1. 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

2. What is ‘bubbles’?

3. Nature (2005)

4. Subject is shown one of 40 faces chosen at random …
Happy
Sad
Fearful
Neutral

5. … but face is only revealed through random ‘bubbles’
First trial: “Sad” expression
Subject is asked the expression: “Neutral”
Response: Incorrect
75 random bubble centres
Smoothed by a
Gaussian ‘bubble’
What the
subject sees
Sad

6. Your turn …
Trial 2
Subject response:
“Fearful”
CORRECT

7. Your turn …
Trial 3
Subject response:
“Happy”
INCORRECT
(Fearful)

8. Your turn …
Trial 4
Subject response:
“Happy”
CORRECT

9. Your turn …
Trial 5
Subject response:
“Fearful”
CORRECT

10. Your turn …
Trial 6
Subject response:
“Sad”
CORRECT

11. Your turn …
Trial 7
Subject response:
“Happy”
CORRECT

12. Your turn …
Trial 8
Subject response:
“Neutral”
CORRECT

13. Your turn …
Trial 9
Subject response:
“Happy”
CORRECT

14. Your turn …
Trial 3000
Subject response:
“Happy”
INCORRECT
(Fearful)

15. E.g. Fearful (3000/4=750 trials):
Bubbles analysis
E.g. Fearful (3000/4=750 trials):
Trial
… + 750
= Sum
Correct
trials
Thresholded at
proportion of
correct trials=0.68,
scaled to [0,1]
Use this
as a
bubble
mask
Proportion of correct bubbles
=(sum correct bubbles)
/(sum all bubbles)

16. Happy Sad Fearful Neutral
Results
Mask average face
But are these features real or just noise?
Need statistics …
Happy Sad Fearful Neutral

17. Very similar to the proportion of correct bubbles:
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:
Z~N(0,1)
statistic
Trial …
Response …

18. Happy Sad Fearful Neutral
Results
Thresholded at Z=1.64 (P=0.05)
Multiple comparisons correction?
Need random field theory …
Z~N(0,1)
statistic
Average face
Happy Sad Fearful Neutral

19. Euler Characteristic Heuristic
Euler characteristic (EC) = #blobs - #holes (in 2D)
Excursion set Xt = {s: Z(s) ≥ t}, e.g. for neutral face:
EC = 30
Heuristic:
At high thresholds t,
the holes disappear,
EC ~ 1 or 0,
E(EC) ~ P(max Z ≥ t).
Observed
Expected 20 10
EC(Xt)
-10
Exact expression for E(EC) for all thresholds,
E(EC) ~ P(max Z ≥ t) is extremely accurate.
-20 -4 -3 -2 -1 1 2 3 4
Threshold, t

20. EC densities of Z above t
The result I f Z ( s ) N ; 1 i a n o t r p c G u d m e l , 2
< w h V @ = P x S E C \ : g z + A 3 v F W H M 4 ( Z t ) 1 2 L ( S ) 1 2
Lipschitz-Killing
curvatures of S
(=Resels(S)×c)
EC densities of Z above t
Z(s)
white noise
filter = *
FWHM

21. Results, corrected for search
Random field theory threshold: Z=3.92 (P=0.05)
Saddle-point approx (2007): Z=↑ (P=0.05)
Bonferroni: Z=4.87 (P=0.05) – nothing
Z~N(0,1)
statistic
Average face
Happy Sad Fearful Neutral

22. EC densities of Z above t
The result I f Z ( s ) N ; 1 i a n o t r p c G u d m e l , 2
< w h = V @ P x S E C \ : g z + A 3 v F W H M 4 ( Z t ) 1 2 L ( S ) 1 2
Lipschitz-Killing
curvatures of S
(=Resels(S)×c)
EC densities of Z above t
Z(s)
white noise
filter = *
FWHM

23. Theorem (1981, 1995) F L e t T ( s ) , 2 S ½ < b a m o h i r p c nD b a m o h i r p c n d l . X = f : g x u E C \ N w Z - y G 1 ; V @ I Â H R z j

24. Example: chi-bar random field = m a x 2 Z 1 c o s + i n
Z1~N(0,1)
Z2~N(0,1) s2 s1
Excursion sets,
Rejection regions, X t = f s : Â g R t = f Z : Â g
Threshold t Z2
Search
Region, S Z1

25. L i p s c h t z - K l n g u r v a e ( S ) E C d e n s i t y ½ ( R ) E\ X t ) = D d L R L i p s c h t z - K l n g u r v a e d ( S ) E C d e n s i t y ( R )
Steiner-Weyl Tube Formula (1930)
Morse Theory method (1981, 1995)
Put a tube of radius r about the search
region λS
EC has a point-set representation: = S d @ Z s E C ( S \ X t ) = s 1 f T g @ i n 2 + b o u d a r y D R e P r
Tube(λS,r) λS
Find volume, expand as a power series
in r, pull off coefficients:
For a Gaussian random field: j T u b e ( S ; r ) = D X d 2 + 1 L d ( Z t ) = 1 p 2 @ P

26. Beautiful symmetry: L i p s c h t z - K l n g u r v a e ( S ) E C d e\ X t ) = D d L R
Adler & Taylor
(2007), Ann. Math,
(submitted)
Beautiful symmetry: L i p s c h t z - K l n g u r v a e d ( S ) E C d e n s i t y ( R )
Steiner-Weyl Tube Formula (1930)
Taylor Gaussian Tube Formula (2003)
Put a tube of radius r about the search region λS and rejection region Rt: = S d @ Z s
Z2~N(0,1) Rt r
Tube(λS,r)
Tube(Rt,r) r λS
Z1~N(0,1)
t-r t
Find volume or probability, expand as a power series in r, pull off coefficients: j T u b e ( S ; r ) = D X d 2 + 1 L P ( T u b e R t ; r ) = 1 X d 2 !

27. L i p s c h t z - K l n g u r v a e ( S ) o f d A r e a ( T u b ¸ S ;
Tube(λS,r) r λS = S d @ Z s
Steiner-Weyl Volume of Tubes Formula (1930) A r e a ( T u b S ; ) = D X d 2 + 1 L P i m t E C
Lipschitz-Killing curvatures are just “intrinisic volumes” or “Minkowski functionals”
in the (Riemannian) metric of the variance of the derivative of the process

28. L i p s c h t z - K l n g u r v a e ( S ) o f y S S S d L ( ² ) = 1 ,
Edge length × λ
Lipschitz-Killing curvature of triangles L ( ) = 1 , N E d g e l n t h 2 P r i m A a
Lipschitz-Killing curvature of union of triangles L ( S ) = P + N 1 2

29. Non-isotropic data L ( ² ) = 1 , ¡ N E d g e l n t h P r i m A a L ( S
Z~N(0,1)
Z~N(0,1) s2 ( s ) = S d @ Z s1
Edge length × λ(s)
Lipschitz-Killing curvature of triangles L ( ) = 1 , N E d g e l n t h 2 P r i m A a
Lipschitz-Killing curvature of union of triangles L ( S ) = P + N 1 2

30. E s t i m a n g L p c h z - K l u r v e ( S ) d R e p l a c o r d i n1 2 3 4 5 6 7 8 9 n E s t i m a n g L p c h z - K l u r v e d ( S )
We need independent & identically distributed random fields
e.g. residuals from a linear model …
Lipschitz-Killing curvature of triangles R e p l a c o r d i n t s f h g 2
< b y m u Z j ; = ( 1 : ) L ( ) = 1 , N E d g e l n t h 2 P r i m A a
Lipschitz-Killing curvature of union of triangles L ( S ) = P + N 1 2

31. 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 …
Trial …
fMRI

32. Thresholding? Cross correlation random field
Correlation between 2 fields at 2 different locations,
searched over all pairs of locations, one in S, one in T:
Bubbles data: P=0.05, n=3000, c=0.113, T=6.22 P m a x s 2 S ; t T C ( ) c E f : g = d i X j L n h 1 ! + b k l
Cao & Worsley, Annals of Applied Probability (1999)

33. 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 5
4.5 4
Average cortical thickness (mm)
3.5 3
2.5
Correlation = ,
T = (423 df) 2
Charil et al,
NeuroImage (2007)
1.5 10 20 30 40 50 60 70 80
Total lesion volume (cc)

34. 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

35. 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
by resels
Distribution:
fit a quadratic to the
peak:
Distribution of maximum cluster extent:
Bonferroni on N = #clusters ~ E(EC). Z
D=1 L D ( c l u s t e r )
extent t
Peak
height L D ( c l u s t e r ) Y Â k s
Cao and Worsley, Advances in Applied Probability (1999)