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Detecting connectivity between images: MS lesions, cortical thickness, and the 'bubbles' task in an fMRI experiment. Keith Worsley, Math + Stats, Arnaud Charil, Montreal Neurological Institute, McGill Philippe Schyns, Fraser Smith, Psychology, Glasgow Jonathan Taylor , - PowerPoint PPT Presentation
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Detecting connectivity between images: MS lesions, cortical thickness, and the 'bubbles'
task in an fMRI experiment
Keith Worsley, Math + Stats, Arnaud Charil, Montreal Neurological
Institute, McGillPhilippe Schyns, Fraser Smith,
Psychology, GlasgowJonathan 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
Subject is asked the expression: “Neutral”
Response: Incorrect
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Sad75 random
bubble centresSmoothed by a
Gaussian ‘bubble’What the
subject sees
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)
0
0.5
1
100
200
300
50100
150
200250
0.65
0.7
0.75
0
0.5
1
Bubbles analysis E.g. Fearful (3000/4=750 trials):
Trial1 + 2 + 3 + 4 + 5 + 6 + 7 + … + 750 = Sum
Correcttrials
Proportion of correct bubbles=(sum correct bubbles)
/(sum all bubbles)
Thresholded atproportion of
correct trials=0.68,scaled to [0,1]
Use thisas a bubblemask
Results
Mask average face
But are these features real or just noise? Need statistics …
Happy Sad Fearful Neutral
0.65
0.7
0.75
-2
0
2
4
0
0.5
1
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:
Response0 1 1 0 1 1 1 … 1
Trial 1 2 3 4 5 6 7 … 750Z~N(0,1)statistic
0.65
0.7
0.75
-2
0
2
4
Both depend on average correct bubbles, rest is ~ constant
Comparison
Proportion correct bubbles= Average correct bubbles / (average all bubbles * 4)
Z=(Average correct bubbles -average incorrect bubbles)
/ pooled sd
1.64
2.13
2.62
3.11
3.6
4.09
4.58
Results
Thresholded at Z=1.64 (P=0.05)
Multiple comparisons correction? Need random field theory …
Average faceHappy Sad Fearful Neutral
Z~N(0,1)statistic
-4 -3 -2 -1 0 1 2 3 4-20
-10
0
10
20
30
Threshold
Eul
er C
hara
cter
istic
Observed
Expected
Euler Characteristic = #blobs - #holesExcursion set {Z > threshold} for neutral face
Heuristic:At high thresholds t,the holes disappear,
EC ~ 1 or 0, E(EC) ~ P(max Z > t).
• Exact expression for E(EC) for all thresholds,• E(EC) ~ P(max Z > t) is extremely accurate.
EC = 0 0 -7 -11 13 14 9 1 0
The details …
2
S
Tube(S,r)r
3
A B
6
S
TubeΛ(S,r)
r
Λ is small
Λ is big
S S s1
s2 s3
U(s1)
U(s2)U(s3)
Tube Tube
2 ν
R
Tube(R,r)r
N2(0,I)Z1
Z2
Tube(R,r)
t-r t
z
Tube(R,r)
R
z1
z2
z3
R
R
r
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)
~ E( Euler characteristic of thresholded set ) = Resels × Euler characteristic density (+ boundary)
Resels (=Lipschitz-Killing curvature/c) is Image area / (bubble FWHM)2 = 146.2
Euler characteristic density(×c) is (4 log(2))D/2 zD-1 exp(-z2/2) / (2π)(D+1)/2
See forthcoming book Adler, Taylor (2007)
3.92
4.03
4.14
4.25
4.36
4.47
4.58
Results, corrected for search
Thresholded at Z=3.92 (P=0.05)Average face
Happy Sad Fearful Neutral
Z~N(0,1)statistic
0
10000
0
0.5
1
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 …
Trial1 2 3 4 5 6 7 … 3000
fMRI
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 Lesion density, smoothed 10mm Cortical thickness, smoothed 20mm Find connectivity i.e. find voxels in 3D, nodes
in 2D with high correlation(lesion density, cortical thickness)
Look for high negative correlations …
0 10 20 30 40 50 60 70 80
1.5
2
2.5
3
3.5
4
4.5
5
5.5
Average lesion volume
Ave
rag
e co
rtic
al t
hic
kne
ssn=425 subjects, correlation = -0.568
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
MS lesion data: P=0.05, c=0.325Cao & Worsley, Annals of Applied Probability (1999)
Normalization
LD=lesion density, CT=cortical thickness Simple correlation:
Cor( LD, CT )
Subtracting global mean thickness: Cor( LD, CT – avsurf(CT) )
And removing overall lesion effect: Cor( LD – avWM(LD), CT – avsurf(CT) )
0
0.5
1
1.5
2
2.5x 10
5
corr
elat
ion
Same hemisphere
0 50 100 150-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0
0.2
0.4
0.6
0.8
1
distance (mm)
corr
elat
ion
0 50 100 150-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0
0.5
1
1.5
2
2.5
x 105
corr
elat
ion
Different hemisphere
0 50 100 150-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0
0.2
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0.6
0.8
1
distance (mm)
corr
elat
ion
0 50 100 150-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
threshold
thresholdthreshold
threshold
Histogram
‘Conditional’ histogram: scaled to same max at each distance
Scien
ce (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)
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
Cao & Worsley, Annals of Applied Probability (1999)
Detecting Connectivity between Images: the
'Bubbles' Task in fMRI
Keith Worsley, McGill
Phillipe Schyns, Fraser Smith, Glasgow
Subject is shown one of 40 faces chosen at random …
Happy
Sad
Fearful
Neutral
… but face is only revealed through random ‘bubbles’ E.g. first trial: “Sad” expression:
Subject is asked the expression: “Neutral”
Response: Incorrect=0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Sad75 random
bubble centresSmoothed by a
Gaussian ‘bubble’What the
subject sees