Correlation random fields, brain connectivity, and astrophysics

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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

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fMRI data 120 scans, 3 scans each of hot, rest,
warm, rest, hot, rest,
T (hot warm effect) / S.d. t110 if no
effect
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Effective connectivity

• Measured by the correlation between residuals at
  pairs of voxels

Activation only
Correlation only
Voxel 2


Voxel 2







Voxel 1
Voxel 1



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Focal correlation
3
0
1
2
3
cor0.58
2
1
4
5
6
7
0
-1
8
9
10
11
n 120 frames
-2
-3
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• 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?

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T sqrt(df) cor / sqrt (1 - cor2)
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Seed
0
1
2
3
T max 7.81 P0.00000004
4
2
4
5
6
7
0
-2
8
9
10
11
-4
-6
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• 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.

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• 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

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Thresholds are not as high as you might think
E.g. 1000cc search region, 10mm smoothing, 100
df, P0.05

dimensions D1 D2 Cor T Voxel1 - Voxel2
0 0 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
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Practical details

• Find threshold first, then keep only correlations
  gt threshold
• Then keep only local maxima i.e.
• cor(voxel1, voxel2)
• gt cor(voxel1, 6 neighbours of
  voxel2),
• gt cor(6 neighbours of voxel1,
  voxel2),

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• 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 (UU I, VV I, D
  diag)
• Regions with high score on a spatial component
  (column of V) are correlated or connected

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Extensive correlation
3
0
1
2
3
cor0.13
2
1
4
5
6
7
0
-1
8
9
10
11
-2
-3
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PCA, component 1
1
0
1
2
3
0.8
0.6
0.4
4
5
6
7
0.2
0
-0.2
-0.4
8
9
10
11
-0.6
-0.8
-1
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Which is better thresholding T statistic (
correlations), or PCA?
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T, extensive correlation
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Seed
0
1
2
3
T max 4.17 P 0.59
4
2
4
5
6
7
0
-2
8
9
10
11
-4
-6
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PCA, focal correlation
1
0
1
2
3
0.8
0.6
0.4
4
5
6
7
0.2
0
-0.2
-0.4
8
9
10
11
-0.6
-0.8
-1
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Summary
Extensive correlation
Focal correlation
6
6
0
1
2
3
0
1
2
3
4
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Thresholding T statistic (correlations)
2
2
4
5
6
7
4
5
6
7
0
0
-2
-2
8
9
10
11
8
9
10
11
-4
-4
-6
-6
1
1
0
1
2
3
0
1
2
3
0.8
0.8
0.6
0.6
0.4
0.4
PCA
4
5
6
7
4
5
6
7
0.2
0.2
0
0
-0.2
-0.2
-0.4
8
9
10
11
-0.4
8
9
10
11
-0.6
-0.6
-0.8
-0.8
-1
-1
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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
  taskseed
• Data seed1 seed2
  seed1seed2

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PCA of time ? space
Temporal components (sd, variance explained)
1 exclude first frames
0
1
0.68, 46.9
2
0.29, 8.6
2 drift
Component
3
0.17, 2.9
4
0.15, 2.4
5
0
20
40
60
80
100
120
Frame
3 long-range correlation or anatomical effect
remove by converting to of brain
Spatial components
1
1
0.5
2
0
Component
3
-0.5
4
-1
4 signal?
0
2
4
6
8
10
12
Slice (0 based)
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Fit a linear model for fMRI time series with
AR(p) errors

• Linear model
• ?
  ?
• Yt (stimulust HRF) b driftt c errort
• AR(p) errors
• ? ?
  ?
• errort a1 errort-1 ap errort-p s WNt
• Subtract linear model to get residuals.
• Look for connectivity.

unknown parameters
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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.

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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)

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Expressive or not expressive (EXNEX)?
Male or female (GENDER)?
Correct bubbles
All bubbles
Image masked by bubbles as presented to the
subject
Correct / all bubbles
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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