Our path. Complex brain network analysis. Levels of Investigation. Background

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1 Complex brain network analysis Jesse Brown UCLA Bookheimer Lab July Our path Background foundations Networks general properties Defining brain networks Graph theory measures and applications Current research - what s new and where we might be going Levels of Investigation Background

2 Levels of Investigation All references are links >>>> Mesulam The figure that launched a Felleman and Van Sporns et al. PLoS

3 What is a network? Networks A set of discrete, interconnected nodes and edges (a graph ) Quantitative assessment of networks We can represent the graph as a matrix and use graph theory and its branch, complex network analysis, to characterize the network s connectivity Directed edgesundirected edges Types of graphs Binary edges Weighted edges

4 Classes of networks Dense Fragmented Regular Defining Brain Networks Sparse Connected Random Defining nodes in the brain Defining nodes in the brain 1. Anatomical atlas 2. Structural parcellation 3. Voxels 4. Coordinates of interest 5. Functional parcellation 6. Random parcellation 1. Anatomical atlas Harvard Oxford (FSL) 2. Structural parcellation LPBA40 3. Voxels Talairach 4. Coordinates of interest 5. Functional parcellation 6. Random parcellation

5 Defining nodes in the brain Defining nodes in the brain 1. Anatomical atlas Freesurfer 2. Structural parcellationautomated Anatomical 3. Voxels Labeling (AAL) 4. Coordinates of interest 5. Functional parcellation 6. Random parcellation 1. Anatomical atlas 2. Structural parcellation 3. Voxels 4. Coordinates of interest 5. Functional parcellation 6. Random parcellation From functional analysis From meta-analysis/ literature Power et. al Defining nodes in the brain Defining nodes in the brain 1. Anatomical atlas ICA 2. Structural parcellationspatially constrained 3. Voxels clustering 4. Coordinates of interest 5. Functional parcellation 6. Random parcellation Smith et al. PNAS 2012 Craddock et al. 1. Anatomical atlas 2. Structural parcellation 3. Voxels 4. Coordinates of interest 5. Functional parcellation 6. Random parcellation Zalesky et al.

6 Defining edges in the brain Structural data (eg DTI) Defining edges in the brain 1. fmri data 2. Define nodes 3. Extract nodal timeseries 4. Pairwise statistical similarity 5. Connectivity matrix Now we re getting Matrix processing issues DTI network fmri network Normally when networks are compared, thresholding is performed to enforce the same connection density for each network. Networks can be weighted or binarized For functional networks, negative weights must be adjusted since most graph theory measures disregard

7 Degree (k) Clustering (C) Graph Theory Properties # of edges connected to a node # of triangles/ # of connected triples (from a given node) Degree (k) Clustering (C) Functional Degree # of edges connected to a node # of triangles/ # of connected triples (from a given node) Regions with high rs-fmri degree have high spatial correspondence Buckner et al. J with sites of greatest Aβ deposition in

8 Structural Clustering Connectivity Connected Fragmented 1 component 31.1% dense (14 / 45) 2 components 28.9% dense (13 / 45) Mean clustering coefficient and mean cortical thickness both decline Brown et al. more rapidly with age in APOE-4 gene # of components = # of non-connected fragments density = # of edges / # of possible edges Path length Path length Shortest path from A to B

9 Path length Path length Shortest path from A to C Non-shortest path from A to B Tabulate shortest path length between all nodes in network Characteristic path length (CPL): avg. path length for the network (grand mean of distance matrix) Store shortest path length between each pair of nodes in a distance matrix Global Efficiency (E glob ): grand mean of inverse distance matrix Local Efficiency (E loc ) (inverse of) Avg. path length of all shortest paths in B-D-E-F subnetwork Regional Efficiency (E reg ) (inverse of) Avg. path length from B to any other node in whole network

10 Functional Path Length Betweenness Centrality % of shortest paths in whole network that traverse a given node Subjects rs-fmri functional characteristic path lengths correlate negatively with Van den Heuvel et Intelligence Quotient (i.e. greater network Structural Centrality Modularity (Q) Participation Coefficient Structural networks derived from Diffusion Spectrum Imaging data have hub Read this paper! >>>> Hagmann et al. regions (high centrality and efficiency) that Division of the network into sub-networks that have maximal within-network connections and minimal between-network connections For a given node, the diversity of intermodular connections

11 Functional Modularity Modularity (Q) Participation Coefficient Less modular For a given node, the diversity of intermodular connections Normally developing children have higher rs-fmri modularity than children with childhood-onset schizophrenia Alexander-Bloch et al. Randomization Many network measures for example, small worldness and modularity are defined by comparing the true network to an equivalently sized but randomly wired network. This random network serves as a null model. Randomization Before rewiring After rewiring Thus, we must randomize.

12 Randomization Before rewiring, this particular network (and most networks derived from neuroim data) has: Structure Well defined modules High clustering Short path lengths Randomization After rewiring, this particular network (and most networks derived from neuroim data) has: Equivalent density Less structure Almost no modularity Low clustering Short path lengths (random networks have lots of shortcuts ) Randomization Before rewiring True CPL True MCC After rewiring Random CPL Random MCC γ = true mcc / random mcc λ = true CPL / random CPL σ = γ / λ; this is the small worldness σ > 1.2: small world regime Other network measures Degree centrality Eigenvector centrality Katz centrality PageRank Closeness centrality Edge centrality Transitivity Assortativity Synchronizability Robustness to targeted attack Robustness to random attack Cost Rent s exponent

13 Software Brain Connectivity Toolbox (graph theory) ( bct/) MATLAB Boost Graph Library (graph theory) ( NetworkX (graph theory) ( igraph (graph theory) ( UMCP (connectivity matrix derivation) ( UCLA_Multimodal_Connectivity_Package) Connectome Viewer (network visualization) ( Current Research Rich club: K-core decomposition Rich club Original network K-core K-core K = 2 φ: rich club coefficient φ = density (K-core) K = 2 φ =.361 (13 / 36) All nodes in K-core have degree >= 2 For this network, no K-cores larger than K = 2 Van den Heuvel et Rich club coefficient can be compared to pseudo value from random networks to determine statistical significance Van den Heuvel et

14 Rich club Average DTI network calculated from 21 healthy subjects Rich club of six bilateral cortical and six bilateral subcortical regions discovered Serves as structural core that is vital for global efficiency (40% of structural cost, 69% of shortest paths Van den Heuvel et al. PNAS Van 2012 den Heuvel et Testing disease models For five neurodegenerative diseases, perform structural MRI on subjects with fully manifested disease Calculate average atrophy maps compared to normal subjects, identify 1) amount of atrophy at every node, 2) epicenters of maximum atrophy Calculate average healthy rs-fmri connectivity matrix from 16 healthy subjects For each node, calculate: 1) Total flow (aka degree) 2) Clustering coefficient Seeley et al. Neuron 2009 Zhou et al. Testing disease models Testing disease models Nodal stress: wear and tear Transneuronal spread: toxic protein propagates along network connections Trophic failure: decreased inter-nodal trophic factor support Zhou et al. For all diseases, nodal shortest path to epicenter (measure 3) had strongest relationship to disease-specific atrophy pattern Zhou et al.

15 Dynamic networks Structure/Function Network Comparison With increasing cognitive effort in a N-back working memory task, MEG functional Kitzbichler et networks become more efficient, less DTI and rs-fmri data for 196 subjects used to derive structural and functional connectivity matrices Matrices assessed for three global measures (path length, mean clustering, and modularity) and five nodal properties (strength, clustering, betweenness, regional efficiency, and participation coefficient) Brown et al., Structure/Function Network Comparison For group-average networks, fcmri network had longer path lengths, lower clustering, higher modularity Brown et al., Structure/Function Network Comparison For global measures compared across individuals, only modularity had a significant correlation between structural and functional networks For nodal measures, no measure had significant correlation Brown et al.,

16 Data sharing Data sharing ebsite from brain network analysis and data sha ading of connectivity matrices along with requis regions included in network, subject pool (norm networks that runs in the cloud, rapidly produ alizations (3D and 2D), global and Brown regional et al., netw Generates graph theory report of any publicly shared network Share your data! We hope the site will enable larger scale meta-analyses, larger set of samples with which to generate hypotheses or train classifiers Brown et al., Open questions Additional References! Are path lengths meaningful in functional networks? What do negative connection strengths mean and how should we deal with them? Does efficiency as defined by graph theory really relate to information processing efficiency? How do functional modules evolve and overlap at rapid timescales? How spatially and temporally independent are they really? Will we eventually get a nice pretty equation

17 Acknowledgements Bookheimer Lab Susan Bookheimer, Alison Burggren, Brian Renner, Mike Jones, Kevin Terashima, Laurel Martin-Harris, Leo Moore, Tessa Harrison NITP Mark Cohen, Edward Lau, Andrew Cho Others Jeff Rudie, Pam Douglas, John Colby, Jack Van Horn, Dani Bassett Real world networks Bookheimer 2012