Elsevier

NeuroImage

Volume 124, Part A, 1 January 2016, Pages 379-393
NeuroImage

The contribution of geometry to the human connectome

https://doi.org/10.1016/j.neuroimage.2015.09.009Get rights and content

Highlights

  • Geometry is crucial to the study of the structural connectome.

  • We developed a novel method for generating geometry-preserving surrogate networks.

  • Spatial null models show that geometry strongly contributes to network topology.

  • Rich club nodes are more peripheral than if they were dictated solely by geometry.

Abstract

The human connectome is a topologically complex, spatially embedded network. While its topological properties have been richly characterized, the constraints imposed by its spatial embedding are poorly understood. By applying a novel resampling method to tractography data, we show that the brain's spatial embedding makes a major, but not definitive, contribution to the topology of the human connectome. We first identify where the brain's structural hubs would likely be located if geometry was the sole determinant of brain topology. Empirical networks show a widespread shift away from this geometric center toward more peripheral interconnected skeletons in each hemisphere, with discrete clusters around the anterior insula, and the anterior and posterior midline regions of the cortex. A relatively small number of strong inter-hemispheric connections assimilate these intra-hemispheric structures into a rich club, whose connections are locally more clustered but globally longer than predicted by geometry. We also quantify the extent to which the segregation, integration, and modularity of the human brain are passively inherited from its geometry. These analyses reveal novel insights into the influence of spatial geometry on the human connectome, highlighting specific topological features that likely confer functional advantages but carry an additional metabolic cost.

Introduction

Network theory offers unique insights into the principles underlying the structural connectivity of the brain (Bullmore and Sporns, 2009, Fornito et al., 2013, Sporns, 2011). Treating the brain as a network of interconnected nodes has uncovered important structural principles in healthy adult brains, such as its highly clustered, well integrated topology (Sporns and Zwi, 2004, Stephan et al., 2000), the presence of an anatomical “backbone” (Hagmann et al., 2008), the extent to which the brain optimizes its wiring costs (Bullmore and Sporns, 2012) and the existence of rich-club hub regions that assume a greater role in network structure than other regions (van den Heuvel and Sporns, 2011). Network theory now also plays an important role in the study of major neurological and psychiatric diseases (Fornito et al., 2015, Stam, 2014).

Many of the methods in network science originate in physics and mathematics, where networks are described as graphs composed of nodes and edges. Such descriptions concentrate on the connectivity between nodes, rather than their spatial locations. Indeed, the spatial distributions of many complex networks, such as linked articles on Wikipedia, are of little relevance since the nodes are not tied to physical locations and forming new links carries no cost. But brains are spatially embedded, and the distances between nodes – the spatial geometry – likely play an important constraining role on network topology (Bassett et al., 2010, Bullmore and Sporns, 2012, Collin et al., 2014, Fornito et al., 2011, Fornito et al., 2015, Rubinov et al., 2015, Vértes et al., 2012). Spatial embedding might in principle contribute strongly to network properties such as clustering, modularity (Henderson and Robinson, 2011, Henderson and Robinson, 2013, Henderson and Robinson, 2014), and richness (Collin et al., 2014, Samu et al., 2014). Put simply, nodes in a spatially embedded network are more likely to connect to their spatial neighbors, particularly if there is an associated cost with long edges (Barthélemy, 2011). Even if there are no topological rules beyond this simple geometric principle (i.e., network edges tend to connect spatially proximal nodes), such networks will be more clustered and lattice-like than random networks with no geometric constraints. In addition, hub regions will simply be those that are, on average, closer to the rest of the brain than non-hub regions (Henderson and Robinson, 2014).

Most network statistics depend on low-level network properties such as the number of nodes and edges. For this reason, metrics from empirical networks must be benchmarked against surrogate networks that preserve these low level properties while being otherwise random. There are two main approaches to generating surrogate networks: generating random networks de novo using basic wiring rules (building a network with a pre-specified number of nodes and edges, Klimm et al., 2014); or randomly rewiring an empirical network (destroying topological structure, while preserving node and edge number). The Erdős–Rényi random graph is a widely-used example of the former (Erdős and Rényi, 1961), while the degree-preserving Maslov–Sneppen algorithm is a widely-used example of the latter (Maslov and Sneppen, 2002). A key illustration of their usefulness is in the definition of the rich-club metric, whose interpretation requires normalization against surrogate networks that preserve node degree (number of edges at each node) or node strength (sum of weights at each node) (Colizza et al., 2006). However, while random networks allow normalization of network metrics, they lack geometry and thus do not allow one to understand the extent to which topological properties might be inherited from spatial embedding.

Here we investigate the role of spatial embedding (geometry) on network structure (topology). We first quantify the relationship between geometry and network connections. By constraining the randomization of network edges so that this relationship is preserved, we create reference graphs that preserve the basic properties of spatial embedding but lack any additional structure. Benchmarking empirical data against these surrogates reveals that proportion of topology that comes passively from geometry and identifies any additional structure that specific wiring rules have imposed on top of that. We first study basic network properties of segregation and integration, then turn to network hubs, which are currently the focus of substantial scientific interest (Achard et al., 2006, Power et al., 2013, van den Heuvel and Sporns, 2011).

Section snippets

Methods

In this section we give details on the participants, the parameters of the diffusion MRI acquisition, standard image pre-processing steps, and the probabilistic tractography algorithm used to generate fiber tracks (streamlines). We then describe how these tracks are collated to form structural brain networks by first dividing the brain into regions (nodes) and counting how many streamlines connect each pair of nodes.

Results

Using probabilistic tractography, we derived estimates of whole brain structural connectivity from diffusion images of 75 healthy adults. Our connectivity matrices were reconstructed from densely seeded tractography between a relatively fine representation of 513 uniformly sized cortical and sub-cortical regions (257 in the left and 256 in the right hemisphere). The resulting weighted, undirected matrices W were nearly fully connected in each subject.

Using these networks, we first analyzed the

Discussion

By generating random reference networks that preserve the brain's background spatial embedding, we assessed the influence of geometry on the topology of the human connectome. We find that if the location and connectivity of network hubs were determined purely by their embedding, the backbone of the human connectome would consist of a densely interconnected morass of topologically central regions in the geometric center of the brain. That is, the unopposed action of geometry would mean that the

Acknowledgments

This work was supported by the National Health and Medical Research Council (Program Grants 1037196 and 628952), the Australian Research Council (Centre of Excellence for Integrative Brain Function CE140100007 and Future Fellowship FT110100726), The Landsdowne Foundation, and the Victorian Government's Operational Infrastructure Support Grant.

References (79)

  • G. Roberts et al.

    Reduced inferior frontal gyrus activation during response inhibition to emotional stimuli in youth at high risk of bipolar disorder

    Biol. Psychiatry

    (2013)
  • M. Rubinov et al.

    Complex network measures of brain connectivity: uses and interpretations

    NeuroImage

    (2010)
  • T. Schreiber et al.

    Surrogate time series

    Physica D

    (2000)
  • K. Setsompop et al.

    Improving diffusion MRI using simultaneous multi-slice echo planar imaging

    NeuroImage

    (2012)
  • K. Setsompop et al.

    Pushing the limits of in vivo diffusion MRI for the Human Connectome Project

    NeuroImage

    (2013)
  • S.M. Smith et al.

    Advances in functional and structural MR image analysis and implementation as FSL

    NeuroImage

    (2004)
  • R.E. Smith et al.

    Anatomically-constrained tractography: improved diffusion MRI streamlines tractography through effective use of anatomical information

    NeuroImage

    (2012)
  • R.E. Smith et al.

    SIFT: spherical-deconvolution informed filtering of tractograms

    NeuroImage

    (2013)
  • R.E. Smith et al.

    The effects of SIFT on the reproducibility and biological accuracy of the structural connectome

    NeuroImage

    (2015)
  • R.E. Smith et al.

    SIFT2: enabling dense quantitative assessment of brain white matter connectivity using streamlines tractography

    NeuroImage

    (2015)
  • J. Theiler et al.

    Testing for nonlinearity in time series: the method of surrogate data

    Physica D

    (1992)
  • J.-D. Tournier et al.

    Robust determination of the fibre orientation distribution in diffusion MRI: non-negativity constrained super-resolved spherical deconvolution

    NeuroImage

    (2007)
  • J.-D. Tournier et al.

    Resolving crossing fibres using constrained spherical deconvolution: validation using diffusion-weighted imaging phantom data

    NeuroImage

    (2008)
  • N. Tzourio-Mazoyer et al.

    Automated anatomical labeling of activations in SPM using a macroscopic anatomical parcellation of the MNI MRI single-subject brain

    NeuroImage

    (2002)
  • X. Wu et al.

    Genetic white matter fiber tractography with global optimization

    J. Neurosci. Methods

    (2009)
  • A. Zalesky et al.

    Whole-brain anatomical networks: does the choice of nodes matter?

    NeuroImage

    (2010)
  • S. Achard et al.

    A resilient, low-frequency, small-world human brain functional network with highly connected association cortical hubs

    J. Neurosci.

    (2006)
  • J. Alstott et al.

    A unifying framework for measuring weighted rich clubs

    Sci. Rep.

    (2014)
  • D.S. Bassett et al.

    Cognitive fitness of cost-efficient brain functional networks

    Proc. Natl. Acad. Sci.

    (2009)
  • D.S. Bassett et al.

    Efficient physical embedding of topologically complex information processing networks in brains and computer circuits

    PLoS Comput. Biol.

    (2010)
  • D.S. Bassett et al.

    Robust detection of dynamic community structure in networks

    Chaos

    (2013)
  • R.F. Betzel et al.

    Generative Models of the Human Connectome

    (2015)
  • M. Breakspear et al.

    Spatiotemporal wavelet resampling for functional neuroimaging data

    Hum. Brain Mapp.

    (2004)
  • E. Bullmore et al.

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

    Nat. Rev. Neurosci.

    (2009)
  • E. Bullmore et al.

    The economy of brain network organization

    Nat. Rev. Neurosci.

    (2012)
  • V. Colizza et al.

    Detecting rich-club ordering in complex networks

    Nat. Phys.

    (2006)
  • G. Collin et al.

    Structural and functional aspects relating to cost and benefit of rich club organization in the human cerebral cortex

    Cereb. Cortex

    (2014)
  • S. Coombes et al.

    Modeling electrocortical activity through improved local approximations of integral neural field equations

    Phys. Rev. E

    (2007)
  • H.D. Critchley et al.

    Neural systems supporting interoceptive awareness

    Nat. Neurosci.

    (2004)
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