Variational Bayesian Inference for fMRI time series Will Penny, Stefan Kiebel and Karl Friston The Wellcome Department of Imaging Neuroscience, UCL http//:www.fil.ion.ucl.ac.uk/~wpenny

Overview Introduction to fMRI GLM-AR models fMRI data analysis Introduction to Bayes Introduction to fMRI GLM-AR models fMRI data analysis

Gaussian Bayes

GLM Bayes

Variational Bayes

Model order selection Model Evidence Free Energy

fMRI: Data Processing Stream Image time-series Kernel Design matrix Posterior Probability Map (PPM) Realignment Smoothing General linear model Normalisation Template Parameter estimates

Functional MRI Neural Activity Blood Oxygenation Magnetic Properties of Oxygenated Blood BOLD

Box car regression: design matrix… (voxel time series) data vector parameters design matrix error vector a =  +  Y = X   + 

Low frequency nuisance effects… Drifts physical physiological Aliased high frequency effects cardiac (~1 Hz) respiratory (~0.25 Hz) Discrete cosine transform basis functions

…design matrix = + Y = X   +  error vector parameters design matrix data vector a m 3 4 5 6 7 8 9 = + Y = X   + 

Errors are autocorrelated Physiological factors Physics of the measurement process Hence AR, AR+white noise model or ARMA model

of sufficient statistics GLM-AR models GLM AR Priors Approximate Posteriors Recursive estimation of sufficient statistics

Synthetic GLM-AR(3) Data

This is an event-related study Face Data This is an event-related study BOLD Signal Face Events 60 secs

Face Data: design matrix

AR model order map

AR order by tissue type GRAY CSF WHITE

Map of first AR coefficient

First AR coefficient by tissue type

Angiograms

Posterior Probability Map Bilateral Fusiform cortex

Comparison with OLS Iterative re-estimation of coeffients increase accuracy of estimation of effect sizes significantly – on real and synthetic data Typical improvement of 15% - commensurate with degree of autocorrelation

Map of first AR coefficient: other subjects

Map of first AR coefficient: more subjects Unmodelled signal

Map of first AR coefficient: last 3 subjects

Unmodelled signal BOLD time series GLM Estimate (dotted line) (solid line) 60 secs

Conclusions Low-order AR processes are sufficient to model residual correlation in fMRI time series VB criterion identifies exact order required Iterative estimation of parameters takes into account correlation Non-homogeneity of residual correlation reflects vasculature, tissue-type and unmodelled signal