The contribution of geometry to the human connectome
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.
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