Connectivity of
aMRI and fMRI data
Keith Worsley
Arnaud Charil
Jason Lerch
Francesco Tomaiuolo
Department of Mathematics and Statistics,
McConnell Brain Imaging Centre,
Montreal Neurological Institute,
McGill University
Effective connectivity
• Measured by the correlation between residuals at
pairs of voxels:
Activation only
Voxel 2
++
+ +++
Correlation only
Voxel 2
+
++
Voxel 1
+ +
Voxel 1
+
Types of connectivity
• Focal
• Extensive
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
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
Methods
1.
2.
3.
4.
Seed
Iterated seed
Thresholding correlations
PCA
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?
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’
Which is better:
thresholding correlations,
or
PCA?
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
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
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?
MS lesions and cortical thickness
(Arnaud et al., 2004)
•
•
•
•
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
cor(lesion density, cortical thickness)
n=425 subjects, correlation = -0.56826
Average cortical thickness
5.5
5
4.5
4
3.5
3
2.5
2
1.5
0
10
20
30
40
50
60
Average lesion volume
70
80
Normalization
• 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) )
Same hemisphere
0.1
1
-0.3
threshold
-0.4
-0.5
0
50
100
150
distance (mm)
Correlation = 0.091943
0.1
correlation
0
0
50
100
150
distance (mm)
1.5
-0.3
1
-0.5
1
0.1
0.4
threshold
-0.2
0
-0.2
-0.4
2
-0.4
0.6
-0.3
-0.1
0.5
threshold
0
50
100
150
distance (mm)
Correlation = -0.1257
0.5
0
1
0
0.8
-0.1
-0.5
correlation
1.5
-0.2
5
x 10
2.5
0
2
-0.1
Different hemisphere
0.1
correlation
correlation
0
5
x 10
2.5
0.8
-0.1
0.6
-0.2
0.4
-0.3
0.2
-0.4
0
-0.5
threshold
0
50
100
150
distance (mm)
0.2
0
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.
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
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
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
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
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.