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

2. Overview Basics of the GLM Improving the model SPM files

3. Basics of the GLM

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

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. Make inferences about effects of interest
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. = + + error 1 2 x1 x2 e Single voxel regression model Time
BOLD signal

8. X y + = Mass-univariate analysis: voxel-wise GLM
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. Voxel-wise time series analysis
single voxel
time series
BOLD signal
Time
Model
specification
Parameter
estimation
Hypothesis
Statistic
SPM

10. Improving the model

11. What are the problems of this model?
BOLD responses have a delayed and dispersed form.
HRF
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 response Solution: Convolution model
HRF
Expected BOLD
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. discrete cosine transform (DCT) set
Problem 2: Low-frequency noise Solution: High pass filtering
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
discrete cosine transform (DCT) set

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

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

17. = 1 + 2 V Q1 Q2 Multiple covariance components
enhanced noise model at voxel i
error covariance components Q
and hyperparameters V Q1 Q2 = 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)
ResMS.nii
RPV.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