Neuroscience Research Institute University of Manchester

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Presentation transcript:

Neuroscience Research Institute University of Manchester Generative Models for fMRI Will Penny Wellcome Department of Imaging Neuroscience, UCL http://www.fil.ion.ucl.ac.uk/~wpenny Neuroscience Research Institute University of Manchester 13 September 2007

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

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

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]

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

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.

Variational Bayes L F KL

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

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.

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

Synthetic Data 1 : from Laplacian Prior x t

F Iteration Number

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

Synthetic Data II : blobs True Smoothing Global prior Laplacian prior

Sensitivity 1-Specificity

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

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

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

Generative model

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

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.