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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

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

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Types of connectivity Focal Extensive

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-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

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

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

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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?

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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.

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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

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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

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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 ),

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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’

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Which is better: thresholding correlations, or PCA?

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-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

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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

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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

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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)

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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

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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) )

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threshold

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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.

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-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

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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

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-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

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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

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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

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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

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