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