Oury’s course, lecture 2
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
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
Sad
75 random
Smoothed by a
bubble centres Gaussian ‘bubble’
What the
subject sees
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0


Subject is asked the expression:
Response:
“Neutral”
Incorrect
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)
Bubbles analysis

1
E.g. Fearful (3000/4=750 trials):
+
2
+
3
+
Trial
4 + 5
+
6
+
7 + … + 750
1
= Sum
300
0.5
200
0
100
250
200
150
100
50
Correct
trials
Proportion of correct bubbles
=(sum correct bubbles)
/(sum all bubbles)
0.75
Thresholded at
proportion of
0.7
correct trials=0.68,
0.65
scaled to [0,1]
1
Use this
as a
0.5
bubble
mask
0
Results

Mask average face
Happy

Sad
Fearful
But are these features real or just noise?
 Need statistics …
Neutral
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:


Trial 1
2
3
4
5
6
7 …
750
1
0.5
0
1
1
Response
0
1
Z~N(0,1)
statistic
4
2
0
-2
0
1
1 …
1
0.75

Very similar to the proportion of correct bubbles:
0.7
0.65
Results

Thresholded at Z=1.64 (P=0.05)
Happy
Average face
Sad
Fearful
Neutral
Z~N(0,1)
statistic
4.58
4.09
3.6
3.11
2.62
2.13
1.64

Multiple comparisons correction?
 Need random field theory …
Results, corrected for search

Random field theory threshold: Z=3.92 (P=0.05)
Happy
Average face
Sad
Fearful
Neutral
Z~N(0,1)
statistic
4.58
4.47
4.36
4.25
4.14
4.03
3.92


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
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)
Scale space, no signal
w = FWHM (mm, on log scale)
34
8
6
4
2
0
-2
22.7
15.2
10.2
6.8
-60
-40
34
-20
0
20
One 15mm signal
40
60
8
6
4
2
0
-2
22.7
15.2
10.2
6.8
-60
-40
-20
0
s (mm)
20
40
60
15mm signal is best detected with a 15mm smoothing filter
Z(s,w)
Matched Filter Theorem (= Gauss-Markov Theorem):
“to best detect signal + white noise,
filter should match signal”
10mm and 23mm signals
w = FWHM (mm, on log scale)
34
8
6
4
2
0
-2
22.7
15.2
10.2
6.8
-60
-40
34
-20
0
20
Two 10mm signals 20mm apart
40
60
8
6
4
2
0
-2
22.7
15.2
10.2
6.8
-60
-40
-20
0
20
40
60
s (mm)
But if the signals are too close together they are
detected as a single signal half way between them
Z(s,w)
Scale space can even separate
two signals at the same location!
8mm and 150mm signals at the same location
10
5
w = FWHM (mm, on log scale)
0
-60
170
-40
-20
0
20
40
60
20
76
15
34
10
15.2
6.8
5
-60
-40
-20
0
s (mm)
20
40
60
Z(s,w)
Bubbles task in fMRI scanner

Correlate bubbles with BOLD at every voxel:
Trial
1
2
3
4
5
6
7 …
3000
1
0.5
0
fMRI
10000
0

Calculate Z for each pair (bubble pixel, fMRI voxel)

a 5D “image” of Z statistics …
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 downstream cortex
Data: n = 425 mild MS patients
5.5
Average cortical thickness (mm)

5
4.5
4
3.5
3
2.5
Correlation = -0.568,
T = -14.20 (423 df)
2
1.5
0
10
20
30
40
50
Total lesion volume (cc)
60
70
80
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



Cluster extent rather than peak height
(Friston, 1994)

Choose a lower level, e.g. t=3.11 (P=0.001)

Find clusters i.e. connected components of excursion set

Measure cluster
extent
L D (clust
er)
by resels
Z
D=1
extent

L D (clust er) » c

t
®
k
Distribution of maximum cluster extent:
 Bonferroni on N = #clusters ~ E(EC).
Peak
height
Distribution:
 fit a quadratic to the
peak:
Y
s