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Measuring Complex Brain Networks Structure

Ester Bonmati1*, Anton Bardera1 and Imma Boada1
  • 1 University of Girona, Institute of Informatics and Applications, Spain

I. INTRODUCTION The human brain has roughly one hundred billion neurons forming a network with trillions of intra-connections. The mapping of structure and functionality of brain networks is therefore an important challenge in understanding the functioning. Connectivity matrices are used to represent brain networks, also called connectome [1,2], as a graph [3–5], where nodes correspond to brain regions and edges to structural or functional connections [6–8]. Different measures have been applied to describe topological features of brain networks [9–11]. For instance, the independence of large areas, denoted as integration, has been studied by the path length measure, the characteristic path length [12], or the global efficiency [13]. Independence of small subsets, defined as segregation, can be analyzed by the clustering coefficient [12], the transitivity [14] or the modularity [15]. The importance of individual nodes can be defined with centrality measures such as the degree [16]. A good summary of the measures can be found in [10]. In this work, we present a global and two local measures, based on the mutual information measure, to quantify brain networks structure. II. MATERIALS AND METHOD A. Materials Synthetic model networks were created using the Brain Connectivity Toolbox (BCT) [10]. Random, lattice, ring lattice and small-world model networks with 128 and 256 nodes with edges ranging from 128 to 8192 with a step of 128 edges were used. Additionally, networks with nodes ranging from 32 to 512 with a step of 32 and a density of 0.4 were also created. As human structural networks, we used the normalized connection matrices created from MRI tractography described in [17]. As human functional networks, we used the HCP 500-PTN functional dataset [18–20]. All networks were weighted and non-directed. B. Method In the proposed approach, brain networks are modelled as a Markov process where neuronal impulses randomly walk from one node to another node. This new interpretation provides a solid theoretical framework from which we derive a global (i.e., a single value for the whole network) and two local (i.e., a value for each node) measures based on mutual information. Mutual information (MI) measures the shared information between two random variables. From our Markov process-based brain model, we propose as a global connectivity measure the mutual information between two consecutive states of the process. Mutual information can also be seen as the difference between the uncertainty of the states without any knowledge and the uncertainty of the states when the past is known (or information gained when the previous node is known). The higher the MI, the less random the connections. Thus, mutual information can be used to quantify the overall brain structure. The mutual information can be decomposed in order to characterize the degree of informativeness of each state. When applied to the connectome, since each state corresponds to an anatomical or functional region, this measure can be seen as the contribution of each node to the whole graph structure. In this work, we propose two local measures. On the one hand, we use the mutual surprise (I1) [21], that expresses how ‘surprising’ are the connections of a node. Nodes that are connected with more likely nodes will lead to low values of mutual surprise, while those with very specific connections or connected with few unlikely nodes will have high mutual surprise. On the other hand, we use the mutual predictability (I2) [21], that expresses the uncertainty of a node taking into account the mean connectivity of all the network. I2 measures the capacity of prediction for a given brain region. III. RESULTS AND DISCUSSION Using model networks with different number of edges, an optimum point was found for lattice and ring lattice networks when increasing the density. This is due to the fact that for low densities, there are regions not connected, thus, the overall mutual information is low. This fact may help to find a minimum number of fibers needed to study brain networks for a given brain parcellation. Overall, higher values were obtained for lattice and ring lattice models, showing a clear evidence of more organized networks compared to random and small-world networks. When the number of edges was increased, the mutual information tended to decrease, since the higher number of connections, the lower correlation between consecutive states. Preserving the density, the mutual information was not very sensitive to random and small-world networks, since the structure is similar. Higher values were obtained for ring lattice networks when comparing with lattice networks, since in lattice networks two nodes are not connected and have a less structured network. Using anatomical and functional connectomes at different scales, a similar behavior was observed for all patients. Local measures were evaluated using the human connectomes. The mutual surprise highlighted regions connected to regions not highly connected, such as the right hemisphere transverse temporal. Low values were obtained for regions connected to highly connected regions such as the left hemisphere thalamus proper. The mutual predictability associated regions with a low number of connections and high weights with a high predictability, such us the right hemisphere temporal pole. Low values were obtained in regions with more uncertainly in predicting the next node, such as the right hemisphere putamen. All measures were consistent for structural and functional human networks. IV. CONCLUSIONS In this work, new measures to quantify structure of complex brain networks are proposed. Brain connectivity graphs are interpreted as a stochastic process where neural impulses are modeled as a random walk. This interpretation provides a solid theoretical framework from which different measures based on the mutual information measure have been applied. The measures have been tested on synthetic model networks and structural and functional human networks at different scales. Results show that the mutual information is able to quantify the structure of different model networks. The mutual surprise, allows the identification of nodes whose neighbors have a high connectivity taking into account all connections. The mutual predictability, shows that regions with a high clustering tend to be more predictable.

Acknowledgements

This work was supported by the Spanish Government (Grant No. TIN2013-47276-C6-1-R) and by the Catalan Government (Grant No. 2014-SGR-1232).

Data were provided, in part, by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University.

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Keywords: Brain network analysis, complex networks of the brain, graph theory analysis, connectome, mutual information (MI), Information Theory

Conference: Neuroinformatics 2016, Reading, United Kingdom, 3 Sep - 4 Sep, 2016.

Presentation Type: Investigator presentations

Topic: General neuroinformatics

Citation: Bonmati E, Bardera A and Boada I (2016). Measuring Complex Brain Networks Structure. Front. Neuroinform. Conference Abstract: Neuroinformatics 2016. doi: 10.3389/conf.fninf.2016.20.00012

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Received: 28 Apr 2016; Published Online: 18 Jul 2016.

* Correspondence: Dr. Ester Bonmati, University of Girona, Institute of Informatics and Applications, Girona, Spain, e.bonmati@ucl.ac.uk