1. Variational Bayesian Inference for fMRI time series
Will Penny, Stefan Kiebel
and Karl Friston
The Wellcome Department of Imaging Neuroscience, UCL
http//:

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

3. Gaussian Bayes

4. GLM Bayes

5. Variational Bayes

6. Model order selection
Model
Evidence
Free
Energy

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

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

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

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

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

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

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

18. Synthetic GLM-AR(3) Data

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

20. Face Data: design matrix

21. AR model order map

22. AR order by tissue type
GRAY
CSF
WHITE

23. Map of first AR coefficient

24. First AR coefficient by tissue type

25. Angiograms

26. Posterior Probability Map
Bilateral
Fusiform
cortex

27. 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

28. Map of first AR coefficient: other subjects

29. Map of first AR coefficient: more subjects
Unmodelled
signal

30. Map of first AR coefficient: last 3 subjects

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

32. 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