Correlation random fields, brain
connectivity, and cosmology
Keith Worsley
Department of Mathematics and Statistics, and
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
Savic et al. (2005). Brain response to
putative pheromones in homosexual
men. Proceedings of the National
Academy of Sciences, 102:7356-7361
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
Scale space: smooth X(t) with a range of filter widths, s
= continuous wavelet transform
adds an extra dimension to the random field: X(t, s)
Scale space, no signal
S = FWHM (mm, on log scale)
34
8
6
4
2
0
-2
22.7
15.2
10.2
6.8
-60
-40
-20
0
20
One 15mm signal
40
60
34
8
6
4
2
0
-2
22.7
15.2
10.2
6.8
-60
-40
-20
0
t (mm)
20
40
60
15mm signal best detected with a ~15mm smoothing filter
Matched Filter Theorem (= Gauss-Markov Theorem):
“to best detect a signal + white noise, filter should match signal”
10mm and 23mm signals
S = FWHM (mm, on log scale)
34
8
6
4
2
0
-2
22.7
15.2
10.2
6.8
-60
-40
-20
0
20
40
Two 10mm signals 20mm apart
60
34
8
6
4
2
0
-2
22.7
15.2
10.2
6.8
-60
-40
-20
0
t (mm)
20
40
60
But if the signals are too close together they are
detected as a single signal half way between them
Scale space can even separate two signals at the same location!
8mm and 150mm signals at the same location
10
5
S = FWHM (mm, on log scale)
0
-60
170
-40
-20
0
20
40
60
113.7
20
76
50.8
15
34
10
22.7
15.2
5
10.2
6.8
-60
-40
-20
0
t (mm)
20
40
60
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
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
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
BrainStat
- the details
Jonathan Taylor, Stanford
Keith Worsley, McGill
What is BrainStat?
Based on FMRISTAT (Matlab)
 Written in Python (open source)
 Part of BrainPy (Poster 763 T-AM)
 Concentrates on statistics
 Analyses both magnitudes and delays
(latencies)
 P-values for peaks and clusters uses
latest random field theory

Details




Input data is motion corrected and preferably slice
timing corrected
Output is complete hierarchical mixed effects ReML
analysis (local AR(p) errors at first stage)
Spatial regularization of (co)variance ratios chosen
to target 100 df (Poster 610 M-PM)
P-values for peaks and clusters are best of

Bonferroni
random field theory

discrete local maxima (Poster 539 T-AM)

Methods






Slice timing and motion correction by FSL
AR(1) errors on each run
For each subject, 2 runs combined using
fixed effects analysis
Spatial registration to 152 MNI by FSL
Subjects combined using mixed effects
analysis
Repeated for all contrasts of both magnitudes
and delays
Magnitude (%BOLD), diff - same sentence
0
1
3
4
6
7
Subject id, block experiment
8
9
10
11
12
13
14
15
Mixed
effects
2
1
Ef
0
-1
Random
/fixed
effects sd
smoothed
11.5625mm
1.5
-2
Contour is: average anatomy > 2000
1
Sd
df
205
206
203
206
206
204
203
201
205
200
200
201
201
205
0.5
1
0
0.5
5
FWHM (mm)
20
100
T
0
-5
P=0.05 threshold for peaks is +/- 5.1375
x (mm)
-50
15
0
10
50
5
-60
-40
-20 0
y (mm)
0
Delay shift (secs), diff - same sentence
0
1
3
4
6
7
Subject id, block experiment
8
9
10
11
12
13
14
15
Mixed
effects
2
1
Ef
0
-1
Random
/fixed
effects sd
smoothed
14.3802mm
1.5
-2
Contour is: magnitude, stimulus average, T statistic > 5
2
1.5
Sd
1
1
0.5
df
205
206
203
206
206
204
203
201
205
200
200
201
201
205
0
0.5
4
FWHM (mm)
20
100
-50
T
0
-2
-4
P=0.05 threshold for peaks is +/- 4.0888
x (mm)
2
15
0
10
50
5
-60
-40
-20 0
y (mm)
0
Conclusions

Strong overall %BOLD increase of 3±0.5%





Substantial subject variability (sd ratio ~8)
Evidence for greater %BOLD response for
different sentences (0.5±0.1%)
Evidence for greater latency for different
sentences (0.16±0.04 secs)
Event design is better for delays
Block design is better for overall magnitude