Detecting Sparse Connectivity:
MS Lesions, Cortical
Thickness, and the ‘Bubbles’
Task in an fMRI Experiment
Keith Worsley, Nicholas Chamandy, McGill
Jonathan Taylor, Stanford and Université de Montréal
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 …
Euler Characteristic Heuristic
Euler characteristic (EC) = #blobs - #holes (in 2D)
Excursion set Xt = {s: Z(s) ≥ t}, e.g. for neutral face:
EC = 0
30
20
0
-7
-11
13
14
9
0
Heuristic:
At high thresholds t,
the holes disappear,
EC ~ 1 or 0,
E(EC) ~ P(max Z ≥ t).
Observed
Expected
10
EC(Xt)
1
0
-10
-20
-4
-3
-2
-1
0
1
Threshold, t
2
• Exact expression for
E(EC) for all thresholds,
• E(EC) ~ P(max Z ≥ t) is
3
4
extremely
accurate.
The»result
2 <2 ,
If Z(s) ¡ N(0;
¢ 1) is an isotropic Gaussian random ¯eld, s
with V @Z = ¸2 I2£2 ,
@s
µ
¶
P max Z(s) ¸ t ¼ E(EC(S \ fs : Z(s) ¸ tg))
s2S
Z 1
1
£
L (S)
= EC(S)
e¡z2 =2 dz
0
(2¼)1=2
t
L (S)
£ 1 e¡t2 =2
1
+
¸
Perimeter(S)
Lipschitz-Killing
1
2
2¼
curvatures of S
1
L (S)
¡t2 =2
2 Area(S) £
(=Resels(S)×c)
+
¸
te
2
(2¼)3=2
If Z(s) is white noise convolved
with an isotropic Gaussian
Z(s)
¯lter of Full Width at Half
Maximum
FWHM then
p
¸ = 4 log 2 :
FWHM
½0 (Z ¸ t)
½1 (Z ¸ t)
½2 (Z ¸ t)
EC densities
of Z above t
white noise
=
filter
*
FWHM
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 (2007): Z=↑ (P=0.05)
Bonferroni: Z=4.87 (P=0.05) – nothing
The»result
If Z(s) N(0; 1) ¡is an¢ isotropic Gaussian random ¯eld, s 2 <2 ,
with ¸2 I2£2 = V @Z ,
@s
µ
¶
P max Z(s) ¸ t ¼ E(EC(S \ fs : Z(s) ¸ tg))
s2S
Z 1
1
£
L (S)
= EC(S)
e¡z2 =2 dz
0
(2¼)1=2
t
L (S)
£ 1 e¡t2 =2
1
+
¸
Perimeter(S)
Lipschitz-Killing
1
2
2¼
curvatures of S
1
L (S)
¡t2 =2
2 Area(S) £
(=Resels(S)×c)
+
¸
te
2
(2¼)3=2
If Z(s) is white noise convolved
with an isotropic Gaussian
Z(s)
¯lter of Full Width at Half
Maximum
FWHM then
p
¸ = 4 log 2 :
FWHM
½0 (Z ¸ t)
½1 (Z ¸ t)
½2 (Z ¸ t)
EC densities
of Z above t
white noise
=
filter
*
FWHM
Theorem (1981, 1995)
F Let T (s), s 2 S ½ <D , be a smooth isotropic random ¯eld.
F Let X = fs : T (s) ¸ tg be the the excursion set inside S.
t
F Then
E(EC(S \ Xt )) =
X
D
L (S)½ (T ¸ t):
d
d
d=0
F Now suppose that T (s) = f (Z(s)) is a function of independent and identically distributed Gaussian random ¯elds Z(s) = (Z1 (s); : : : ; Zn (s)), each
with Zi (s) » N(0; 1) and V( @Zi ) = ¸2 ID£D .
@s
F Example: T (s) is a Â2 , T, F, Hotelling0 s T 2 , . . .
F Let Rt = fz : f (z) ¸ tg be the rejection region of T .
Â
¹=
max Z1 cos µ + Z2 sin µ
0·µ·¼=2
Example: chi-bar random field
Z1~N(0,1)
Z2~N(0,1)
s2
3
2
1
0
-1
-2
Excursion sets,
Xt = fs : Â
¹ ¸ tg s1
-3
Threshold
t 4
Rejection regions,
Z2
Rt = fZ : Â
¹ ¸ tg
2
3
Search
Region,
S
2
1
0
Z1
-2
-2
0
2
E(EC(S \ Xt )) =
X
D
L (S)½ (R )
d
d
t
d=0
Lipschitz-Killing curvature Ld (S)
EC density ½d (Rt )
Steiner-Weyl Tube Formula (1930)
Morse Theory method (1981, 1995)
µ
¶
• Put a tube of radius r about
@Z the search
¸ = Sd
region λS
@s
• EC has a point-set representation:
14
10
Tube(λS,r)
8
λS
6
4
2
2
4
6
8 10 12 14
• Find volume, expand as a power series
in r, pull off coefficients:
jTube(¸S; r)j =
X
D
d=0
1fT ¸tg 1f@T =@s=0g
s
r
12
EC(S \ Xt ) =
X
¼d
¡(d=2 + 1)
L
(S)r d
D ¡d
µ
¶
£ sign ¡ @ 2 T
+ boundary
0
@s@s
µ
µ
¶
2
1
@ T
E 1f ¸ g det ¡
½D (Rt ) =
T t
¸D
@s@s0
¯
¶ µ
¶
¯ @T
@T
¯
P
=0
¯ @s = 0
@s
µ ¡ random
¶ field:
• For a Gaussian
½d (Z ¸ t) =
p1 @
2¼ @t
d
P(Z ¸ t)
E(EC(S \ Xt )) =
Beautiful symmetry:
X
D
L (S)½ (R )
d
d
t
Adler & Taylor
(2007), Ann. Math,
(submitted)
d=0
Lipschitz-Killing curvature Ld (S)
Steiner-Weyl Tube Formula (1930)
EC density ½d (Rt )
Taylor Gaussian Tube Formula (2003)
µ
¶
• Put a tube of radius r about
@Z the search region λS and rejection region Rt:
¸ = Sd
@s
Z2~N(0,1)
14
r
12
10
Rt
Tube(λS,r)
8
Tube(Rt,r)
r
λS
6
t-r
t
Z1~N(0,1)
4
2
2
4
6
8 10 12 14
• Find volume or probability, expand as a power series in r, pull off1coefficients:
jTube(¸S; r)j =
X
D
d=0
¼d
L
P(Tube(Rt ; r)) =
¡d (S)r d
D
¡(d=2 + 1)
X (2¼)d=2
d!
d=0
½d (Rt )rd
Lipschitz-Killing
curvature Ld (S)
of a triangle
r
Tube(λS,r)
λS
¸ = Sd
µ
@Z
@s
¶
Steiner-Weyl Volume of Tubes Formula (1930)
Area(Tube(¸S; r)) =
X
D
¼ d=2
L
¡d (S)r d
D
¡(d=2 + 1)
d=0
= L2 (S) + 2L1 (S)r + ¼ L0 (S)r2
= Area(¸S) + Perimeter(¸S)r + EC(¸S)¼r2
L (S) = EC(¸S)
0
L (S) = 1 Perimeter(¸S)
1
2
L (S) = Area(¸S)
2
Lipschitz-Killing curvatures are just “intrinisic volumes” or “Minkowski functionals”
in the (Riemannian) metric of the variance of the derivative of the process
Lipschitz-Killing curvature Ld (S) of any set
S
S
S
Edge length × λ
12
10
8
6
4
2
.
.. . .
.
. . .
.. . .
.. . .
.
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.. . .
. . .
... .
.
4
..
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6
..
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8
..
. .
. ...
. ..
. . .
. .
. . .....
. . .
. ....
..
..
10
of triangles
L (Lipschitz-Killing
²) = 1, L (¡) curvature
L (N
=
1,
)=1
0
0
0
L (¡) = Edge length, L (N) = 1 Perimeter
1
2
L1 (N) = Area
2
P Lcurvature
P L
Lipschitz-Killing
union
L
² ¡ Pof L
¡ of triangles
N
(S) = P² 0 ( )
¡ 0( ) +
P
L (S) =
L (¡) ¡
L (N)
¡
N 1
L1 (S) = P L 1(N)
2
N 2
0
N
0
( )
Non-isotropic data
¸(s) = Sd
Z~N(0,1)
s2
3
µ
@Z
@s
¶
2
1
0.14
0.12
0
-1
-2
Edge length × λ(s)
12
10
8
6
4
2
..
.
.
.
. .
. .
.
.. .
.
.. .
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.. .
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...
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4
6
..
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8
.
..
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. ...
. ..
. . .
. .
. . .....
. . .
. ....
...
10
s1
0.1
0.08
0.06
-3
of triangles
L (Lipschitz-Killing
²) = 1, L (¡) curvature
L (N
=
1,
)=1
0
0
0
L (¡) = Edge length, L (N) = 1 Perimeter
1
2
L1 (N) = Area
2
P Lcurvature
P L
Lipschitz-Killing
union
L
² ¡ Pof L
¡ of triangles
N
(S) = P² 0 ( )
¡ 0( ) +
P
L (S) =
L (¡) ¡
L (N)
¡
N 1
L1 (S) = P L 1(N)
2
N 2
0
N
0
( )
Estimating Lipschitz-Killing curvature Ld (S)
We need independent & identically distributed random fields
e.g. residuals from a linear model
Z1
Z2
Z3
Z4
Replace coordinates of
the triangles 2 <2 by
normalised residuals
Z 2<
n;
jjZ jj
Z5
Z7
Z8
Z9 … Zn
of triangles
L (Lipschitz-Killing
²) = 1, L (¡) curvature
L (N
=
1,
)=1
0
0
0
L (¡) = Edge length, L (N) = 1 Perimeter
1
2
L1 (N) = Area
2
P Lcurvature
P L
Lipschitz-Killing
union
L
² ¡ Pof L
¡ of triangles
N
(S) = P² 0 ( )
¡ 0( ) +
P
L (S) =
L (¡) ¡
L (N)
¡
N 1
L1 (S) = P L 1(N)
2
N 2
0
Z = (Z1 ; : : : ; Zn ):
Z6
N
0
( )
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? Cross correlation random field
Correlation between 2 fields at 2 different locations,
searchedµ
over all pairs of locations,
¶ one in S, one in T:

P
max C(s; t) ¸ c
s2S;t2T
=
¼ E(EC fs 2 S; t 2 T : C(s; t) ¸ cg)
dim(S)
X dim(T
X)
i=0
2n¡2¡h (i ¡ 1)!j!
¸
½ij (C c) =
¼h=2+1
L (S)L (T )½ (C ¸ c)
i
j
ij
j=0
b(hX
¡1)=2c
(¡1)k ch¡1¡2k (1 ¡ c2 )(n¡1¡h)=2+k
k=0
X
k X
k
l=0 m=0
¡( n¡i + l)¡( n¡j + m)
2
2
¡
¡
¡
¡
l!m!(k l m)!(n 1 h + l + m + k)!(i ¡ 1 ¡ k ¡ l + m)!(j ¡ k ¡ m + l)!

Bubbles data: P=0.05, n=3000, c=0.113, T=6.22
Cao & Worsley,
Annals of Applied
Probability (1999)
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 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

L (cluster)
Measure cluster
extent
by resels D
Z
D=1
extent

L (cluster) » c
D

t
®
k
Distribution of maximum cluster extent:
 Bonferroni on N = #clusters ~ E(EC).
Peak
height
Distribution:
 fit a quadratic to the
peak:
Y
s
Cao and Worsley,
Advances in Applied
Probability (1999)