Detecting connectivity: MS lesions, cortical thickness, and the

“bubbles” task in the fMRI scanner

Keith Worsley, McGill (and Chicago)

Nicholas Chamandy, McGill and Google

Jonathan Taylor, Université de Montréal and Stanford

Robert Adler, Technion

Philippe Schyns, Fraser Smith, Glasgow

Frédéric Gosselin, Université de Montréal

Arnaud Charil, Alan Evans, Montreal Neurological Institute

Oury’s course, lecture 2

What is ‘bubbles’?

Nature (2005)

Subject is shown one of 40 faces chosen at random …

Happy

Sad

Fearful

Neutral

… but face is only revealed through random ‘bubbles’

First trial: “Sad” expression

Subject is asked the expression: “Neutral”

Response: Incorrect

0

0.1

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Sad 75 random bubble centres

Smoothed by aGaussian ‘bubble’

What the subject sees

Your turn …

Trial 2

Subject response:

“Fearful”

CORRECT

Your turn …

Trial 3

Subject response:

“Happy”

INCORRECT(Fearful)

Your turn …

Trial 4

Subject response:

“Happy”

CORRECT

Your turn …

Trial 5

Subject response:

“Fearful”

CORRECT

Your turn …

Trial 6

Subject response:

“Sad”

CORRECT

Your turn …

Trial 7

Subject response:

“Happy”

CORRECT

Your turn …

Trial 8

Subject response:

“Neutral”

CORRECT

Your turn …

Trial 9

Subject response:

“Happy”

CORRECT

Your turn …

Trial 3000

Subject response:

“Happy”

INCORRECT(Fearful)

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100

200

300

50100150200250

0.65

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Bubbles analysis

E.g. Fearful (3000/4=750 trials):Trial

1 + 2 + 3 + 4 + 5 + 6 + 7 + … + 750 = Sum

Correcttrials

Proportion of correct bubbles=(sum correct bubbles)

/(sum all bubbles)

Thresholded atproportion of

correct trials=0.68,scaled to [0,1]

Use thisas a bubblemask

Results

Mask average face

But are these features real or just noise? Need statistics …

Happy Sad Fearful Neutral

0.65

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0.75

-2024

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Statistical analysis

Correlate bubbles with response (correct = 1, incorrect = 0), separately for each expression

Equivalent to 2-sample Z-statistic for correct vs. incorrect bubbles, e.g. Fearful:

Very similar to the proportion of correct bubbles:

Response0 1 1 0 1 1 1 … 1

Trial 1 2 3 4 5 6 7 … 750Z~N(0,1)statistic

1.64

2.13

2.62

3.11

3.6

4.09

4.58

Results

Thresholded at Z=1.64 (P=0.05)

Multiple comparisons correction? Need random field theory …

Average faceHappy Sad Fearful Neutral

Z~N(0,1)statistic

3.92

4.03

4.14

4.25

4.36

4.47

4.58

Results, corrected for search

Random field theory threshold: Z=3.92 (P=0.05)

3.82 3.80 3.81 3.80 Saddle-point approx (Chamandy, 2007): Z=↑ (P=0.05) Bonferroni: Z=4.87 (P=0.05) – nothing

Average faceHappy Sad Fearful Neutral

Z~N(0,1)statistic

Scale

Separate analysis of the bubbles at each scale

Scale space: smooth Z(s) with range of filter widths w= continuous wavelet transform

adds an extra dimension to the random field: Z(s,w)

15mm signal is best detected with a 15mm smoothing filter

-20 2 4 6 8

Scale space, no signal

6.8

10.2

15.2

22.7

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-20 2 4 6 8

One 15mm signal

6.8

10.2

15.2

22.7

34

-60 -40 -20 0 20 40 60

w =

FW

HM

(mm

, on

log

scal

e)

s (mm)Z(s,w)

-20 2 4 6 8

10mm and 23mm signals

6.8

10.2

15.2

22.7

34

-60 -40 -20 0 20 40 60

-20 2 4 6 8

Two 10mm signals 20mm apart

6.8

10.2

15.2

22.7

34

-60 -40 -20 0 20 40 60

w =

FW

HM

(mm

, on

log

scal

e)

s (mm)But if the signals are too close together they are

detected as a single signal half way between them

Matched Filter Theorem (= Gauss-Markov Theorem): “to best detect signal + white noise,

filter should match signal”

Z(s,w)

-60 -40 -20 0 20 40 600

5

108mm and 150mm signals at the same location

5

10

15

20

6.8

15.2

34

76

170

-60 -40 -20 0 20 40 60

w =

FW

HM

(mm

, on

log

scal

e)

s (mm)

Scale space can even separate two signals at the same location!

Z(s,w)

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10000

Bubbles task in fMRI scanner

Correlate bubbles with BOLD at every voxel:

Calculate Z for each pair (bubble pixel, fMRI voxel) a 5D “image” of Z statistics …

Trial1 2 3 4 5 6 7 … 3000

fMRI

Thresholding?

Thresholding in advance is vital, since we cannot store all the ~1 billion 5D Z values Resels = (image resels = 146.2) × (fMRI resels =

1057.2) for P=0.05, threshold is Z = 6.22 (approx)

Only keep 5D local maxima Z(pixel, voxel) > Z(pixel, 6 neighbours of voxel) > Z(4 neighbours of pixel, voxel)

Generalised linear models?

The random response is Y=1 (correct) or 0 (incorrect), or Y=fMRI The regressors are Xj=bubble mask at pixel j, j=1 … 240x380=91200 (!) Logistic regression or ordinary regression:

logit(E(Y)) or E(Y) = b0+X1b1+…+X91200b91200

But there are only n=3000 observations (trials) … Instead, since regressors are independent, fit them one at a time:

logit(E(Y)) or E(Y) = b0+Xjbj

However the regressors (bubbles) are random with a simple known distribution, so turn the problem around and condition on Y: E(Xj) = c0+Ycj

Equivalent to conditional logistic regression (Cox, 1962) which gives exact inference for b1 conditional on sufficient statistics for b0

Cox also suggested using saddle-point approximations to improve accuracy of inference …

Interactions? logit(E(Y)) or E(Y)=b0+X1b1+…+X91200b91200+X1X2b1,2+ …

MS lesions and cortical thickness Idea: MS lesions interrupt neuronal signals, causing thinning in down-

stream cortex Data: n = 425 mild MS patients

0 10 20 30 40 50 60 70 801.5

2

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5.5

Ave

rage

cor

tical

thic

knes

s (m

m)

Total lesion volume (cc)

Correlation = -0.568, T = -14.20 (423 df)

MS lesions and cortical thickness at all pairs of points

Dominated by total lesions and average cortical thickness, so remove these effects as follows:

CT = cortical thickness, smoothed 20mm ACT = average cortical thickness LD = lesion density, smoothed 10mm TLV = total lesion volume

Find partial correlation(LD, CT-ACT) removing TLV via linear model: CT-ACT ~ 1 + TLV + LD test for LD

Repeat for all voxels in 3D, nodes in 2D ~1 billion correlations, so thresholding essential! Look for high negative correlations … Threshold: P=0.05, c=0.300, T=6.48

Choose a lower level, e.g. t=3.11 (P=0.001)

Find clusters i.e. connected components of excursion set

Measure cluster extent by resels

Distribution: fit a quadratic to the peak:

Distribution of maximum cluster extent: Bonferroni on N = #clusters ~ E(EC).

Cluster extent rather than peak height (Friston, 1994)

Z

s

t Peak

height

extent

D=1

LD (cluster) » cY

®k

LD (cluster)