Detecting sparse connectivity: the 'bubbles' task in the fMRI scanner - PowerPoint PPT Presentation


Title: Detecting sparse connectivity: the 'bubbles' task in the fMRI scanner


1
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

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
Bubbles analysis
  • E.g. Fearful (3000/4750 trials)

Trial 1 2 3 4
5 6 7 750
Sum
Correct trials
Thresholded at proportion of correct
trials0.68, scaled to 0,1
Use this as a bubble mask
Proportion of correct bubbles (sum correct
bubbles) /(sum all bubbles)
16
Results
  • Mask average face
  • But are these features real or just noise?
  • Need statistics

Happy Sad
Fearful Neutral
17
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

ZN(0,1) statistic
Trial 1 2 3 4
5 6 7 750
Response 0 1 1 0
1 1 1 1
18
Comparison
  • Both depend on average correct bubbles, rest is
    constant

Proportion correct bubbles Average correct
bubbles / (average all bubbles 4)
  • Z(Average correct bubbles
  • average incorrect bubbles)
  • / pooled sd

19
Results
  • Thresholded at Z1.64 (P0.05)
  • Multiple comparisons correction?
  • Need random field theory

ZN(0,1) statistic
Average face Happy Sad
Fearful Neutral
20
Euler Characteristic blobs - holes
Excursion set Z gt threshold for neutral face
EC 0 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 gt
t).
  • Exact expression for E(EC) for all thresholds,
  • E(EC) P(max Z gt t) is extremely accurate.

21
The details
Intrinsic volumes or Minkowski functionals
22
The brain mapping version
EC0(S)
Resels0(S)
Resels1(S)
EC1(S)
Resels2(S)
EC2(S)
Resels (Resolution elements)
EC densities
23
Math version
24
The details
25
2
Tube(S,r)
r
S
26

3

27

B
A
28

29
6
? is big
Tube?(S,r)
S
r
? is small
30
2
?
U(s1)
s1
S
Tube
S
Tube
s2
s3
U(s3)
U(s2)
31
Z2
R
r
Tube(R,r)
Z1
N2(0,I)
32

Tube(R,r)
R
z
t-r
t
z1
Tube(R,r)
r
R
R
z2
z3
33

34

35
Summary
36
(No Transcript)
37
Random field theory results
  • For searching in D (2) dimensions, P-value of
    max Z is (Adler, 1981 W, 1995)
  • P(max Z gt z)
  • E( Euler characteristic of thresholded set )
  • Resels Euler characteristic density (
    boundary terms)
  • Resels (Lipschitz-Killing curvature/c) is
  • Image area / (bubble FWHM)2 146.2 (unitless)
  • Euler characteristic density(c) is
  • (4 log(2))D/2 zD-1 exp(-z2/2) / (2p)(D1)/2
  • See forthcoming book Adler, Taylor (2007)

38
Results, corrected for search
  • Thresholded at Z3.92 (P0.05)

ZN(0,1) statistic
Average face Happy Sad
Fearful Neutral
39
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 1 2 3 4
5 6 7 3000
fMRI
40
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 P0.05, 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) gt Z(pixel, 6 neighbours of voxel)
  • gt Z(4 neighbours of
    pixel, voxel)

41
Discussion modeling
  • The random response is Y1 (correct) or 0
    (incorrect), or YfMRI
  • The regressors are Xjbubble mask at pixel j, j1
    240x38091200 (!)
  • Logistic regression or ordinary regression
  • logit(E(Y)) or E(Y) b0X1b1X91200b91200
  • But there are only n3000 observations (trials)
  • Instead, since regressors are independent, fit
    them one at a time
  • logit(E(Y)) or E(Y) b0Xjbj
  • However the regressors (bubbles) are random with
    a simple known distribution, so turn the problem
    around and condition on Y
  • E(Xj) c0Ycj
  • 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)b0X1b1X91200
    b91200X1X2b1,2

42
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 -0.568, T -14.20 (423 df)
2
Charil et al, NeuroImage (2007)
1.5
0
10
20
30
40
50
60
70
80
Total lesion volume (cc)
43
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
  • Subtract average cortical thickness
  • 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
  • 1 billion correlations, so thresholding
    essential!
  • Look for high negative correlations

44
Thresholding? Crosscorrelation 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 P0.05, c0.300, T6.46

Cao Worsley, Annals of Applied Probability
(1999)
45
Cluster extent rather than peak height (Friston,
1994)
  • Choose a lower level, e.g. t3.11 (P0.001)
  • Find clusters i.e. connected components of
    excursion set
  • Measure cluster
  • extent by
  • Distribution
  • fit a quadratic
  • to the peak
  • Distribution of maximum cluster extent
  • Bonferroni on N clusters E(EC).

Z
D1
extent
t
Peak height
s
Cao, Advances in Applied Probability (1999)
46
How do you choose the threshold t for defining
clusters?
  • If signal is focal i.e. FWHM of noise
  • Choose a high threshold
  • i.e. peak height is better
  • If signal is broad i.e. gtgtFWHM of noise
  • 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!

47
Thresholding? Crosscorrelation 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)
  • MS lesion data P0.05, c0.300, T6.46

Cao Worsley, Annals of Applied Probability
(1999)
48
Histogram
threshold
threshold
Conditional histogram scaled to same max at
each distance
threshold
threshold
49
Science (2004)
50
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.
51
What are the subjects watching during high
activation? Faces
52
buildings
53
hands
54
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. resels1000, df100, threshold5, usual
    P0.051, homologous P0.139

Cao Worsley, Annals of Applied Probability
(1999)
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Detecting sparse connectivity: the 'bubbles' task in the fMRI scanner

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Title: Detecting sparse connectivity: the 'bubbles' task in the fMRI scanner


1
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

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
Bubbles analysis
  • E.g. Fearful (3000/4750 trials)

Trial 1 2 3 4
5 6 7 750
Sum
Correct trials
Thresholded at proportion of correct
trials0.68, scaled to 0,1
Use this as a bubble mask
Proportion of correct bubbles (sum correct
bubbles) /(sum all bubbles)
16
Results
  • Mask average face
  • But are these features real or just noise?
  • Need statistics

Happy Sad
Fearful Neutral
17
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

ZN(0,1) statistic
Trial 1 2 3 4
5 6 7 750
Response 0 1 1 0
1 1 1 1
18
Comparison
  • Both depend on average correct bubbles, rest is
    constant

Proportion correct bubbles Average correct
bubbles / (average all bubbles 4)
  • Z(Average correct bubbles
  • average incorrect bubbles)
  • / pooled sd

19
Results
  • Thresholded at Z1.64 (P0.05)
  • Multiple comparisons correction?
  • Need random field theory

ZN(0,1) statistic
Average face Happy Sad
Fearful Neutral
20
Euler Characteristic blobs - holes
Excursion set Z gt threshold for neutral face
EC 0 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 gt
t).
  • Exact expression for E(EC) for all thresholds,
  • E(EC) P(max Z gt t) is extremely accurate.

21
The details
Intrinsic volumes or Minkowski functionals
22
The brain mapping version
EC0(S)
Resels0(S)
Resels1(S)
EC1(S)
Resels2(S)
EC2(S)
Resels (Resolution elements)
EC densities
23
Math version
24
The details
25
2
Tube(S,r)
r
S
26

3

27

B
A
28

29
6
? is big
Tube?(S,r)
S
r
? is small
30
2
?
U(s1)
s1
S
Tube
S
Tube
s2
s3
U(s3)
U(s2)
31
Z2
R
r
Tube(R,r)
Z1
N2(0,I)
32

Tube(R,r)
R
z
t-r
t
z1
Tube(R,r)
r
R
R
z2
z3
33

34

35
Summary
36
(No Transcript)
37
Random field theory results
  • For searching in D (2) dimensions, P-value of
    max Z is (Adler, 1981 W, 1995)
  • P(max Z gt z)
  • E( Euler characteristic of thresholded set )
  • Resels Euler characteristic density (
    boundary terms)
  • Resels (Lipschitz-Killing curvature/c) is
  • Image area / (bubble FWHM)2 146.2 (unitless)
  • Euler characteristic density(c) is
  • (4 log(2))D/2 zD-1 exp(-z2/2) / (2p)(D1)/2
  • See forthcoming book Adler, Taylor (2007)

38
Results, corrected for search
  • Thresholded at Z3.92 (P0.05)

ZN(0,1) statistic
Average face Happy Sad
Fearful Neutral
39
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 1 2 3 4
5 6 7 3000
fMRI
40
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 P0.05, 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) gt Z(pixel, 6 neighbours of voxel)
  • gt Z(4 neighbours of
    pixel, voxel)

41
Discussion modeling
  • The random response is Y1 (correct) or 0
    (incorrect), or YfMRI
  • The regressors are Xjbubble mask at pixel j, j1
    240x38091200 (!)
  • Logistic regression or ordinary regression
  • logit(E(Y)) or E(Y) b0X1b1X91200b91200
  • But there are only n3000 observations (trials)
  • Instead, since regressors are independent, fit
    them one at a time
  • logit(E(Y)) or E(Y) b0Xjbj
  • However the regressors (bubbles) are random with
    a simple known distribution, so turn the problem
    around and condition on Y
  • E(Xj) c0Ycj
  • 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)b0X1b1X91200
    b91200X1X2b1,2

42
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 -0.568, T -14.20 (423 df)
2
Charil et al, NeuroImage (2007)
1.5
0
10
20
30
40
50
60
70
80
Total lesion volume (cc)
43
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
  • Subtract average cortical thickness
  • 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
  • 1 billion correlations, so thresholding
    essential!
  • Look for high negative correlations

44
Thresholding? Crosscorrelation 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 P0.05, c0.300, T6.46

Cao Worsley, Annals of Applied Probability
(1999)
45
Cluster extent rather than peak height (Friston,
1994)
  • Choose a lower level, e.g. t3.11 (P0.001)
  • Find clusters i.e. connected components of
    excursion set
  • Measure cluster
  • extent by
  • Distribution
  • fit a quadratic
  • to the peak
  • Distribution of maximum cluster extent
  • Bonferroni on N clusters E(EC).

Z
D1
extent
t
Peak height
s
Cao, Advances in Applied Probability (1999)
46
How do you choose the threshold t for defining
clusters?
  • If signal is focal i.e. FWHM of noise
  • Choose a high threshold
  • i.e. peak height is better
  • If signal is broad i.e. gtgtFWHM of noise
  • 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!

47
Thresholding? Crosscorrelation 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)
  • MS lesion data P0.05, c0.300, T6.46

Cao Worsley, Annals of Applied Probability
(1999)
48
Histogram
threshold
threshold
Conditional histogram scaled to same max at
each distance
threshold
threshold
49
Science (2004)
50
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.
51
What are the subjects watching during high
activation? Faces
52
buildings
53
hands
54
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. resels1000, df100, threshold5, usual
    P0.051, homologous P0.139

Cao Worsley, Annals of Applied Probability
(1999)
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