1. Connectivity of aMRI and fMRI data Keith Worsley Arnaud Charil Jason Lerch Francesco Tomaiuolo Department of Mathematics and Statistics, McConnell Brain Imaging Centre, Montreal Neurological Institute, McGill University

2. Effective connectivity Measured by the correlation between residuals at pairs of voxels: Voxel 2 Voxel 1 + + + + + + Activation only Voxel 2 Voxel 1 + + + + + + Correlation only

3. Types of connectivity Focal Extensive

4. -3 -2 0 1 2 3 8 4 0 9 5 1 10 6 2 11 7 3 cor=0.58 Focal correlation n = 120 frames

5. -3 -2 0 1 2 3 8 4 0 9 5 1 10 6 2 11 7 3 cor=0.13 Extensive correlation

6. Methods 1.Seed 2.Iterated seed 3.Thresholding correlations 4.PCA

7. Method 1: ‘Seed’ Friston et al. (19??): Pick one voxel, then find all others that are correlated with it: Problem: how to pick the ‘seed’ voxel?

8. Method 2: Iterated ‘seed’ Problem: how to find the rest of the connectivity network? Hampson et al., (2002): Find significant correlations, use them as new seeds, iterate.

9. Method 3: All correlations Problem: how to find isolated parts of the connectivity network? Cao & Worsley (1998): find all correlations (!) 6D data, need higher threshold to compensate

10. Thresholds are not as high as you might think: E.g. 1000cc search region, 10mm smoothing, 100 df, P=0.05: dimensions D 1 D 2 Cor T Voxel 1 - Voxel 2 0 0 0.165 1.66 One seed voxel - volume 0 3 0.448 4.99 Volume – volume (auto-correlation) 3 3 0.609 7.64 Volume 1 – volume 2 (cross-correlation) 3 3 0.617 7.81

11. Practical details Find threshold first, then keep only correlations > threshold Then keep only local maxima i.e. cor(voxel 1, voxel 2 ) > cor(voxel 1, 6 neighbours of voxel 2 ), > cor(6 neighbours of voxel 1, voxel 2 ),

12. Method 4: Principal Components Analysis (PCA) Friston et al: (1991): find spatial and temporal components that capture as much as possible of the variability of the data. Singular Value Decomposition of time x space matrix: Y = U D V’ (U’U = I, V’V = I, D = diag) Regions with high score on a spatial component (column of V) are correlated or ‘connected’

13. Which is better: thresholding correlations, or PCA?

14. -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 8 4 0 9 5 1 10 6 2 11 7 3 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 8 4 0 9 5 1 10 6 2 11 7 3 -6 -4 -2 0 2 4 6 8 4 0 9 5 1 10 6 2 11 7 3 -6 -4 -2 0 2 4 6 8 4 0 9 5 1 10 6 2 11 7 3 Summary Extensive correlationFocal correlation Thresholding T statistic (=correlations) PCA

15. fMRI data: 120 scans, 3 scans each of hot, rest, warm, rest, hot, rest, … T = (hot – warm effect) / S.d. ~ t 110 if no effect

16. 020406080100120 5 4 3 2 1 0 Component Temporal components (sd, % variance explained) 0.68, 46.9% 0.29, 8.6% 0.17, 2.9% 0.15, 2.4% -0.5 0 0.5 1 Slice (0 based) Component Spatial components 024681012 1 2 3 4 PCA of time  space: 1: exclude first frames 2: drift 3: long-range correlation or anatomical effect: remove by converting to % of brain 4: signal? Frame

17. MS lesions and cortical thickness (Arnaud et al., 2004) n = 425 mild MS patients Lesion density, smoothed 10mm Cortical thickness, smoothed 20mm Find connectivity i.e. find voxels in 3D, nodes in 2D with high cor(lesion density, cortical thickness)

18. 01020304050607080 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Average lesion volume Average cortical thickness n=425 subjects, correlation = -0.56826

19. Normalization Simple correlation: Cor( LD, CT ) Subtracting global mean thickness: Cor( LD, CT – av surf (CT) ) And removing overall lesion effect: Cor( LD – av WM (LD), CT – av surf (CT) )

20. threshold

21. Deformation Based Morphometry (DBM) (Tomaiuolo et al., 2004) n 1 = 19 non-missile brain trauma patients, 3-14 days in coma, n 2 = 17 age and gender matched controls Data: non-linear vector deformations needed to warp each MRI to an atlas standard Locate damage: find regions where deformations are different, hence shape change Is damage connected? Find pairs of regions with high canonical correlation.

22. -6 -4 -2 0 2 4 6 8 4 0 9 5 1 10 6 2 11 7 3 Seed T = sqrt(df) cor / sqrt (1 - cor 2 ) T max = 7.81 P=0.00000004

23. PCA, component 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 8 4 0 9 5 1 10 6 2 11 7 3

24. -6 -4 -2 0 2 4 6 8 4 0 9 5 1 10 6 2 11 7 3 Seed T max = 4.17 P = 0.59 T, extensive correlation

25. PCA, focal correlation -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 8 4 0 9 5 1 10 6 2 11 7 3

26. Modulated connectivity Looking for correlations not very interesting – ‘resting state networks’ More intersting: how does connectivity change with - task or condition (external) - response at another voxel (internal) Friston et al., (1995): add interaction to the linear model: Data ~ task + seed + task*seed Data ~ seed 1 + seed 2 + seed 1 *seed 2

27. Fit a linear model for fMRI time series with AR(p) errors Linear model: ? ? Y t = (stimulus t * HRF) b + drift t c + error t AR(p) errors: ? ? ? error t = a 1 error t-1 + … + a p error t-p + s WN t Subtract linear model to get residuals. Look for connectivity. unknown parameters