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Graph Analysis of fMRI data
Sepideh Sadaghiani, PhD
U C L A / S E M E L A D VA N C E D N E U R O I M A G I N G S U M M E R P R O G R A M 2 0 1 5
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Contents Introduction § What is graph analysis? § Why use graphs in fMRI?
Decisions § Which fMRI data? § Which nodes? § Which edges? § Which modules? § Which graph metrics?
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What is a graph? § Mathematical representation of a real-world network with pairwise
relations between objects
undirected unweighted directed directed
weighted
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Why graphs?
Euler 1736: The bridge puzzle of Königsberg
Necessary condition for the walk crossing each bridge exactly once: Zero or two nodes with odd degree.
Physical space Topological space
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Real-world networks Social network
seed person
Protein-protein “interactome”
Goh et al. PNAS 2007
Disease gene network
Brain connectome
Bullmore & Bassett 2010
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Graph Theory § Real-life networks are complex
§ Graph theory allows mathematical study of complex networks
§ Describe properties of a complex system: Quantify topological characteristics of its graph representation
Bullmore & Sporns, 2012
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Why use graphs analyses in fMRI? Quantification of global properties of spatio-temporal network organization
§ Early motivation: A testable theory of consciousness (Edelman & Tononi 2000)
Based on global network integration (information theory) § Structural connectivity ↔ functional connectivity § Comparisons across individuals (e.g. in disorders) § Comparison across mental and functional states
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Graph construction in fMRI - overview
Wang et al., 2010
Time series extraction
Choice of nodes
Thresholding (& optional binarization)
Adjacency matrix Rows & columns: nodes
Entries: edges
Pairwise connectivity (e.g. Pearson’s
correlations)
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Which datasets? M E N T A L S T A T E ,
P R E P R O C E S S I N G
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fMRI Datasets Connections most commonly derived from resting state, but task data
possible in principle Session length most commonly 5-10min § Long enough for multiple cycles of infraslow (<0.1Hz) frequencies § Short enough to minimize mental state change
§ Shorter term time-varying dynamics (e.g. sliding window) Preprocessing: same considerations as any fMRI connectivity study: § What motion correction? § Slice time correction?
§ Physiological nuisance measures? § Compartment signal regression (GM, WM, CSF)?
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Which nodes? A N A T O M I C A L V S . F U N C T I O N A L
A T L A S V S . D A T A - D R I V E N
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Anatomical atlases Nodes: § Internally coherent / homogeneous (connectivity) § Externally independent
Anatomical atlases
§ Automated Anatomical Labeling (AAL) template § Eickhoff-Zilles (Cytoarchitectonic) § FreeSurfer (Gyral. Individual surface-based possible) § Harvard-Oxford § Talairach & Tournoux
§ J Comparability (across subjects and modalities) § L Highly variable node size. Not functionally coherent.
21subcortical 48 cortical Harvard-Oxford
FreeSurfer (Destrieux)
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Functional atlases
Functional atlases
§ Craddock (local homogeneity of connectivity) § Power (resting state seed-based & task) § Stanford Atlas FIND lab (ICA-based) § J Comparability across subjects. § J Functionally coherent (L but suboptimal for individuals)
Functional subject-specific parcellations
§ ICA § (Seed-based) § Connectivity homogeneity: Craddock § J Functionally coherent § L Time-intensive
List of atlases: https://en.wikibooks.org/wiki/SPM/Atlases
Power et al. 2011: 164 peak locations
Craddock et al. 2012
FINDlab, Shirer et al. 2012
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Which edges? C O N N E C T I V I T Y A N D
T H R E S H O L D I N G
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Edges in fMRI Based on magnitude of temporal covariation
§ Pearson’s cross-correlations (by far most common) § Partial correlations § Mutual information
§ à symmetric adjacency matrices (undirected graphs) Directionality problematic in fMRI (but measures of effective connectivity possible)
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Other Data Modalities Structural (e.g. DTI, histological tracing)
§ Nodes: cf. fMRI § Edges: e.g. number of reconstructed fibers
EEG / MEG
§ Nodes: sensors or reconstructed sources § Edges: Correlation in oscillation amplitudes
Oscillation phase synchrony (coherence of phase locking)
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Thresholding Most metrics require sparse graphs Threshold to remove weak connections Use proportional thresholds (vs. absolute thresholds) Use broad range of proportions
Thresholding (& optional
binarization)
Adjacency matrix
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Which Modules? C O M M U N I T Y D E T E C T I O N
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Modules Communities of densely interconnected nodes
Community detection
Wang et al., 2010
Optimization algorithms
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Community Detection Algorithms Modularity-base algorithms Maximize number of within-community edges (compared to random network) § Newman’s Modularity (Newman, 2006) § Louvain method (Blondel et al. 2008) Infomap algorithm (Rosvall and Bergstrom, 2008)
Minimize information theoretic descriptions of random walks on the graph Review: Fortunato, 2010
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Which graph metrics? N O D A L A N D G L O B A L
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Nodal Measures Degree
Number of edges connected to a node
Nodal Clustering Coefficient (è basis for measure of global segregation)
Fraction of all possible edges realized among a node’s neighbors = Fraction of all possible triangles around a node
Shortest Path Length (è basis for measure of global integration)
Number of edges on shortest geodesic path between two nodes
Sporns, 2011
Path Length=3
Degree=6
Degree=1
CC =8/15 =0.53
è Distance matrix
(n(n-1)/2)
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Nodal Measures Measures of centrality: Closeness Centrality
Inverse of the node’s average Shortest Path Length Betweenness Centrality
Fraction of all shortest paths passing through the node Participation Coefficient
Diversity of intermodular connections Within-Module Degree (z-score)
Degreeintramodule z-scored within the node’s module “Provincial hubs”: high within-module degree & low participation coefficient “Connector hubs”: high participation coefficient “Rich club”: densely interconnected connector hubs
Connector hub
Bullmore & Sporns, 2012
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24
Global Measures
Measures of Integration Characteristic Path Length average Shortest Path Length to all
other nodes
Global Efficiency
average inverse Shortest Path Length to all other nodes
Measures of Segregation
Clustering Coefficient
nodal Clustering Coefficient Modularity (Newman’s)
Fraction of edges falling within the module minus expected fraction in a random network
àOften used to detect community structure
1n nodes∑1
n nodes∑
1n nodes∑
Rubinov & Sporns, 2010
Modules∑
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Global Measures Small-worldness Optimal balance between functional segregation and integration
Clustering Coefficientreal / Clustering Coenfficientrandom
Characteristic Path Lengthreal / Characteristic Path Lengthrandom
J Functionally specialized (segregated) modules AND intermodular (integrating) edges
Watts & Strogatz, 1998
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Resources Analysis Software § MATLAB-based: Brain Connectivity Toolbox (Rubinov & Sporns, 2010)
https://sites.google.com/site/bctnet/
§ Python-based: NetworkX (Hagberg et al., 2008) https://networkx.github.io
Visualization § General: Gephi http://gephi.github.io § Anatomical space: Multimodal Connectivity Database
http://umcd.humanconnectomeproject.org § Anatomical space: Connectome Visualization Utility
https://github.com/aestrivex/cvu Reading § Rubinov M, Sporns O. Complex network measures of brain connectivity: Uses
and interpretations. NeuroImage. 2010 Sep;52(3):1059–69.
§ Bullmore ET, Bassett DS. Brain Graphs: Graphical Models of the Human Brain Connectome. Annu Rev Clin Psychol. 2010 Apr;7(1):113–40.
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References § Blondel, V.D., Guillaume, J.-L., Lambiotte, R., Lefebvre, E., 2008. Fast unfolding of
communities in large networks. J. Stat. Mech. 2008, P10008 § Bullmore E, Sporns O. The economy of brain network organization. Nat Rev Neurosci.
2012 May;13(5):336–49. § Craddock RC, James GA, Holtzheimer PE, Hu XP, Mayberg HS. A whole brain fMRI atlas
generated via spatially constrained spectral clustering. Hum Brain Mapp. 2012 Aug 1;33(8):1914–28.
§ Fortunato, S. (2010). Community detection in graphs. Phys. Rep. 486, 75–174. § Newman MEJ. Modularity and community structure in networks. PNAS. 2006 Jun
6;103(23):8577–82. § Hagberg, A.A., Schult, D.A., Swart, P.J., 2008. Exploring network structure, dynamics,
and function using networkx. In: Varoquaux, G., Vaught, T., Millman, J. (Eds.), Proceedings of the 7th Python in Science Conference (SciPy2008). Pasadena, CA USA, pp. 11–15.
§ Power JD, Cohen AL, Nelson SM, Wig GS, Barnes KA, Church JA, et al. Functional Network Organization of the Human Brain. Neuron. 2011 Nov 17;72(4):665–78.
§ Rosvall, M., and Bergstrom, C.T. (2008). Maps of random walks on complex networks reveal community structure. Proc. Natl. Acad. Sci. USA 105, 1118– 1123.
§ Shirer WR, Ryali S, Rykhlevskaia E, Menon V, Greicius MD. Decoding Subject-Driven Cognitive States with Whole-Brain Connectivity Patterns. Cerebral Cortex. 2012 Jan 1;22(1):158–65.
§ Sporns O. The non-random brain: efficiency, economy, and complex dynamics. Front Comput Neurosci. 2011;5:5.
§ Wang J, Zuo X, He Y. Graph-based network analysis of resting-state functional MRI. Frontiers in Systems Neuroscience. 2010. doi: 10.3389/fnsys.2010.00016
§ Watts DJ, Strogatz SH. Collective dynamics of “small-world” networks. Nature. 1998 Jun 4;393(6684):440–2.