The General Linear Model SPM for fMRI Course Peter Zeidman/Christophe Phillips 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 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

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?

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

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

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.

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

Improving the model

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

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

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

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

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

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

= 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).

Spm files

1.Specify the model

1.Specify the model

SPM files (after specifying the model)

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

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 RPV.nii Residual variance estimate Estimated RESELS per voxel

SPM files (after estimation)

SPM files (after contrast estimation)

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