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. Three examples of spatial point data
Multiple Sclerosis
(MS) lesions
Galaxies
‘Bubbles’
I hope to show the ‘connections’ …

3. Astrophysics
우리의 은하

4. Sloan Digital Sky Survey, data release 6, Aug. ‘07

5. What is ‘bubbles’?

6. Nature (2005)

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

8. … 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

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

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

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

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

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

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

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

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

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

18. 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)

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

20. 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 …

21. 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

22. 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

23. 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

24. 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

25. 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 34 8
22.7 6
15.2 4 2
10.2 -2
6.8
-60
-40
-20 20 40 60
w = FWHM (mm, on log scale)
One 15mm signal 34 8
22.7 6
15.2 4 2
10.2 -2
6.8
-60
-40
-20 20 40 60
Z(s,w)
s (mm)
15mm signal is best detected with a 15mm smoothing filter

26. Matched Filter Theorem (= Gauss-Markov Theorem):
“to best detect signal + white noise,
filter should match signal”
10mm and 23mm signals 34 8
22.7 6 4
15.2 2
10.2 -2
6.8
-60
-40
-20 20 40 60
w = FWHM (mm, on log scale)
Two 10mm signals 20mm apart 34 8
22.7 6
15.2 4 2
10.2 -2
6.8
-60
-40
-20 20 40 60
Z(s,w)
s (mm)
But if the signals are too close together they are
detected as a single signal half way between them

27. Scale space can even separate two signals at the same location!
8mm and 150mm signals at the same location 10 5
-60
-40
-20 20 40 60
170 20 76 15
w = FWHM (mm, on log scale) 34 10
15.2 5
6.8
-60
-40
-20 20 40 60
Z(s,w)
s (mm)

28. 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

29. Random field theory for scale-spaceg b t w L z K u v ( ) ? ; 1 2 d x m y = + 4 3

30. Rotation space: Try all rotated elliptical filters
Unsmoothed fMRI data:
T stat for visual stimulus
Threshold
Z=5.25 (P=0.05)
Maximum filter

31. 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

32. Theorem (1981, 1995) L e t T ( s ) , 2 S ½ < b a m o h i r p c n dl . X = f : g x u E C \ N w Z - y G 1 ; V @ I R z j
Example: the chi-bar random field, a special case of a random field of test statistics for the magnitude of an fMRI response in the presence of unknown delay of the hemodynamic response function.

33. 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

34. 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 !

35. 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

36. 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)

37. 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)

38. 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

39. 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:
Find distribution for a single cluster
Bonferroni on N = #clusters ~ E(EC). Z
D=1
extent t
Peak
height s
Cao and Worsley, Advances in Applied Probability (1999)

40. Three examples of spatial point data
Multiple Sclerosis
(MS) lesions
Galaxies
‘Bubbles’
I hope I have shown the connections ...
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