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Variational Bayesian Inference for fMRI time series

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

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Variational Bayesian Inference for fMRI time series

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  1. 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

  2. Overview • 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 Design matrix Image time-series Kernel 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

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

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

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

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

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

  14. Synthetic GLM-AR(3) Data

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

  16. Face Data: design matrix

  17. AR model order map

  18. AR order by tissue type GRAY CSF WHITE

  19. Map of first AR coefficient

  20. First AR coefficient by tissue type

  21. Angiograms

  22. Posterior Probability Map Bilateral Fusiform cortex

  23. 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

  24. Map of first AR coefficient: other subjects

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

  26. Map of first AR coefficient: last 3 subjects

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

  28. 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

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