1. Neuroscience Research Institute University of Manchester
Generative Models for fMRI Will Penny Wellcome Department of Imaging Neuroscience, UCL
Neuroscience Research Institute
University of Manchester
13 September 2007

2. Serial snapshots of blood –oxygenation levels
2D fMRI time-series
Serial snapshots of blood –oxygenation levels
Time
Time

3. Motivation Even without applied spatial smoothing, activation maps
(and maps of eg. AR coefficients) have spatial structure
Contrast
AR(1)
We can increase the sensitivity of our inferences by smoothing
data with Gaussian kernels (SPM2). This is worthwhile, but crude.
Can we do better with a spatial model (SPM5) ?
Aim: For SPM5 to remove the need for spatial smoothing just as
SPM2 removed the need for temporal smoothing

4. The Model Y=XW+E q1 q2 r1 r2 u1 u2 l a b A W Y Voxel-wise AR:
Spatial pooled AR:
Y=XW+E
[TxN] [TxK] [KxN] [TxN]

5. Synthetic Data 1 : from Laplacian Prior
reshape(w1,32,32) t

6. Prior, Likelihood and Posterior
In the prior, W factorises over k and A factorises over p:
The likelihood factorises over n:
The posterior over W therefore does’nt factor over k or n. It is a Gaussian
with an NK-by-NK full covariance matrix. This is unwieldy to even store,
let alone invert ! So exact inference is intractable.

7. Variational Bayes L F KL

8. Variational Bayes If you assume posterior factorises
then F can be maximised by letting
where

9. Variational Bayes
In the prior, W factorises over k and A factorises over p:
In chosen approximate posterior, W and A factorise over n:
So, in the posterior for W we only have to store and invert N K-by-K
covariance matrices.

10. Updating approximate posterior
Regression coefficients, W
AR coefficients, A
Spatial precisions for W
Spatial precisions for A
Observation noise

11. Synthetic Data 1 : from Laplacian Priorx t

12. F
Iteration Number

13. Least Squares VB – Laplacian Prior y y x x Coefficients = 1024
`Coefficient RESELS’ = 366

14. Synthetic Data II : blobs
True
Smoothing
Global prior
Laplacian prior

15. Sensitivity
1-Specificity

16. Event-related fMRI: Faces versus chequerboard
Smoothing
Global prior
Laplacian Prior

17. Event-related fMRI: Familiar faces versus unfamiliar faces
Smoothing
Penny WD, Trujillo-Barreto NJ, Friston KJ. Bayesian fMRI time series analysis with spatial priors.
Neuroimage Jan 15;24(2):
Global prior
Laplacian Prior

18. Bayesian Wavelet Shrinkage
Wavelet coefficients are a priori independent,
The prior density of each coefficient is given by a mixture of two zero-mean Gaussian.
Consider each level separately
Applied only to detail levels
Negligible coeffs.
Significant coeffs.
Estimation of the parameters via an Empirical Bayes algorithm

19. Generative model

20. Iteratively update to maximise a lower bound on evidence
Approximate posteriors
Iteratively update to maximise a lower bound on evidence

21. Event-related fMRI: overall effect of faces
Standard GLM smoothing
Bayesian wavelet shrinkage
Flandin G, Penny WD. Bayesian fMRI data analysis with sparse spatial basis function priors.
Neuroimage Feb 1;34(3):