1. The General Linear Model Guillaume Flandin Wellcome Trust Centre for Neuroimaging University College London SPM fMRI Course London, October 2012

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

3. 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? Stimulus function One session A very simple fMRI experiment

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

5. BOLD signal Time =  1 22 + + error x1x1 x2x2 e Single voxel regression model

6. Mass-univariate analysis: voxel-wise GLM = + y y X X Model is specified by 1.Design matrix X 2.Assumptions about e Model is specified by 1.Design matrix X 2.Assumptions about e N: number of scans p: number of regressors N: number of scans p: number of regressors The design matrix embodies all available knowledge about experimentally controlled factors and potential confounds.

7. one sample t-test two sample t-test paired t-test Analysis of Variance (ANOVA) Analysis of Covariance (ANCoVA) correlation linear regression multiple regression GLM: a flexible framework for parametric analyses

8. Parameter estimation = + Ordinary least squares estimation (OLS) (assuming i.i.d. error): Objective: estimate parameters to minimize y X

9. A geometric perspective on the GLM y e Design space defined by X x1x1 x2x2 Smallest errors (shortest error vector) when e is orthogonal to X Ordinary Least Squares (OLS)

10. Problems of this model with fMRI time series 1.The BOLD response has a delayed and dispersed shape. 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.

11. Boynton et al, NeuroImage, 2012. Scaling Additivity Shift invariance Problem 1: BOLD response Hemodynamic response function (HRF): Linear time-invariant (LTI) system: u(t)x(t) hrf(t) Convolution operator:

12. Problem 1: BOLD response Solution: Convolution model

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

14. 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

15. autocovariance function Problem 3: Serial correlations i.i.d:

16. Multiple covariance components = 1 + 2 Q1Q1 Q2Q2 Estimation of hyperparameters with ReML (Restricted Maximum Likelihood). V enhanced noise model at voxel i error covariance components Q and hyperparameters 

17. Weighted Least Squares (WLS) Let Then where WLS equivalent to OLS on whitened data and design WLS equivalent to OLS on whitened data and design

18. A mass-univariate approach Time Summary

19. Estimation of the parameters noise assumptions: WLS: Summary

20. References  Statistical parametric maps in functional imaging: a general linear approach, K.J. Friston et al, Human Brain Mapping, 1995.  Analysis of fMRI time-series revisited – again, K.J. Worsley and K.J. Friston, NeuroImage, 1995.  The general linear model and fMRI: Does love last forever?, J.-B. Poline and M. Brett, NeuroImage, 2012.  Linear systems analysis of the fMRI signal, G.M. Boynton et al, NeuroImage, 2012.

21. Contrasts & statistical parametric maps Q: activation during listening ? c = 1 0 0 0 0 0 0 0 0 0 0 Null hypothesis:

22. stimulus function 1.Decompose data into effects and error 2.Form statistic using estimates of effects and error Make inferences about effects of interest Why? How? data statistic Modelling the measured data linear model linear model effects estimate error estimate

23. Summary  Mass univariate approach.  Fit GLMs with design matrix, X, to data at different points in space to estimate local effect sizes,  GLM is a very general approach  Hemodynamic Response Function  High pass filtering  Temporal autocorrelation

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

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

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