the general linear model spm for fmri course peter zeidman methods group wellcome trust centre for...

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