1. Bayesian Methods in Brain Imaging
Will Penny
Thanks to Karl Friston &
Wellcome Department of Imaging Neuroscience,
University College London, UK
Institute for Adaptive and Neural Computation,
University of Edinburgh, 27 January 2004.

2. Overview 1. Bayesian Inference
2. Brain Imaging: Functional Segregation
3. Brain Imaging: Functional Integration

3. Overview 1. Bayesian Inference
2. Brain Imaging: Functional Segregation
3. Brain Imaging: Functional Integration

4. First level of Bayesian Inference
We have data, y, and some parameters, b

5. One parameter Likelihood and Prior Posterior Likelihood Prior
Relative Precision Weighting

6. Two parameters

7. First level of Bayesian Inference
We have data, y, and some parameters, b
Parameters are of model, M, ….

8. First and Second Levels
The first level again, writing in dependence on M:
Second level of Inference: What’s the best model ?

9. Model Selection We need to compute the Bayesian Evidence:
We can’t always compute it exactly, but we
can approximate it: Log p(y|M) ~ F(M)
Laplace
Approximations
Variational
Bayes

10. Overview 1. Bayesian Inference
2. Brain Imaging: Functional Segregation
3. Brain Imaging: Functional Integration

11. realignment & motion correction
image data
fMRI data
kernel
Parameter
estimates
realignment & motion correction
General Linear Model
model fitting
statistic image
smoothing
normalisation
design matrix
anatomical reference
Probabilistic map
of activations

12. Face processing data Press left key if famous, right key if not
R. Henson et al. (2001)
Every face presented twice
Part of larger study looking
at factors influencing
priming
Press left key if famous, right key if not
24 Transverse Slices acquired with TR=2s
Time series of 351 images

13. Modelling the Signal
Assumption: Neuronal Event Stream is
Identical to the Experimental Event Stream
Convolve event-stream with basis functions to account for
the hemodynamic response function

14. FIR models
Size of
signal
Time
after
event 5s

15. Separate smoothness priors for each event type
FIR model
Design matrix
for FIR model with
8 time bins in a
20-second window
Separate smoothness priors for each event type

16. fMRI time series model Use a General Linear Model y = X b + e
Priors factorise into groups:
p(b) = p(b1) p(b2) p(b3)
Priors in each group may be smoothness
priors or Gaussians

17. Remove with ICA/PCA – but non-automatic
Noise sources in fMRI
1. Slow drifts due to instrumentation instabilities
2. Subject movement
3. Vasomotor oscillation ~ 0.1 Hz
4. Respiratory activity ~ 0.25 Hz
5. Cardiac activity ~ 1 Hz
Remove with ICA/PCA – but non-automatic

18. fMRI time series model – with Stefan Kiebel
Use a General Linear Model
y = X b + e;
Priors factorise into groups:
p(b) = p(b1) p(b2) p(b3)
Priors in each group may be smoothness
priors or Gaussians

19. Activation Map
Structural MRI
Main Effect of Faces
p(cTb > 0.5%)

20. FIR model average responses
FIR basis set
Right fusiform cortex
(x=45, y=-60, z=-18)
FIR model average responses
Larger response to first
presentation – priming effect

21. FIR model average responses
FIR basis set
Left occipital cortex
(x=-33, y=-81, z=-24)
FIR model average responses
Priming effect only for
unfamiliar faces

22. RFX-Event model 97 parameters ! But only 24 effective parameters
Responses to each event of type A are randomly distributed
about some typical “type A” response
Design Matrix
97 parameters ! But only 24 effective parameters

23. Synthetic GLM-AR(3) Data

24. Map of AR model order, p
p=0,1,2,3

25. Angiograms

26. Current work – with Nelson Trujillo-Baretoq1 q2 b L a z W Y
[TxN] = [TxK][KxN] + [TxN]
Spatial Priors – Laplace priors

27. Maps of regression coefficients – Image (wk)
TRUE
ESTIMATE – w/o Spatial Prior
1024 regression coefficients
but only 280 `effective’
coefficients.
ESTIMATE

28. Overview 1. Bayesian Inference
2. Brain Imaging: Functional Segregation
3. Brain Imaging: Functional Integration

29. Attention to Visual Motion
Stimuli
250 radially moving dots at 4.7 degrees/s
Pre-Scanning
5 x 30s trials with 5 speed changes (reducing to 1%)
Task - detect change in radial velocity
Scanning (no speed changes)
6 normal subjects, scan sessions;
each session comprising 10 scans of 4 different condition
e.g. F A F N F A F N S
F – fixation
S – stationary dots
N – moving dots
A – attended moving dots
Buchel et al. 1997
Experimental Factors
Photic Stimulation, S,N,A
Motion, N,A
Attention, A

30. Motion Sensitive Areas
Mass Univariate Analyses
y = Xw + e
Where is effect, w, eg.
of motion, significantly
non-zero.
Analysis of 360 images each
containing 100,000 voxels, ie.
100,000 time series. New image
every 3 seconds.

31. Network Analysis IFG SPC V5 V1 Photic Inputs and Motion Outputs
Attention

32. DCM: A network model for fMRI
Set u2
Stimuli u1
Input
State
Output
PRIORS ….
Friston et al. 2003

33. Bilinear Dynamics: Positive transients
Stimuli u1
Set u2 u 1 Z 2 - + Z1 - + + Z2 - -

34. Impulse
response
Hemodynamics
BOLD is
sluggish

35. …with Andrea Mechelli
and Klaas Stephan
Bayes factors:

36. Current work Stochastic Neurodynamics – with Zoubin G.
DCMs for EEG/ERPs – with Olivier David/Lee H.
EEG-fMRI sensor fusion …………..