The General Linear Model

1. The General Linear Model SPM for fMRI Course Peter Zeidman Methods Group Wellcome Trust Centre for Neuroimaging

2. Overview • Basics of the GLM • Improving the model • SPM files http://www.fil.ion.ucl.ac.uk/~pzeidman/teaching/GLM.ppt

3. Basics of the GLM

4. Image time-series Statistical Parametric Map Design matrix Spatial filter Realignment Smoothing General Linear Model StatisticalInference RFT Normalisation p <0.05 Anatomicalreference Parameter estimates

5. A very simple fMRI experiment One session Passive word listening versus rest 7 cycles of rest and listening Blocks of 6 scans with 7 sec TR Question: Is there a change in the BOLD response between listening and rest?

6. Modelling the measured data Make inferences about effects of interest Why? • Decompose data into effects and error • Form statistic using estimates of effects and error How? effects estimate linear model statistic data error estimate

7. Single voxel regression model error = + + 1 2 Time e x1 x2 BOLD signal

8. Mass-univariate analysis: voxel-wise GLM X + y = • Model is specified by • Design matrix X • Assumptions about e N: number of scans p: number of regressors The design matrix embodies all available knowledge about experimentally controlled factors and potential confounds.

9. Model specification Parameter estimation Hypothesis Statistic Voxel-wise time series analysis Time Time BOLD signal single voxel time series SPM

10. Improving the model

11. HRF What are the problems of this model? • BOLD responses have a delayed and dispersed form. • The BOLD signal includes substantial amounts of low-frequency noise (eg due to scanner drift). • Due to breathing, heartbeat & unmodeled neuronal activity, the errors are serially correlated. This violates the assumptions of the noise model in the GLM

12. Problem 1: Shape of BOLD responseSolution: Convolution model Expected BOLD HRF Impulses  = expected BOLD response = input function impulse response function (HRF)

13. Convolution model of the BOLD response Convolve stimulus function with a canonical hemodynamic response function (HRF):  HRF

14. blue= data black = mean + low-frequency drift green= predicted response, taking into account low-frequency drift red= predicted response, NOT taking into account low-frequency drift Problem 2: Low-frequency noise Solution: High pass filtering discrete cosine transform (DCT) set

15. High pass filtering discrete cosine transform (DCT) set

16. Problem 3: Serial correlations with 1st order autoregressive process: AR(1) autocovariance function

17. Multiple covariance components enhanced noise model at voxel i error covariance components Q and hyperparameters V Q2 Q1 1 + 2 = Estimation of hyperparameters  with ReML (Restricted Maximum Likelihood).

18. Spm files

19. 1.Specify the model

20. 1.Specify the model

21. SPM files (after specifying the model)

22. SPM.mat(after specifying the model) SPM.xY – Filenames of fMRI volumes SPM.Sess – Per-session experiment timing SPM.xX – Design matrix For documentation on these structures, type: help spm_spm

23. SPM.xX (Design matrix) Design matrix imagesc(SPM.xX.X);

24. SPM.xX (Design matrix) Confounds (HPF) imagesc(SPM.xX.K.X0);

25. 2. Estimate the model

26. SPM files (after estimation)

27. SPM files (after estimation) beta_0001.nii – beta_0004.nii mask.nii

28. SPM files (after estimation) RPV.nii ResMS.nii Residual variance estimate Estimated RESELS per voxel

29. SPM files (after estimation)

30. SPM files (after contrast estimation)

31. Summary • We specify a general linear model of the data • The model is combined with the HRF, high-pass filtered and serial correlations corrected • The model is applied to every voxel, producing beta images. • Next we’ll compare betas to make inferences http://www.fil.ion.ucl.ac.uk/~pzeidman/teaching/GLM.ppt