The (human) brain describes a complex system of anatomically interlinked and functionally interacting elements. At the micro cellular scale, neurons are connected to other neurons by means of dendrites, axons and synapses and form a vast network of neuron-to-neuron conections. Similarly, at the meso- or macroscale –at the level we acquire with Magnetic Resonance Imaging (MRI) -, neural columns and large-scale brain regions are anatomically interconnected by long-range axonal projections, facilitating neural communication and functional interaction. A species’ ‘connectome’ (Sporns, et al., 2005) describes a comprehensive map of all connections of an organism’s nervous system, and is believed to act as an anatomical basis for functional dynamics and functional interactions between brain regions to occur, and thus a requisite for brain function (1, 2). Connectomics describes the detailed mapping and reconstruction of the neural connections of nervous systems, combined with the extensive studying of the topological structure (i.e. the organization of a network invariant to its spatial organization) of the derived brain wiring maps.

Several techniques are available to map connectomes. In animals, Electron Microscopy (e.g. (3) and tract-tracing approaches enable high-resolution reconstruction of detailed (partial) connectome maps of the nervous systems of several species. Examples of microsale connectomes include the seminal work on the neuron level connectome map of the worm the C. elegans (4). Examples of tract-tracing maps of connectomes of mammalian species include those of (among others) cat (5), macaque (6), mouse (7) and rat (8). In humans -in which invasive approaches are highly limited-, advances in diffusion-weighted, functional MRI and EEG/MEG have led the way to in vivo mapping of structural and functional brain connections, and therewith reconstructions of the human macroscale connectome (e.g. (9-11)).

Once a connectome map is reconstructed, the question arises of a how the connections of a given connectome map are organized. Can we extract features that describe the complex organization of the system? Are there characteristic attributes that describe its topological structure? This introduces the second aim of the connectome field: elucidating key architectural features of connectome maps and determine how they play a role in the emergence of brain function. In the last decade, network science or graph theory have become more and more used as a fruitful, formal framework to study the global and local organisational structure of reconstructed brain networks (2, 12-15). The goal of this morning workshop is to provide an introductory course on the use of graph theory to study the topological organizaiton of brain networks. Graph theory as a tool to extract key organizational features of brain networks One way to approach the brain as a system is by describing it as a ‘network’ or ‘graph’. Within this mathematical framework, the structure of a neural system is described as a collection of ‘nodes’ (which can be neurons, neural columns and/or large-scale brain regions) and ‘edges’ describing the connections interlinking these nodes (which can be axons, white matter bundles, functional interactions).

Once such a formal, mathematical description of our connectome map is established, graph theory can be used to describe the overall topological architecture of the network, allowing for the investigation of organizational features that would otherwise remain hidden when we would exclusively focus on information form single brain regions and/or single connections. One appealing aspect of the use of network science is that graph theory provides a vast array of data-driven metrics to describe the topology of networks (16, 17), with a number of such graph theoretical attributes particularly useful in describing the organization of neural networks (18). [Figure 1 provides a schematic overview of a number of graph metrics commonly used in the field of MRI connectomics]. In the morning workshop we will discuss some of the most commonly used metrics. ‘Clustering’ describes the tendency of nodes to locally link together, describing a high level of connectivity in the direct surrounding of a node. In brain networks, high levels of clustering (or its cousins ‘local efficiency’ and ‘transitivity’) and ‘modular organization’ -the tendency of groups of nodes to form densely connected subclusters or communities within the overall network- are thought to reflect functionally linked neuronal assembles, and thus to form an anatomical substrate for local information processing and functional segregation (19). Providing insight into a network’s global organization, the ‘characteristic path length’ (or its inverse cousin ‘global efficiency’) reflects the ease of to which information can be transported across a network, summarizing the number of steps that -on average- have to be taken to travel from one node to another node in the network. Short path lengths (or high global network efficiency levels) in brain networks have been suggested to reflect high levels of communication efficiency (19). Networks with a high level of clustering (thus reflecting a high level of local organization), combined with a relative short average path length (thus reflecting high levels of global communication capacity) are referred to as ‘small-world networks’ (20), a class of networks known to exhibit properties of an efficient topological structure.

Evaluating the above described graph theoretical metrics, network studies of the human, but also of the macaque, cat, mouse and rat brain, have shown evidence of connectomes to show an efficient small-world modular architecture, with high levels of clustering, short communication pathways and a pronounced functional and structural community structure (e.g. (9, 11, 21-24)). Converging evidence suggests that these network attributes play a role in cognitive brain functioning (25-27) and importantly, a growing number of patient-control studies tend to show that disruptions in these network attributes may play an important role in neurological and psychiatric disorders (see for review for example (26, 28-35)).

In addition to metrics that describe the global structure of a network, graph theory also provides the opportunity to describe and examine the role of individual nodes (involving a ‘node-centric’ analysis of networks). In this respect, ‘degree’ and ‘centrality’ graph metrics provide insight into the role of a node in the overall architecture of the network, for example elucidating nodes that show higher connectivity and a more central role in communication paths (e.g. the metric of betweenness centrality) than other nodes. Studies have noted that many connectomes have a so-called heavy-tailed degree distribution (that is, the distribution of the number of connections attached to each node)(e.g.(36-39)), suggesting the formation of a small, but prominent present, group of highly connected ‘hubs’. Across different connectomes (ranging from C. elegans, to the Drosophila fly to mammalian species, including humans) these hubs have been shown to display a dense level of mutual connectivity, forming a central core or ‘rich club’ (10)(40)(16). Due to their high-degree and central embedding in the overall network, rich club regions and their connections have been suggested to form a backbone for neural communication and to potentially form an anatomical substrate for neural integration in the human and animal brain (16).

Shifting gear from a ‘node-perspective’ to a more ‘edge-perspective’ view of networks, graph theory also provides the opportunity to provide more insight into the role of edges. Examples of edge-centric approaches include the evaluation of the contribution of edges to the communication capacity of networks (41), the examination of recurrent classes of communication paths (called ‘path motifs’) (33) and approaches that compute the formation of so-called ‘edge communities’, being subsets of edges that play a similar role in the network (41). Potentially powerful approaches in the examination of graph theoretical aspects of neural networks and their behavior over time include ‘spectral graph theory’ examinations (42), the examination of dynamical networks (43), the examination of network morphospace (44) and the use of dynamic simulation models (45, 46). The use and understanding of these graph attributes with respect to brain networks is still in its infancy, but include promising new ways to get a better understanding of the structure and dynamics of brain networks, further underscoring an ongoing expansion of the field of brain connectomics.

In this introductory workshop we will discuss the use and application of graph theoretical metrics (e.g. degree, clustering, path length, modules) in the field of brain networks, talk about different types of global networks organization (e.g. small-world, scale-free, rich club organization) and their potential implication with respect to neural networks. If time allows, we will also briefly discuss novel graph theoretical metrics such as edge based graph metrics (e.g. edge-classes, rich clubs, path motifs) and their potential application in the examination of brain networks.