1. Using Graph Theory to Study Neural Networks (Watrous, Tandon, Conner, Pieters & Ekstrom, 2012)

2. Origins of graph theory (1736) Leonard Euler Is it possible to cross every bridge only once and return to your starting point? No

3. Origins of graph theory Impossible: If there are more than 2 odd landmasses (node degree). Possible: If there are exactly 2 odd landmasses. No odd landmasses.

4. Definitions Node degree: # of edges connected to a node (high degree = hub). Cluster: when the nearest neighbors of a node are directly connected to each other (complex networks have high clustering). Path length: minimum # of edges to get from one node to another (complex networks have short average paths). Modules: contain many densely interconnected nodes. There are few connections between nodes in different modules.

5. Networks Random network: each pair of nodes have an equal probability of being connected to each other (used as a comparison for the observed network). Small-world network: common in biological and technological systems. – High levels of local clustering among nodes (greater than random network) – Short paths that link all nodes in the network (about equal to random network). The ways specific networks deviate from randomness reflects functionality.

6. Hubs Provincial Hub: high- degree nodes within the same module (V4 and MT). Connector hub: high- degree nodes with diverse connectivity to several modules (area 46/DLPFC).

7. 1. Define network nodes EEG, MEG, MRI, DTI, fMRI Parcellation schemes matter! One fMRI study used 2 different atlases that divided the brain into 70 and 90 regions of interest. Both found small-world properties, but… Significant differences existed at the local and global level. (Wang et al., 2009) 1

8. 2. Estimate a continuous association measure between nodes MEG: spectral coherence or granger causality between sensors. DTI: connection probability between 2 regions. MRI: inter-regional correlations of volume or cortical thickness. fMRI: correlation between any possible pair of regional residual time series, obtain a correlation matrix to threshold. 2 2

9. 3. Create an association matrix by compiling all pairwise associations Apply a threshold to each element of the association matrix to produce a binary adjacency matrix or an undirected graph. The threshold chosen can affect the sparsity or connection density. Alternatively, you can weight graphs instead. 3 3

10. 4. Calculate network parameters, compare them to random networks A number of parameters can be identified: Node degree Clustering Path length Connection density Connector hubs Provincial hubs Modularity 4

11. Computational models based on structural networks Structural networks serve as a matrix of coupling coefficients that link nodes. Time-course of activity is modeled by dynamic equations that model neural populations with physiologically plausible parameters. Functional networks come from measures of association between the simulated time series or cross-correlations of activity from simulated BOLD data. Matrices can then be thresholded to yield binary networks which can be used to obtain network measures.

12. Computational models based on structural networks 47 nodes and 505 edges were compiled from anatomical tract- tracing of the macaque cortex. Long time-scale: functional activity overlaps with structural network. Short time-scale: functional activity is less constrained by structural networks. (Honey, Kotter, Breakspear and Sporns, 2007)

13. Computational models based on structural networks Two major anticorrelated clusters were found. 1: mostly visual areas in OL and TL (blue). 2: somatomotor areas in FL and PL (green). “Association areas” participate in both clusters while other areas stay within their respective cluster.

14. Histological data from the macaque cortex Dorsal module(yellow) Ventral module (grey) V4 is a hub (red) High clustering Short path length Sparse connectivity between modules Areas linked through hubs

15. V4 and MT are provincial hubs. V4 connects mostly to areas in visual cortex. Dorsal white, ventral black Most other hubs including area 46 in the PFC are connector hubs. 46 has connections with visual, somatosensory, and motor regions. (Sporns, Honey, and Kotter, 2007)

16. Schizophrenics show high clustering among high-degree nodes # = Brodmann area ‘ = left hemisphere Healthy: low clustering of high-degree nodes. Schizophrenia: high clustering of high- degree nodes and long distance connections. Networks are more similar to random graphs.