The General Linear Model SPM for fMRI Course Peter Zeidman Methods Group Wellcome Trust Centre for Neuroimaging
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The General Linear Model
SPM for fMRI Course
Peter Zeidman
Methods Group
Wellcome Trust Centre for Neuroimaging
Overview
• Basics of the GLM
• Improving the model
• SPM files
http://www.fil.ion.ucl.ac.uk/~pzeidman/teaching/GLM.ppt
BASICS OF THE GLM
Statistical Parametric MapStatistical Parametric Map
StatisticalStatisticalInferenceInference
RFTRFT
p <0.05p <0.05
NormalisationNormalisation
Image time-seriesImage time-series
RealignmentRealignment SmoothingSmoothing
AnatomicalAnatomicalreferencereference
Spatial filterSpatial filter
Parameter estimatesParameter estimates
General Linear ModelGeneral Linear Model
Design matrix
Passive word listeningversus rest
7 cycles of rest and listening
Blocks of 6 scanswith 7 sec TR
Question: Is there a change in the BOLD response between listening and rest?
One session
A very simple fMRI experiment
1. Decompose data into effects and error
2. Form statistic using estimates of effects and error
Make inferences about effects of interest
Why?
How?
datastatisti
c
Modelling the measured data
linearmode
l
effects estimate
error estimate
BOLD signal
Tim
e =1 2+ +
err
or
x1 x2 e
Single voxel regression model
exxy 2211 exxy 2211
Mass-univariate analysis: voxel-wise GLM
=
e+yy XX
N
1
N N
1 1p
p
Model is specified by1. Design matrix X2. Assumptions
about e
Model is specified by1. Design matrix X2. Assumptions
about e
N: number of scansp: number of regressors
N: number of scansp: number of regressors
eXy eXy
The design matrix embodies all available knowledge about experimentally controlled factors and potential
confounds.
),0(~ 2INe ),0(~ 2INe
Voxel-wise time series analysis
Time
single voxeltime series
single voxeltime series
BOLD signal
Tim
e
Modelspecification
Modelspecification
Parameterestimation
Parameterestimation
HypothesisHypothesis
StatisticStatistic
SPMSPM
IMPROVING THE MODEL
What are the problems of this model?
1. BOLD responses have a delayed and dispersed form.
HRF
2. The BOLD signal includes substantial amounts of low-frequency noise (eg due to scanner drift).
3. Due to breathing, heartbeat & unmodeled neuronal activity, the errors are serially correlated. This violates the assumptions of the noise model in the GLM
t
dtgftgf0
)()()(
Problem 1: Shape of BOLD responseSolution: Convolution model
expected BOLD response = input function impulse response function (HRF)
=
Impulses HRF Expected BOLD
Convolution model of the BOLD response
Convolve stimulus function with a canonical hemodynamic response function (HRF):
HRF
t
dtgftgf0
)()()(
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
discrete cosine transform (DCT) set
discrete cosine transform (DCT) set
High pass filtering
withwithttt aee 1 ),0(~ 2 Nt
1st order autoregressive process: AR(1)
)(eCovautocovariance
function
N
N
Problem 3: Serial correlations
Multiple covariance components
= 1 + 2
Q1 Q2
Estimation of hyperparameters with ReML (Restricted Maximum Likelihood).
V
enhanced noise model at voxel i
error covariance components Qand hyperparameters
jj
ii
QV
VC
2
),0(~ ii CNe
SPM FILES
1.Specify the model
1.Specify the model
SPM files (after specifying the model)
SPM.mat (after specifying the model)
SPM.xX – Design matrix
For documentation on these structures, type: help spm_spm
SPM.xY – Filenames of fMRI volumes
SPM.Sess – Per-session experiment timing
SPM.xX (Design matrix)
Design matrix
imagesc(SPM.xX.X);
SPM.xX (Design matrix)
Confounds (HPF)
imagesc(SPM.xX.K.X0);
2. Estimate the model
SPM files (after estimation)
SPM files (after estimation)
beta_0001.nii – beta_0004.nii mask.nii
SPM files (after estimation)
ResMS.nii
Residual variance estimate
RPV.nii
Estimated RESELS per voxel
SPM files (after estimation)
SPM files (after contrast estimation)
Summary
1. We specify a general linear model of the data
2. The model is combined with the HRF, high-pass filtered and serial correlations corrected
3. The model is applied to every voxel, producing beta images.
4. Next we’ll compare betas to make inferences
http://www.fil.ion.ucl.ac.uk/~pzeidman/teaching/GLM.ppt
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