Report

Correlation random fields, brain connectivity, and astrophysics Keith Worsley Arnaud Charil Jason Lerch Francesco Tomaiuolo Department of Mathematics and Statistics, McConnell Brain Imaging Centre, Montreal Neurological Institute, McGill University CfA red shift survey, FWHM=13.3 100 80 Euler Characteristic (EC) 60 "Meat ball" topology 40 20 "Bubble" topology 0 -20 -40 "Sponge" topology -60 -80 -100 -5 CfA Random Expected -4 -3 -2 -1 0 1 Gaussian threshold 2 3 4 5 fMRI data: 120 scans, 3 scans each of hot, rest, warm, rest, hot, rest, … First scan of fMRI data Highly significant effect, T=6.59 1000 hot rest warm 890 880 870 500 0 100 200 300 No significant effect, T=-0.74 820 hot rest warm 0 800 T statistic for hot - warm effect 5 0 -5 T = (hot – warm effect) / S.d. ~ t110 if no effect 0 100 0 100 200 Drift 300 810 800 790 200 Time, seconds 300 Effective connectivity • Measured by the correlation between residuals at pairs of voxels: Activation only Voxel 2 ++ + +++ Correlation only Voxel 2 + ++ Voxel 1 + + Voxel 1 + 3 Focal correlation 2 1 0 0 1 2 3 3 -1 -2 2 cor=0.58 -3 -2 0 4 2 5 6 7 1 0 8 n = 120 frames 9 10 11 -1 -2 -3 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? T = sqrt(df) cor / sqrt (1 - cor2) 0 Seed 1 2 3 T max = 7.81 P=0.00000004 4 6 4 5 6 7 2 0 8 9 10 11 -2 -4 -6 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. 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 Thresholds are not as high as you might think: E.g. 1000cc search region, 10mm smoothing, 100 df, P=0.05: dimensions D1 D2 Voxel1 - Voxel2 0 0 Cor T 0.165 1.66 One seed voxel - volume 0 3 0.448 4.99 Volume – volume (auto-correlation) 3 3 0.609 7.64 Volume1 – volume2 (cross-correlation) 3 3 0.617 7.81 Practical details • Find threshold first, then keep only correlations > threshold • Then keep only local maxima i.e. cor(voxel1, voxel2) > cor(voxel1, 6 neighbours of voxel2), > cor(6 neighbours of voxel1, voxel2), 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’ Extensive correlation 3 2 1 0 0 1 2 3 3 -1 -2 -3 2 cor=0.13 -2 0 4 2 5 6 7 1 0 8 9 10 11 -1 -2 -3 PCA, component 1 0 1 2 3 1 0.8 0.6 0.4 4 5 6 7 0.2 0 -0.2 8 9 10 11 -0.4 -0.6 -0.8 -1 Which is better: thresholding T statistic (= correlations), or PCA? T, extensive correlation 0 Seed 1 2 3 T max = 4.17 P = 0.59 6 4 4 5 6 7 2 0 8 9 10 11 -2 -4 -6 PCA, focal correlation 0 1 2 3 1 0.8 0.6 0.4 4 5 6 7 0.2 0 -0.2 8 9 10 11 -0.4 -0.6 -0.8 -1 Summary Focal correlation 0 1 2 3 Extensive correlation 6 0 1 2 3 4 Thresholding T statistic (=correlations) 4 5 6 7 2 4 4 5 6 7 0 8 0 9 1 10 2 11 3 -2 5 6 7 PCA 8 9 10 11 2 0 8 9 10 11 -2 -4 -4 -6 -6 1 0 1 2 3 1 0.8 0.8 0.6 0.6 0.4 4 6 0.2 0.4 4 5 6 7 0.2 0 0 -0.2 -0.2 -0.4 8 9 10 11 -0.4 -0.6 -0.6 -0.8 -0.8 -1 -1 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 ~ seed1 + seed2 + seed1*seed2 PCA of time space: Temporal components (sd, % variance explained) Component 0 1 0.68, 46.9% 2 0.29, 8.6% 3 0.17, 2.9% 4 0.15, 2.4% 5 0 20 40 60 80 100 1 Component 1 0.5 2 0 3 -0.5 4 2 4 6 8 Slice (0 based) 10 2: drift 120 Frame Spatial components 0 1: exclude first frames 12 -1 3: long-range correlation or anatomical effect: remove by converting to % of brain 4: signal? Fit a linear model for fMRI time series with AR(p) errors • Linear model: ? ? Yt = (stimulust * HRF) b + driftt c + errort • AR(p) errors: unknown parameters ? ? ? errort = a1 errort-1 + … + ap errort-p + s WNt • Subtract linear model to get residuals. • Look for connectivity. Deformation Based Morphometry (DBM) (Tomaiuolo et al., 2004) • n1 = 19 non-missile brain trauma patients, 3-14 days in coma, • n2 = 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. MS lesions and cortical thickness (Arnaud et al., 2004) • • • • N = 347 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) Male or female (GENDER)? Expressive or not expressive (EXNEX)? Correct bubbles All bubbles Image masked by bubbles as presented to the subject Correct / all bubbles Fig. 1. Results of Experiment 1. (a) the raw classification images, (b) the classification images filtered with a smooth low-pass (Butterworth) filter with a cutoff at 3 cycles per letter, and (c) the best matches between the filtered classification images and 11,284 letters, each resized and cut to fill a square window in the two possible ways. For (b), we squeezed pixel intensities within 2 standard deviations from the mean. Subject 1 Subject 2 Subject 3