Graph Theory. Steffie Tomson UCLA NITP 2013

Transcription

1 Graph Theory Steffie Tomson UCLA NITP 2013

2 What is graph theory? Mathematical descriptions of relationships between regions Node Opsahl et al. 2010

3 Why use graphs? Nodal Measures: Participation Coefficient Clustering Coefficient Degree Local efficiency Global Measures: Global efficiency Modularity Hubs

4 Graph Pipeline Network organization Aleman Gomez et al Functional MRI Structural MRI A B Brain Regions Brain Regions

5 Parcellation How do you divide up the brain? Anatomical atlas AAL Havard Oxford Talairach Functional regions Craddock et al Power et al Craddock et al. 2012

6 Graph Pipeline Network organization Aleman Gomez et al Functional MRI Structural MRI A B Brain Regions Brain Regions

7 Structural connectivity Fractional anisotropy (FA) Measure of white matter diffusivity at each voxel Tractography Directional pattern of white matter tracts Structural CoCoMac

8 Functional connectivity Looooooong timeseries Resting state Listening to audio Watching video Acquire twice as many timepoints as nodes in your graph (Braun et al. 2012) Prepocessing Regress out motion Small smoothing kernel Functional MRI EEG MEG NIRS

9 Functional connectivity Functional MRI Full correlation Partial correlation Bayes nets LiNGAM EEG/MEG Coherence Granger Functional MRI EEG MEG NIRS Smith et al Network modeling methods for fmri. NeuroImage.

10 Connectivity measures Functional MRI Correlation A B C Partial Correlation A B C All three regions appear to be correlated because they all covary. One region, however, could be driving activity in two other regions, inducing a false When the correlation between A and B is evaluated given C s activity, C is actually better at explaining the variance of A and B than either are of each other. C is the common driver. correlation. Marrelec et al. 2006, Smith et al. 2011

11 Graph Pipeline Network organization Aleman Gomez et al Functional MRI Structural MRI A B Brain Regions Brain Regions

12 Graph pipeline Functional MRI A Structural MRI B zz Quantify relationship between nodes Brain Regions Brain Regions

13 Threshold ? Brain Regions Brain Regions

14 Threshold Keep only 10, 20, 30% of connections Keep everything above a value Keep everything above an absolute value Discard values around zero Use range of thresholds

15 Network Metric Group 1 Group Matrix Threshold

16 Stability selection 1 subject Dr. Genevera Allen Manjari Narayan λ 1 λ 2 λ n 1 λ 2-1 Average Average Average Sparse network best supported by data Liu et al., 2010.

17 MONET Markov Network Estimation Toolbox Genevera Allen, Manjari Naryan, Jonathan Stewart at Rice University

18 Graph pipeline Aleman Gomez et al Functional MRI Structural MRI A B Brain Regions Brain Regions

19 Types of graphs Weighted Binary

20 Rubinov and Sporns 2010

21 Graph pipeline Network organization Aleman Gomez et al Functional MRI Structural MRI A B Brain Regions Brain Regions

22 NODE Brain Regions Brain Regions EDGE Node = Brain region Edge = Connection between brain regions

23 Graph theory metrics Rubinov and Sporns. Complex network measures of brain connectivity: uses and interpretations Neuroimage.

24 Graph theory metrics Degree High degree Low degree Number of edges emanating from a single node

25 Graph theory metrics Clustering coefficient High Low How many of your nearest neighbors are connected to one another?

26 Graph theory metrics Local efficiency High Low Average shortest path connecting all neighbors of a given node

27 Graph theory metrics Characteristic path length Low High Average shortest path length between all node pairs

28 Graph theory metrics Global efficiency High Low Inverse of the average path length

29 Graph theory metrics Betweenness centrality Hubness High Low Number of shortest paths that pass through a given node

30 Graph theory metrics Hub Buckner et al. 2009

31 Graph theory metrics Modularity

32 Graph theory metrics Modularity Control Child onset schizophrenia Alexander Bloch et al. 2010

33 Brain connectivity toolbox

34 Testing for differences Statistical inference Bootstrap procedure Permutation tests Tomson et al In Press.

35 Testing for differences Statistical inference Bootstrap procedure Permutation tests Tomson et al In Press.

36 Multiple comparisons False discovery rate Benjamini Hochberg Family wise error rate Gaussian random field theory Bonferroni Choose graph metrics you care about Choose graph metrics least mathematically related

37 Visualization Connectome Viewer

38 Resources Brain connectivity toolbox MONET UMCD Brain Net viewer ABIDE ADNI COINS Connectome Project Open fmri Rubinov and Sporns 2010 Smith et al Bullmore and Sporns Nat Rev Neuro.

39 Acknowledgements Bookheimer lab Dappretto lab Bearden lab Dr. Genevera Allen Manjari Narayan NITP 2011 Neurobehavioral Genetics grant

40 Questions steffietomson.com