discussion is equally applicable to smaller scale connectivity, with appropriate redefinitions. For instance, small-scale brain networks could consist of neurons linked by synapses or of cortical columns linked by intercolumnar connections.
Section snippets
Construction of brain networks
A network is a mathematical representation of a real-world complex system and is defined by a collection of nodes (vertices) and links (edges) between pairs of nodes (Fig. 1). Nodes in large-scale brain networks usually represent brain regions, while links represent anatomical, functional, or effective connections (Friston, 1994), depending on the dataset. Anatomical connections typically correspond to white matter tracts between pairs of brain regions. Functional connections correspond to
Measures of brain networks
An individual network measure may characterize one or several aspects of global and local brain connectivity. In this section, we describe measures that variously detect aspects of functional integration and segregation, quantify importance of individual brain regions, characterize patterns of local anatomical circuitry, and test resilience of networks to insult. Fig. 3 illustrates some basic concepts underlying these measures, while Table A1 contains mathematical definitions of all measures.
Network comparison
Complex network analysis may be useful for exploring connectivity relationships in individual subjects or between subject groups. In individual subjects, comparisons of structural and functional networks may provide insights into structural–functional connectivity relationships (e.g., Honey et al., 2007). Across subject populations, comparisons may detect abnormalities of network connectivity in various brain disorders (Bassett and Bullmore, 2009). The increased emphasis on structure–function
Brain connectivity analysis software
Multiple network analysis software packages are freely available on the Web. These packages include command-line toolboxes in popular languages, such as Matlab (Gleich, 2008) and Python (Hagberg et al., 2008), as well as standalone graphical user interface software (Batagelj and Mrvar, 2003, NWB-Team, 2006). Some of these packages are especially suitable for the analysis of large networks containing thousands of nodes, while others have powerful network visualization capabilities.
To accompany
Conclusion
Complex network analysis has emerged as an important tool for characterization of anatomical and functional brain connectivity. We described a collection of measures that quantify local and global properties of complex brain networks. The accompanying brain connectivity toolbox allows researchers to start exploring network properties of complex structural and functional datasets. We hope that the brain mapping community will be able to benefit from and contribute to these tools.
Acknowledgments
We thank Rolf Kötter, Patric Hagmann, Aviad Rubinstein, and Chris Honey for their contributions to the brain connectivity toolbox; Jonathan Power and Vassilis Tsiaras for suggesting valuable improvements to our toolbox functions; and Alain Barrat for definitional clarifications. M.R. is grateful to Michael Breakspear for his supervision and support during this project. M.R. and O.S. were supported by the J.S. McDonnell Foundation Brain NRG JSMF22002082. M.R. was supported by CSIRO ICT Centre
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