Correlation random fields, brain
connectivity, and astrophysics
Keith Worsley
Arnaud Charil
Jason Lerch
Francesco Tomaiuolo
Department of Mathematics and Statistics,
McConnell Brain Imaging Centre,
Montreal Neurological Institute,
McGill University
CfA red shift survey, FWHM=13.3
100
80
Euler Characteristic (EC)
60
"Meat ball"
topology
40
20
"Bubble"
topology
0
-20
-40
"Sponge"
topology
-60
-80
-100
-5
CfA
Random
Expected
-4
-3
-2
-1
0
1
Gaussian threshold
2
3
4
5
fMRI data: 120 scans, 3 scans each of hot, rest, warm, rest, hot, rest, …
First scan of fMRI data
Highly significant effect, T=6.59
1000
hot
rest
warm
890
880
870
500
0
100
200
300
No significant effect, T=-0.74
820
hot
rest
warm
0
800
T statistic for hot - warm effect
5
0
-5
T = (hot – warm effect) / S.d.
~ t110 if no effect
0
100
0
100
200
Drift
300
810
800
790
200
Time, seconds
300
Effective connectivity
• Measured by the correlation between residuals at
pairs of voxels:
Activation only
Voxel 2
++
+ +++
Correlation only
Voxel 2
+
++
Voxel 1
+ +
Voxel 1
+
3
Focal correlation
2
1
0
0
1
2
3
3
-1
-2
2
cor=0.58
-3
-2
0
4
2
5
6
7
1
0
8
n = 120
frames
9
10
11
-1
-2
-3
Method 1: ‘Seed’
Friston et al. (19??): Pick one voxel, then find all others
that are correlated with it:
Problem: how to pick the ‘seed’ voxel?
T = sqrt(df) cor / sqrt (1 - cor2)
0
Seed
1
2
3
T max = 7.81
P=0.00000004
4
6
4
5
6
7
2
0
8
9
10
11
-2
-4
-6
Method 2: Iterated ‘seed’
• Problem: how to find the rest of the connectivity network?
• Hampson et al., (2002): Find significant correlations, use them
as new seeds, iterate.
Method 3: All correlations
• Problem: how to find isolated parts of the connectivity network?
• Cao & Worsley (1998): find all correlations (!)
• 6D data, need higher threshold to compensate
Thresholds are not as high as you
might think:
E.g. 1000cc search region, 10mm smoothing, 100 df, P=0.05:
dimensions D1 D2
Voxel1 - Voxel2
0 0
Cor
T
0.165 1.66
One seed voxel - volume
0 3 0.448 4.99
Volume – volume (auto-correlation)
3 3
0.609 7.64
Volume1 – volume2 (cross-correlation) 3 3
0.617 7.81
Practical details
• Find threshold first, then keep only
correlations > threshold
• Then keep only local maxima i.e.
cor(voxel1, voxel2)
> cor(voxel1, 6 neighbours of voxel2),
> cor(6 neighbours of voxel1, voxel2),
Method 4: Principal Components
Analysis (PCA)
• Friston et al: (1991): find spatial and temporal
components that capture as much as possible of the
variability of the data.
• Singular Value Decomposition of time x space matrix:
Y = U D V’ (U’U = I, V’V = I, D = diag)
• Regions with high score on a spatial component (column
of V) are correlated or ‘connected’
Extensive correlation
3
2
1
0
0
1
2
3
3
-1
-2
-3
2
cor=0.13
-2
0
4
2
5
6
7
1
0
8
9
10
11
-1
-2
-3
PCA, component 1
0
1
2
3
1
0.8
0.6
0.4
4
5
6
7
0.2
0
-0.2
8
9
10
11
-0.4
-0.6
-0.8
-1
Which is better:
thresholding T statistic (= correlations),
or PCA?
T, extensive correlation
0
Seed
1
2
3
T max = 4.17
P = 0.59
6
4
4
5
6
7
2
0
8
9
10
11
-2
-4
-6
PCA, focal correlation
0
1
2
3
1
0.8
0.6
0.4
4
5
6
7
0.2
0
-0.2
8
9
10
11
-0.4
-0.6
-0.8
-1
Summary
Focal correlation
0
1
2
3
Extensive correlation
6
0
1
2
3
4
Thresholding
T statistic
(=correlations)
4
5
6
7
2
4
4
5
6
7
0
8
0
9
1
10
2
11
3
-2
5
6
7
PCA
8
9
10
11
2
0
8
9
10
11
-2
-4
-4
-6
-6
1
0
1
2
3
1
0.8
0.8
0.6
0.6
0.4
4
6
0.2
0.4
4
5
6
7
0.2
0
0
-0.2
-0.2
-0.4
8
9
10
11
-0.4
-0.6
-0.6
-0.8
-0.8
-1
-1
Modulated connectivity
• Looking for correlations not very interesting – ‘resting state
networks’
• More intersting: how does connectivity change with
- task or condition (external)
- response at another voxel (internal)
• Friston et al., (1995): add interaction to the linear model:
Data ~ task + seed + task*seed
Data ~ seed1 + seed2 + seed1*seed2
PCA of time  space:
Temporal components (sd, % variance explained)
Component
0
1
0.68, 46.9%
2
0.29, 8.6%
3
0.17, 2.9%
4
0.15, 2.4%
5
0
20
40
60
80
100
1
Component
1
0.5
2
0
3
-0.5
4
2
4
6
8
Slice (0 based)
10
2: drift
120
Frame
Spatial components
0
1: exclude
first frames
12
-1
3: long-range
correlation
or anatomical
effect: remove
by converting
to % of brain
4: signal?
Fit a linear model for fMRI time
series with AR(p) errors
• Linear model:
?
?
Yt = (stimulust * HRF) b + driftt c + errort
• AR(p) errors:
unknown parameters
?
?
?
errort = a1 errort-1 + … + ap errort-p + s WNt
• Subtract linear model to get residuals.
• Look for connectivity.
Deformation Based Morphometry
(DBM) (Tomaiuolo et al., 2004)
• n1 = 19 non-missile brain trauma patients, 3-14
days in coma,
• n2 = 17 age and gender matched controls
• Data: non-linear vector deformations needed to
warp each MRI to an atlas standard
• Locate damage: find regions where deformations
are different, hence shape change
• Is damage connected? Find pairs of regions with
high canonical correlation.
MS lesions and cortical thickness
(Arnaud et al., 2004)
•
•
•
•
N = 347 mild MS patients
Lesion density, smoothed 10mm
Cortical thickness, smoothed 20mm
Find connectivity i.e. find voxels in 3D,
nodes in 2D with high
cor(lesion density, cortical thickness)
Male or female
(GENDER)?
Expressive or not
expressive (EXNEX)?
Correct bubbles
All bubbles
Image masked by bubbles
as presented to the subject
Correct / all bubbles
Fig. 1. Results of Experiment 1. (a) the raw
classification images, (b) the classification images
filtered with a smooth low-pass (Butterworth) filter with
a cutoff at 3 cycles per letter, and (c) the best matches
between the filtered classification images and 11,284
letters, each resized and cut to fill a square window in
the two possible ways. For (b), we squeezed pixel
intensities within 2 standard deviations from the mean.
Subject 1
Subject 2
Subject 3