Hierarchical modularity in human brain functional networks

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

Network Scaling Effects in Graph Analytic Studies of Human Resting-State fMRI Data

Graph analysis has become an increasingly popular tool for characterizing topological properties of brain connectivity networks. Within this approach, the brain is modeled as a graph comprising N nodes connected by M edges. In functional magnetic resonance imaging (fMRI) studies, the nodes typically represent brain regions and the edges some measure of interaction between them. These nodes are commonly defined using a variety of regional parcellation templates, which can vary both in the volume sampled by each region, and the number of regions parcellated. Here, we sought to investigate how such variations in parcellation templates affect key graph analytic measures of functional brain organization using resting-state fMRI in 30 healthy volunteers. Seven different parcellation resolutions (84, 91, 230, 438, 890, 1314, and 4320 regions) were investigated. We found that gross inferences regarding network topology, such as whether the brain is small-world or scale-free, were robust to the template used, but that both absolute values of, and individual differences in, specific parameters such as path length, clustering, small-worldness, and degree distribution descriptors varied considerably across the resolutions studied. These findings underscore the need to consider the effect that a specific parcellation approach has on graph analytic findings in human fMRI studies, and indicate that results obtained using different templates may not be directly comparable

Statistical Analysis of Tract-Tracing Experiments Demonstrates a Dense, Complex Cortical Network in the Mouse.

Anatomical tract tracing methods are the gold standard for estimating the weight of axonal connectivity between a pair of pre-defined brain regions. Large studies, comprising hundreds of experiments, have become feasible by automated methods. However, this comes at the cost of positive-mean noise making it difficult to detect weak connections, which are of particular interest as recent high resolution tract-tracing studies of the macaque have identified many more weak connections, adding up to greater connection density of cortical networks, than previously recognized. We propose a statistical framework that estimates connectivity weights and credibility intervals from multiple tract-tracing experiments. We model the observed signal as a log-normal distribution generated by a combination of tracer fluorescence and positive-mean noise, also accounting for injections into multiple regions. Using anterograde viral tract-tracing data provided by the Allen Institute for Brain Sciences, we estimate the connection density of the mouse intra-hemispheric cortical network to be 73% (95% credibility interval (CI): 71%, 75%); higher than previous estimates (40%). Inter-hemispheric density was estimated to be 59% (95% CI: 54%, 62%). The weakest estimable connections (about 6 orders of magnitude weaker than the strongest connections) are likely to represent only one or a few axons. These extremely weak connections are topologically more random and longer distance than the strongest connections, which are topologically more clustered and shorter distance (spatially clustered). Weak links do not substantially contribute to the global topology of a weighted brain graph, but incrementally increased topological integration of a binary graph. The topology of weak anatomical connections in the mouse brain, rigorously estimable down to the biological limit of a single axon between cortical areas in these data, suggests that they might confer functional advantages for integrative information processing and/or they might represent a stochastic factor in the development of the mouse connectome.Netherlands Organisation for Scientific Research (Rubicon Fellowship), National Institute of Health Research (NIHR) Cambridge Biomedical Research CentreThis is the final version of the article. It first appeared from the Public Library of Science via http://dx.doi.org/10.1371/journal.pcbi.100510

Small-World Brain Networks Revisited.

It is nearly 20 years since the concept of a small-world network was first quantitatively defined, by a combination of high clustering and short path length; and about 10 years since this metric of complex network topology began to be widely applied to analysis of neuroimaging and other neuroscience data as part of the rapid growth of the new field of connectomics. Here, we review briefly the foundational concepts of graph theoretical estimation and generation of small-world networks. We take stock of some of the key developments in the field in the past decade and we consider in some detail the implications of recent studies using high-resolution tract-tracing methods to map the anatomical networks of the macaque and the mouse. In doing so, we draw attention to the important methodological distinction between topological analysis of binary or unweighted graphs, which have provided a popular but simple approach to brain network analysis in the past, and the topology of weighted graphs, which retain more biologically relevant information and are more appropriate to the increasingly sophisticated data on brain connectivity emerging from contemporary tract-tracing and other imaging studies. We conclude by highlighting some possible future trends in the further development of weighted small-worldness as part of a deeper and broader understanding of the topology and the functional value of the strong and weak links between areas of mammalian cortex.DSB acknowledges support from the John D. and Catherine T. MacArthur Foundation, the Alfred P. Sloan Foundation, the Army Research Laboratory and the Army Research Office through contract numbers W911NF-10-2-0022 and W911NF-14-1-0679, the National Institute of Health (2-R01-DC-009209-11, 1R01HD086888-01, R01-MH107235, R01-MH107703, and R21-M MH-106799), the Office of Naval Research, and the National Science Foundation (BCS-1441502, CAREER PHY-1554488, and BCS-1631550).This is the final version of the article. It first appeared from Sage at http://dx.doi.org/10.1177/1073858416667720

A wavelet-based estimator of the degrees of freedom in denoised fMRI time series for probabilistic testing of functional connectivity and brain graphs.

Connectome mapping using techniques such as functional magnetic resonance imaging (fMRI) has become a focus of systems neuroscience. There remain many statistical challenges in analysis of functional connectivity and network architecture from BOLD fMRI multivariate time series. One key statistic for any time series is its (effective) degrees of freedom, df, which will generally be less than the number of time points (or nominal degrees of freedom, N). If we know the df, then probabilistic inference on other fMRI statistics, such as the correlation between two voxel or regional time series, is feasible. However, we currently lack good estimators of df in fMRI time series, especially after the degrees of freedom of the "raw" data have been modified substantially by denoising algorithms for head movement. Here, we used a wavelet-based method both to denoise fMRI data and to estimate the (effective) df of the denoised process. We show that seed voxel correlations corrected for locally variable df could be tested for false positive connectivity with better control over Type I error and greater specificity of anatomical mapping than probabilistic connectivity maps using the nominal degrees of freedom. We also show that wavelet despiked statistics can be used to estimate all pairwise correlations between a set of regional nodes, assign a P value to each edge, and then iteratively add edges to the graph in order of increasing P. These probabilistically thresholded graphs are likely more robust to regional variation in head movement effects than comparable graphs constructed by thresholding correlations. Finally, we show that time-windowed estimates of df can be used for probabilistic connectivity testing or dynamic network analysis so that apparent changes in the functional connectome are appropriately corrected for the effects of transient noise bursts. Wavelet despiking is both an algorithm for fMRI time series denoising and an estimator of the (effective) df of denoised fMRI time series. Accurate estimation of df offers many potential advantages for probabilistically thresholding functional connectivity and network statistics tested in the context of spatially variant and non-stationary noise. Code for wavelet despiking, seed correlational testing and probabilistic graph construction is freely available to download as part of the BrainWavelet Toolbox at www.brainwavelet.org.This work was supported by the Wellcome Trust- and GSK-funded Translational Medicine and Therapeutics Programme (085686/Z/08/C, AXP) and the University of Cambridge MB/PhD Programme (AXP). The Behavioral and Clinical Neuroscience Institute is supported by the Wellcome Trust (093875/Z/10/Z) and the Medical Research Council (G1000183). ETB works half-time for GlaxoSmithKline and half-time for the University of Cambridge; he holds stock in GSK.This is the final version of the article. It was first available from Elsevier via http://dx.doi.org/10.1016/j.neuroimage.2015.04.05

Modular and Hierarchically Modular Organization of Brain Networks

Brain networks are increasingly understood as one of a large class of information processing systems that share important organizational principles in common, including the property of a modular community structure. A module is topologically defined as a subset of highly inter-connected nodes which are relatively sparsely connected to nodes in other modules. In brain networks, topological modules are often made up of anatomically neighboring and/or functionally related cortical regions, and inter-modular connections tend to be relatively long distance. Moreover, brain networks and many other complex systems demonstrate the property of hierarchical modularity, or modularity on several topological scales: within each module there will be a set of sub-modules, and within each sub-module a set of sub-sub-modules, etc. There are several general advantages to modular and hierarchically modular network organization, including greater robustness, adaptivity, and evolvability of network function. In this context, we review some of the mathematical concepts available for quantitative analysis of (hierarchical) modularity in brain networks and we summarize some of the recent work investigating modularity of structural and functional brain networks derived from analysis of human neuroimaging data

Wiring cost and topological participation of the mouse brain connectome.

Brain connectomes are topologically complex systems, anatomically embedded in 3D space. Anatomical conservation of "wiring cost" explains many but not all aspects of these networks. Here, we examined the relationship between topology and wiring cost in the mouse connectome by using data from 461 systematically acquired anterograde-tracer injections into the right cortical and subcortical regions of the mouse brain. We estimated brain-wide weights, distances, and wiring costs of axonal projections and performed a multiscale topological and spatial analysis of the resulting weighted and directed mouse brain connectome. Our analysis showed that the mouse connectome has small-world properties, a hierarchical modular structure, and greater-than-minimal wiring costs. High-participation hubs of this connectome mediated communication between functionally specialized and anatomically localized modules, had especially high wiring costs, and closely corresponded to regions of the default mode network. Analyses of independently acquired histological and gene-expression data showed that nodal participation colocalized with low neuronal density and high expression of genes enriched for cognition, learning and memory, and behavior. The mouse connectome contains high-participation hubs, which are not explained by wiring-cost minimization but instead reflect competitive selection pressures for integrated network topology as a basis for higher cognitive and behavioral functions.Funding was provided by a NARSAD Young Investigator award and Isaac Newton Trust (to M.R.); a Rubicon Fellowship (to R.J.F.Y.); the Medical Research Council and the Wellcome Trust (Behavioural & Clinical Neuroscience Institute); and the National Institute for Health Research Cambridge Biomedical Research Centre (high-performance computing facilities).This is the accepted manuscript of a paper published in the Proceedings of the National Academy of Sciences (Rubinov M, Ypma RJF, Watson C, Bullmore ET, PNAS, 2015, 112, 10032-10037, doi:10.1073/pnas.1420315112). The final version is available at http://dx.doi.org/10.1073/pnas.142031511

Comparative Connectomics.

We introduce comparative connectomics, the quantitative study of cross-species commonalities and variations in brain network topology that aims to discover general principles of network architecture of nervous systems and the identification of species-specific features of brain connectivity. By comparing connectomes derived from simple to more advanced species, we identify two conserved themes of wiring: the tendency to organize network topology into communities that serve specialized functionality and the general drive to enable high topological integration by means of investment of neural resources in short communication paths, hubs, and rich clubs. Within the space of wiring possibilities that conform to these common principles, we argue that differences in connectome organization between closely related species support adaptations in cognition and behavior.We thank Lianne Scholtens, Jim Rilling, Tom Schoenemann for discussions and comments. MPvdH was supported by a VENI (# 451-12-001) grant from the Netherlands Organization for Scientific Research (NWO) and a Fellowship of MQ.This is the author accepted manuscript. The final version is available from Elsevier via https://doi.org/10.1016/j.tics.2016.03.00

Do Not Adjust Your Mind: The Fault Is in Your Glia.

Glia have been implicated in schizophrenia, although whether they play a primary role is uncertain. In this issue of Cell Stem Cell, Windrem et al. (2017) transplant human glial progenitors from schizophrenia patients into mouse brains, which develop abnormalities and behaviors characteristic of schizophrenia, thereby suggesting a primary role for glia in the complex disease pathogenesis

Topological Isomorphisms of Human Brain and Financial Market Networks

Although metaphorical and conceptual connections between the human brain and the financial markets have often been drawn, rigorous physical or mathematical underpinnings of this analogy remain largely unexplored. Here, we apply a statistical and graph theoretic approach to the study of two datasets – the time series of 90 stocks from the New York stock exchange over a 3-year period, and the fMRI-derived time series acquired from 90 brain regions over the course of a 10-min-long functional MRI scan of resting brain function in healthy volunteers. Despite the many obvious substantive differences between these two datasets, graphical analysis demonstrated striking commonalities in terms of global network topological properties. Both the human brain and the market networks were non-random, small-world, modular, hierarchical systems with fat-tailed degree distributions indicating the presence of highly connected hubs. These properties could not be trivially explained by the univariate time series statistics of stock price returns. This degree of topological isomorphism suggests that brains and markets can be regarded broadly as members of the same family of networks. The two systems, however, were not topologically identical. The financial market was more efficient and more modular – more highly optimized for information processing – than the brain networks; but also less robust to systemic disintegration as a result of hub deletion. We conclude that the conceptual connections between brains and markets are not merely metaphorical; rather these two information processing systems can be rigorously compared in the same mathematical language and turn out often to share important topological properties in common to some degree. There will be interesting scientific arbitrage opportunities in further work at the graph-theoretically mediated interface between systems neuroscience and the statistical physics of financial markets