Neuroscience Research Institute University of Manchester 13 September 2007

1. Generative Models for fMRIWill PennyWellcome Department of Imaging Neuroscience, UCLhttp://www.fil.ion.ucl.ac.uk/~wpenny Neuroscience Research Institute University of Manchester 13 September 2007

2. 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. q1 q2 a b u1 u2 l W A Y The Model r2 r1 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. L KL F Variational Bayes

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

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

11. Synthetic Data 1 : from Laplacian Prior y t x

12. F Iteration Number

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

14. Synthetic Data II : blobs Smoothing True 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. 2005 Jan 15;24(2):350-62. 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. 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. 2007 Feb 1;34(3):1108-25.