Functional Brain Signal Processing: EEG & fMRI Lesson 15

1. M.Tech. (CS), Semester III, Course B50 Functional Brain Signal Processing: EEG & fMRILesson 15 Kaushik Majumdar Indian Statistical Institute Bangalore Center kmajumdar@isibang.ac.in

2. Poldrack et al., 2011, Section 6.1 Group Statistics: An Example

3. Fixed Effect vs. Mixed Effect Model is fixed effect variance in average hair length in a gender group. is within group variance, which is assumed to be 1 here. is the average mixed effect variance, where is the between the groups variance.

4. Poldrack et al., 2011 Mixed Effect Model in fMRI Here β = a.

5. http://en.wikipedia.org/wiki/File:Linear_regression.svg Linear Regression Analysis

6. Multilevel GLM for Group Analysis Yk is a vector of T time points. k є{1,…..,N}, where N is the number of subjects in the group. Mk is the design matrix of model functions and ek is the error vector (for the kth subject with one element in the vector for each time point.

7. Beckmann et al., 2003 Two Level Model • Level 1: individual analysis • Level 2: group analysis (1) MG is the group level design matrix, aG is a vector of group level parameters d is the residual vector of group level parameters.

8. Two Level Model as A Single Level Model This is equivalent to the two level model described in the previous slide.

9. Parameter Estimation at Two Levels • Linear spaces generated by the columns of M and MTM are the same (Rao, 1974, p. 222). Proof: Let λ be an eigenvalue of M. Then there is an eigenvector v such that Mv = λv or M = λI MT = λI. So, MTMv = MTλv = λ2v. In other words M and MTM have same eigenvectors and therefore generate the same eigen space.

10. Parameter Estimation (cont) In general Y = Ma may be inconsistent (may not have unique solution), but MTMa = MTY always has a unique solution in a, because MTY is in the space generated by columns of MTM. Let â be a solution of MTMa = MTY, then (Y – Ma)T(Y – Ma) = [Y – Mâ + M(â – a)]T[Y – Mâ + M(â – a)] = (Y – Mâ)T(Y – Mâ) + (â – a)TMTM(â – a) ≥ (Y – Mâ)T(Y – Mâ). This shows that the minimum of (Y – Ma)T(Y – Ma) is (Y – Mâ)T(Y – Mâ) and is attained for

11. Parameter Estimation (cont) a = â, which is unique for all solutions â of MTMa = MTY.

12. Solution for Two Level GL Model For individual. For group. This together with (1) gives the second level estimation of the parameters.

13. Poldrack et al., 2011 Inference of BOLD Activation

14. Buxton, 2009 Nature of BOLD Signal CBF = Cerebral blood flow. CMRO2 = Cerebral metabolic rate of O2. CBV = Cerebral blood volume.

15. BOLD Components • BOLD response is primarily driven by CBF, but also strongly modulated by two other factors: • , and • M, which reflects level of deoxyhemoglobin at the baseline.

16. Buxton, 2009 BOLD is Best Captured in Gradient Recall Echo (GRE) Imaging

17. References • R. A. Poldrack, J. A. Mumford and T. E. Nichols, Handbook of Functional MRI Data Analysis, Cambridge University Press, Cambridge, New York, 2011. Chapter 6. • C. F. Beckmann, M. Jenkinson and S. M. Smith, General multilevel linear modeling for group analysis in fMRI, NeuroImage, 20: 1052 – 1063, 2003.

18. References (cont) • C. R. Rao, Linear Statistical Inference and Its Applications, 2e, Wiley Eastern Ltd., New Delhi, 1974, Chapter 4 (Theory of least squares and analysis of variance).

19. THANK YOUThis lecture is available at http://www.isibang.ac.in/~kaushik