Bayesian Modelling of Functional Imaging Data

1. Bayesian Modelling of Functional Imaging Data Will Penny The Wellcome Department of Imaging Neuroscience, UCL http//:www.fil.ion.ucl.ac.uk/~wpenny

2. Overview • Multiple levels of Bayesian Inference • A model of fMRI time series: The Noise • A model of fMRI time series: The Signal • The fMRI Inverse Problem

3. First level of Bayesian Inference We have data, y, and some parameters, b First level of Inference: What are the best parameters ? Parameters are of model, M, ….

4. First and Second Levels The first level again, writing in dependence on M: Second level of Inference: What’s the best model ?

5. Model Selection We need to compute the Bayesian Evidence: We can’t always compute it exactly, but we can approximate it: Log p(y|M) ~ F(M) Evidence = Accuracy - Complexity

6. Model Averaging Revisiting the first level: Model-dependent posteriors are weighted according to the posterior probability of each model

7. Multiple Levels w3 w1 w2 w3 w1 w2 Y Y Evidence Up Posteriors Down

8. Overview • Multiple levels of Bayesian Inference • A model of fMRI time series: The Noise • A model of fMRI time series: The Signal • The fMRI Inverse Problem

9. Noise sources in fMRI 1. Slow drifts due to instrumentation instabilities 2. Subject movement 3. Vasomotor oscillation ~ 0.1 Hz 4. Respiratory activity ~ 0.25 Hz 5. Cardiac activity ~ 1 Hz Remove with ICA/PCA – but non-automatic

10. fMRI time series model • Use a General Linear Model: y = X b + e • The errors are modelled as an AR(p) process • The order can be selected using Bayesian evidence

11. Synthetic GLM-AR(3) Data

12. Map of AR model order, p Face Data p=0,1,2,3

13. Angiograms

14. Other subjects, a1 Ring of voxels with highly correlated error

15. Other subjects, a1 Unmodelled signal or increased cardiac artifact due to increased blood flow?

16. Overview • Multiple levels of Bayesian Inference • A model of fMRI time series: The Noise • A model of fMRI time series: The Signal • The fMRI Inverse Problem

17. fMRI time series model • Use a General Linear Model for the signal : y = X b + e • Priors factorise into groups: p(b) = p(b1) p(b2) p(b3) • Priors in each group may be smoothness priors or Gaussians

18. Rik’s data Every face presented twice Part of larger study looking at factors influencing repetition suppresion Press left key if famous, right key if not 24 Transverse Slices acquired with TR=2s Time series of 351 images

19. Modelling the Signal Assumption: Neuronal Event Stream is Identical to the Experimental Event Stream Convolve event-stream with basis functions to account for the HRF

20. FIR model Design matrix for FIR model with 8 time bins in a 20-second window Separate smoothness priors for each event type Q. Is this a good prior ?

21. FIR basis set Left occipital cortex (x=-33, y=-81, z=-24) FIR model average responses

22. FIR basis set Right fusiform cortex (x=45, y=-60, z=-18) FIR model average responses

23. RFX-Event model Responses to each event of type A are randomly distributed about some typical “type A” response Design Matrix 97 parameters ! But only 24 effective parameters

24. Non-stationary models As RFX-event but smoothness priors Testing for smooth temporal variations statistically …

25. Simpler Designs Canon. + Temp. Deriv Gammas

26. Right Fusiform Left Occipital Gammas Canon. + Temp. Deriv Canon. + Temp. Deriv Gammas RFX-Event RFX-Event FIR FIR Comparing Types of Models Evidence NonStat NonStat Model averaging to get peak post-stimulus response

27. Overview • Multiple levels of Bayesian Inference • A model of fMRI time series: The Noise • A model of fMRI time series: The Signal • The fMRI Inverse Problem

28. The fMRI Inverse Problem • In EEG there is an ill-posed spatial inverse problem. We wish to recover the electrical activity at a particular voxel from scalp electrical activity. • It is solved via modelling. • In fMRI there is an ill-posed temporal inverse problem. We wish to recover the electrical activity at a voxel from hemodynamic activity at that voxel.

29. HDM & DCM: Conceptual shift • For a given subject and point in brain, the HRF is fixed ! • Need two-stage models (i) How do experimental events affect neurodynamics ? A. Via a bilinear dynamical model (ii) How do neurodynamics affect hemodynamics ? A. Via the balloon model

30. u 1 Bilinear Dynamics Stimuli u1 Set u2 - + Z1 - u 2 + Z 1 + Z2 Z 2 - -

31. Neuronal Transients and BOLD: I Bigger transients produce bigger BOLD signals 300ms 500ms Seconds Seconds More enduring transients produce bigger BOLD signals The interaction changes the shape of the response

32. Seconds Seconds Neuronal Transients and BOLD: II Seconds BOLD is sensitive to frequency content of transients Relative timings of transients are amplified in BOLD

33. Inferences about Neuronal Transients U1,U2,F1,F2 F2 - + Even for a single area we can ask eg.: Does the second presentation of a familiar face (a) increase the magnitude of the neuronal transient ?, (b) increase its time constant ? Z1 - Dt (or fast v. slow responses) Dm

34. Conclusions • Bayesian model selection and averaging can help in the choice of signal and noise models • I have described some useful exploratary tools • Spatial Models • Need to solve fMRI inverse problem

35. Gaussian-smoothed contrast images

36. Wavelet-smoothed contrast images

37. w3 w1 w2 w3 w1 w2 Y Y Analogy: Processing in sensory cortex “Reagan” Evidence Up Posteriors Down