Correlation random fields, brain connectivity, and astrophysics

Keith WorsleyArnaud CharilJason Lerch

Francesco Tomaiuolo

Department of Mathematics and Statistics, McConnell Brain Imaging Centre, Montreal Neurological Institute,

McGill University

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100CfA red shift survey, FWHM=13.3

Gaussian threshold

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"Bubble"topology

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CfARandomExpected

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Drift

Time, seconds

fMRI data: 120 scans, 3 scans each of hot, rest, warm, rest, hot, rest, …

T = (hot – warm effect) / S.d. ~ t110 if no effect

Effective connectivity• Measured by the correlation between residuals at

pairs of voxels:

Voxel 2

Voxel 1

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Activation onlyVoxel 2

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Correlation only

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Focal correlation

n = 120frames

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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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T = sqrt(df) cor / sqrt (1 - cor2)

T max = 7.81P=0.00000004

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 Cor T

Voxel1 - Voxel2 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

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’

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Extensive correlation

PCA, component 1

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Which is better:

thresholding T statistic (= correlations),

or PCA?

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T max = 4.17P = 0.59

T, extensive correlation

PCA, focal correlation

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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 ~ seed1 + seed2 + seed1*seed2

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0.68, 46.9%

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PCA of time space:

1: excludefirst frames

2: drift

3: long-range correlationor anatomicaleffect: removeby converting to % of brain

4: signal?

Frame

Fit a linear model for fMRI time series with AR(p) errors

• Linear model: ? ? Yt = (stimulust * HRF) b + driftt c + errort

• AR(p) errors: ? ? ? errort = a1 errort-1 + … + ap errort-p + s WNt

• Subtract linear model to get residuals.• Look for connectivity.

unknown parameters

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)

Expressive or notexpressive (EXNEX)?

Male or female(GENDER)?

Correct bubbles

Image masked by bubblesas 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 1 Subject 2 Subject 3