FMRI Time Series Analysis

1. FMRI Time Series Analysis Mark Woolrich & Steve Smith Oxford University Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB)

2. FMRI Time Series AnalysisOverview • Noise modelling (autocorrelation) • Signal modelling: • Complex parameterised HRF model • Optimised basis functions for HRF modelling

3. Reminder - Multiple Regression Model (GLM)

4. Temporal Autocorrelation

5. FMRI Noise • Time series from each voxel contains low frequency drifts and high frequency noise • Drifts are scanner-related and physiological (cardiac cycle, breathing etc) • Both high and low frequency noise hide activation Power Spectral Density

6. High-pass Filtering • Removes the worst of the low frequency trends High-pass

7. High-Frequency Noise • Unless high-frequency noise is modelled or corrected for: • Incorrect stats (probably false positives) • Inefficient stats (false negatives)

8. Temporal Filtering and the GLM S is a matrix for temporal filtering Prewhitening (Bullmore et al ‘96) S = K-1 Best Linear Unbiased Estimator Precolouring (Worsley et al ‘95) S = L (L is a low pass filter) Smothers intrinsic autocorrelation

9. Different Regressors Fixed ISI single-event with jitter Randomized ISI single-event Boxcar Regressor Low-pass Low-pass Low-pass FFT

10. Sensitivity

11. FMRIB's Improved Linear Modelling (FILM) Performs prewhitening LOCALLY: • Fit the GLM and estimate the raw autocorrelation on the residuals • Spectrally and spatially smooth autocorrelation estimate • Construct prewhitening filter to "undo" autocorrelation • Use filter on data and design matrix and refit

12. FMRIB's Improved Linear Modelling (FILM) Performs prewhitening LOCALLY: • Fit the GLM and estimate the raw autocorrelation on the residuals • Spectrally and spatially smooth autocorrelation estimate • Construct prewhitening filter to "undo" autocorrelation • Use filter on data and design matrix and refit

13. Spectral Smoothing Autocorrelation estimate Raw autocorrelation IFFT FFT Power Spectral Density Tukey taper smoothed

14. Spatial Smoothing Autocorr EPI

15. Non-linear Spatial Smoothing Gaussian spatial smoother with weights: Ii= EPI signal intensity t = brightness threshold

16. Unbiased Statistics • P-P plots for FILM on 6 null datasets Boxcar Single Event

17. Session Effects Investigation • 3 paradigms x 33 “identical” sessions (McGonigle 2000) • Variety of 1st-level analyses - use group-level mixed-effects-Z to judge efficiency of first-level analysis

18. Session EffectsME-Z Plots

19. Autocorrelation Conclusions • Precolouring is nearly as sensitive as prewhitening for boxcar designs • Single-event designs require prewhitening for increased sensitivity • Local autocorrelation estimation using a Tukey taper with nonlinear spatial smoothing produces close to zero bias when prewhitening • More advanced: need spatiotemporal noise model: • Model-based (Woolrich) • Model-“free” (Beckmann)

20. Signal Modelling • Start with stimulation timings • Several conditions (original Evs)? • Convolve with HRF to blur and delay • What choice of HRF? • Does it vary across subjects? • Does it vary across the brain? • “Advanced” issues: • Allow signal height to change over time (dynamic)? • Use nonlinear convolution (events interact)? • Spatiotemporal modelling

21. HRF Modelling

22. Linear (time invariant) System Experimental Stimulus, e.g. boxcar Parameterised HRF, e.g. Gamma function Assumed response

23. HRF Parameterisation Half-cosine parameterisation Prior samples ?

24. HRF Parameterisation Half-cosine parameterisation Model Selection Model 1(no undershoot): c2 = 0 Model 2(undershoot): c2 ≠ 0 ?

25. Automatic Relevance Determination (ARD) Prior(Mackay 1995) • Relevance of a parameter is automatically determined by the parameter then with high precision : Model 1 then is non-zero : Model 2 MCMC

26. ARD of Undershoot Simulated data with no undershoot No ARD prior ARD prior True value True value

27. Prior samples HRF Results Boxcar Jittered single-event Randomised single-event Boxcar Jittered Single-event Randomised Single-event Response fits Marginal posterior samples Posterior samples Posterior samples Posterior samples

28. HRF Basis Sets

29. Basis Sets for HRF Modelling • Basis functions in the GLM: instead of one fixed HRF we can have several • Here an F across all 3 betas finds the best linear combination of the 3 HRFs • 3 Original EVs now become 6 (2 submodels each with 3 HRFs)

30. 4 HRF basis functions partial model fits full model fits

31. Basis Sets for HRF Modelling MCMC • Instead of a parameterised HRF • We can use a linear basis set to span the space of expected HRF shapes Variational Bayes WHY? : We can then use the basis set in an easier to infer GLM

32. Generating HRF Basis Sets (1) Take samples of the HRF

33. Generating HRF Basis Sets (1) Take samples of the HRF (2) Perform SVD

34. Generating HRF Basis Sets (1) Take samples of the HRF (2) Perform SVD (3) Select the top eigenvectors as the optimal basis set

35. Unconstrained Basis Set BUT: The basis set spans a wider range of HRF shapes than we want to allow: HRF samples from prior Unconstrained basis set

36. Constrained Basis Set We can regress the HRF samples back on to the basis set Basis set HRF samples from prior

37. Constrained Basis Set and fit a multivariate normal to the basis set parameter space Basis set HRF samples from prior

38. Constrained Basis Set This contrains the basis set to give only sensible looking HRF shapes HRF samples from prior Samples from basis set Unconstrained Constrained

39. Using the Constrained Basis Set • The multivariate normal on the basis set parameters can then be used as a prior on those parameters in the GLM Constrained basis set Variational Bayes

40. Constrained Basis Set - Results

41. Constraining Basis Sets Results in Increased Sensitivity

42. Acknowledgements FMRIB Analysis Group UK EPSRC, MRC, MIAS-IRC