Connectivity of aMRI and fMRI data

Keith WorsleyArnaud CharilJason Lerch

Francesco Tomaiuolo

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

McGill University

Effective connectivity• Measured by the correlation between residuals at

pairs of voxels:

Voxel 2

Voxel 1

++ +++ +

Activation onlyVoxel 2

Voxel 1++

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Correlation only

Types of connectivity

• Focal

• Extensive

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Focal correlation

n = 120frames

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Extensive correlation

Methods

1. Seed

2. Iterated seed

3. Thresholding correlations

4. PCA

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?

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’

Which is better:

thresholding correlations,

or

PCA?

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Summary

Extensive correlationFocal correlation

Thresholding T statistic

(=correlations)

PCA

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1000First scan of fMRI data

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T statistic for hot - warm effect

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870880890 hot

restwarm

Highly significant effect, T=6.59

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820hotrestwarm

No significant effect, T=-0.74

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790800810

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

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Temporal components (sd, % variance explained)

0.68, 46.9%

0.29, 8.6%

0.17, 2.9%

0.15, 2.4%

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Slice (0 based)

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Spatial components

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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

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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Average lesion volume

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n=425 subjects, correlation = -0.56826

Normalization

• Simple correlation:

Cor( LD, CT )

• Subtracting global mean thickness:

Cor( LD, CT – avsurf(CT) )

• And removing overall lesion effect:

Cor( LD – avWM(LD), CT – avsurf(CT) )

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distance (mm)

corr

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Same hemisphere

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distance (mm)

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Correlation = 0.091943

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x 105

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Different hemisphere

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Correlation = -0.1257

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threshold

thresholdthreshold

threshold

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.

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3Seed

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

T max = 7.81P=0.00000004

PCA, component 1

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3Seed

T max = 4.17P = 0.59

T, extensive correlation

PCA, focal correlation

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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 ~ seed1 + seed2 + seed1*seed2

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