Detecting sparse
connectivity: the 'bubbles'
task in the fMRI scanner
Keith Worsley, Math + Stats,
Arnaud Charil, Montreal Neurological
Institute, McGill
Philippe Schyns, Fraser Smith,
Psychology, Glasgow
Jonathan Taylor,
Stanford and Université de Montréal
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

E.g. Fearful (3000/4=750 trials):
1
Trial
4 + 5
+
2
+
3
+
+
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
Neutral
But are these features real or just noise?

Need statistics …
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:
Z~N(0,1)


Trial 1
2
3
4
5
6
7 …
750
1
0.5
0
1
1
Response
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
Comparison

Both depend on average correct bubbles,
rest is ~ constant
Z=(Average correct bubbles
-average incorrect bubbles)
/ pooled sd
4
2
0
-2
Proportion correct bubbles
= Average correct bubbles
/ (average all bubbles * 4)
0.75
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 …
Euler Characteristic = #blobs - #holes
Excursion set {Z > threshold} for neutral face
EC = 0
30
Euler Characteristic
20
0
-7
-11
13
14
9
1
0
Heuristic:
At high thresholds t,
the holes disappear,
EC ~ 1 or 0,
E(EC) ~ P(max Z > t).
Observed
Expected
10
0
-10
-20
-4
-3
-2
-1
0
Threshold
1
2
• Exact expression for
E(EC) for all thresholds,
• E(EC) ~ P(max Z > t)
3
4accurate.
is extremely
The details
If ¡Z (s)¢ » N(0; 1) is an isot ropic Gaussian random ¯eld, s 2 < 2 , wit h ¸ I 2£ 2 =
V @Z ,
µ
¶
@s
P max Z (s) ¸ t ¼ E(E C(S \ f s : Z (s) ¸ tg))
s2 S
Z
1
1
= E C(S)
e¡ z 2 =2 dz
(2¼) 1=2
t
Intrinsic volumes or
Minkowski functionals
¸ 1=2
+ 1 Perimet er(S)
e¡ t 2 =2
2
2¼
¸
+ Area(S)
te¡ t 2 =2 :
(2¼) 3=2
If Z (s) is whit e noise convolved wit h an isot ropic Gaussian ¯lt er of Full Widt h
at Half Maximum FWHM t hen ¸ = 4 log 2 .
FW HM
2
The brain mapping version
If Z (s) is whit e noise smoot hed wit h an isot ropic Gaussian ¯lt er of Full Widt h
at Half Maximum FWHM
µ
¶
P max Z (s) ¸ t ¼ E(E C(S \ f s : Z (s) ¸ tg))
s2 S
Z
1
1
= E C(S)
e¡ z 2 =2 dz
EC0(S)
(2¼) 1=2
Resels0(S)
t
Resels1(S)
Resels2(S)
Perimet er(S) (4 log 2) 1=2
e¡
FWHM
2¼
Area(S) 4 log 2
+
te¡ t 2 =2 :
FWHM 2 (2¼) 3=2
+
Resels (Resolution elements)
1
2
t 2 =2
EC1(S)
EC2(S)
EC densities
Math version
If T (s) = f (Z 1 (s); : : : ; Z n (s)) is a funct ion³ of i.i.d.
Gaussian random ¯elds
´
Z i (s) » Z (s) » N(0; 1), s 2 < D , wit h V @Z ( s) = ¤ D £ D (s), t hen replace
@s
resels by Lipschitz-K illing curvature L d (S; ¤ ):
µ
¶
P max T (s) ¸ t
s2 S
XD
¼ E(E C(S \ f s : T (s) ¸ tg)) =
L d (S; ¤ )½d (t);
d= 0
where ½d (t) is t he same EC density for t he isot ropic case wit h ¤ (s) = I D £ D
(Taylor & Adler, 2003). Bot h Lipschit z-K illing curvat ure L d (S; ¤ ) and EC
density ½d (t) are de¯ned implicit ly as coe± cient s of a power series expansion of
t he volume of a t ube as a funct ion of it s radius. In t he case of Lipschit z-K illing
curvat ure, t he t ube is about t he search region S in t he Riemannian met ric
induced by ¤ (s); in t he case of t he EC densit ies, t he t ube is about t he reject ion
region f ¡ 1 ([t; 1 )) and volume is replaced by Gaussian probability. Fort unat ely
t here are simple ways of est imat ing Lipschit z-K illing curvat ure from sample dat a
(Taylor & Worsley, 2007), and simple ways of calculat ing EC densit ies.
The details …
2
Tube(S,r)
r
S
3
A
B
6
Λ is big
TubeΛ(S,r)
S
Λ is small
r
ν
2
U(s1)
S
S
s1
Tube
s2
s3
Tube
U(s2)
U(s3)
Z2
R
r
Tube(R,r)
Z1
N2(0,I)
Tube(R,r)
R
t-r
z
t
z1
RR
Tube(R,r)
r
z2
z3
Summary
Random field theory results

For searching in D (=2) dimensions, P-value of
max Z is (Adler, 1981; W, 1995):

P(max Z > z)



Resels (=Lipschitz-Killing curvature/c) is


Image area / (bubble FWHM)2 = 146.2 (unitless)
Euler characteristic density(×c) is


~ E( Euler characteristic of thresholded set )
= Resels × Euler characteristic density (+ boundary terms)
(4 log(2))D/2 zD-1 exp(-z2/2) / (2π)(D+1)/2
See forthcoming book Adler, Taylor (2007)
Results, corrected for search

Thresholded at 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
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 …
Discussion: 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)
The threshold based on Gaussian RFT can be improved
using new non-Gaussian RFT based on saddle-point
approximations (Chamandy et al., 2006)
 Model the bubbles as a smoothed Poisson point
process
 The improved thresholds are slightly lower, so more
activation is detected
Only keep 5D local maxima

Z(pixel, voxel) > Z(pixel, 6 neighbours of voxel)
> Z(4 neighbours of pixel, voxel)
Discussion: modeling







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
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
Charil et al,
NeuroImage (2007)
MS lesions and cortical
thickness at all pairs of points


Dominated by total lesions and average cortical
thickness, so remove these effects
Cortical thickness CT, smoothed 20mm



Lesion density LD, smoothed 10mm
Find partial correlation(lesion density, cortical
thickness) removing total lesion volume


linear model: CT-av(CT) ~ 1 + TLV + LD, test for LD
Repeat of all voxels in 3D, nodes in 2D


Subtract average cortical thickness
~1 billion correlations, so thresholding essential!
Look for high negative correlations …
Thresholding? Cross
correlation random field

Correlation between 2 fields at 2 different
locations, searched over all pairs of locations


one in R (D dimensions), one in S (E dimensions)
sample size n
Cao & Worsley, Annals of Applied Probability (1999)

MS lesion data: P=0.05, c=0.300, T=6.46
Cluster extent rather than
peak height (Friston, 1994)



fit a quadratic
Y
to
the
peak:
L D (clust er) » c
®
k


Peak
height

Choose a lower level, e.g. t=3.11 (P=0.001)
Find clusters i.e. connected components of excursion
set
Z
D=1
Measure cluster
LD
extent
extent by
t
Distribution:
Distribution of maximum cluster extent:

Bonferroni on N = #clusters ~ E(EC).
s
Cao, Advances
in Applied
Probability (1999)
How do you choose the threshold
t for defining clusters?

If signal is focal i.e. ~FWHM of noise



If signal is broad i.e. >>FWHM of noise





Choose a high threshold
i.e. peak height is better
Choose a low threshold
i.e. cluster extent is better
Conclusion: cluster extent is better for detecting
broad signals
Alternative: smooth data with filter that matches
signal (Matched Filter Theorem)… try range of filter
widths … scale space search … correct using
random field theory … a lot of work …
Cluster extent is easier!
Thresholding? Cross
correlation random field

Correlation between 2 fields at 2 different
locations, searched over all pairs of locations

one in R (D dimensions), one in S (E dimensions)
µ
¶
XD XE
P max Correlat ion > c ¼
L d (R) L e (S) ½d;e (c)
R ;S
½d;e (c) =
d= 0 e= 0
(d ¡ 1)!e!2n ¡
¼d +2 e + 1
d¡ e¡ 2
X
X
(¡ 1) k cd+ e¡
1¡ 2k (1 ¡
c2 )
n ¡ d ¡ e¡ 1
2
+k
k
i ;j
¡ ( n ¡ d + i )¡ ( n ¡ e + j )
2
2
i !j !(k ¡ i ¡ j )!(n ¡ 1 ¡ d ¡ e + i + j + k)!(d ¡ 1 ¡ k ¡ i + j )!(e ¡ k ¡ j + i )!
Cao & Worsley, Annals of Applied Probability (1999)

MS lesion data: P=0.05, c=0.300, T=6.46
0.1
correlation
0
5
x 10
2.5
2
-0.1
1.5
-0.2
1
-0.3
threshold
-0.4
-0.5
0
50
100
150
Different hemisphere
0.1
5
x 10
2.5
0
correlation
Histogram Same hemisphere
-0.1
2
-0.2
1.5
-0.3
1
0.5
-0.4
0
-0.5
0
threshold
50
100
150
0.5
0
‘Conditional’ histogram: scaled to same max at each distance
0.1
1
-0.1
0.6
-0.2
0.4
-0.3
-0.4
-0.5
0
threshold
50
100 150
distance (mm)
1
0
0.8
correlation
correlation
0
0.1
0.8
-0.1
0.6
-0.2
0.4
-0.3
0.2
-0.4
0
-0.5
0
threshold
50
100 150
distance (mm)
0.2
0
Science (2004)
fMRI activation detected by correlation
between subjects at the same voxel
The average nonselective time course across all activated regions
obtained during the first 10 min of the movie for all five subjects.
Red line represents the across subject average time course.
There is a striking degree of synchronization among different
individuals watching the same movie.
Voxel-by-voxel intersubject correlation between the source subject
(ZO) and the target subject (SN). Correlation maps are shown on
unfolded left and right hemispheres (LH and RH, respectively).
Color indicates the significance level of the intersubject correlation
in each voxel. Black dotted lines denote borders of retinotopic
visual areas V1, V2, V3, VP, V3A, V4/V8, and estimated border of
auditory cortex (A1).The face-, object-, and building-related
borders (red, blue, and green rings, respectively) are also
superimposed on the map. Note the substantial extent of
intersubject correlations and the extension of the correlations
beyond visual and auditory cortices.
What are the subjects watching
during high activation? Faces …
… buildings …
… hands
Thresholding? Homologous
correlation random field

Correlation between 2 equally smooth fields at the same location,
searched over all locations in R (in D dimensions)
Cao & Worsley, Annals of
Applied Probability (1999)

P-values are larger than for the usual correlation field (correlation
between a field and a scalar)
 E.g. resels=1000, df=100, threshold=5, usual P=0.051,
homologous P=0.139