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

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

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

cor=0.58 Focal correlation n = 120 frames

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?

Seed T = sqrt(df) cor / sqrt (1 - cor 2 ) T max = 7.81 P=

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 D 1 D 2 Cor T Voxel 1 - Voxel One seed voxel - volume Volume – volume (auto-correlation) Volume 1 – volume 2 (cross-correlation)

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 ),

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’

cor=0.13 Extensive correlation

PCA, component

Which is better: thresholding T statistic (= correlations), or PCA?

Seed T max = 4.17 P = 0.59 T, extensive correlation

PCA, focal correlation

Summary Extensive correlationFocal correlation Thresholding T statistic (=correlations) PCA

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

Component Temporal components (sd, % variance explained) 0.68, 46.9% 0.29, 8.6% 0.17, 2.9% 0.15, 2.4% Slice (0 based) Component Spatial components 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

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

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.

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)

Expressive or not expressive (EXNEX)? Male or female (GENDER)? Correct bubbles Image masked by bubbles as presented to the subject All bubbles 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 1Subject 2Subject 3