Analytical Techniques

1. Analytical Techniques • Hypothesis Driven • Data Driven • Principal Component Analysis (PCA) • Independent Component Analysis (ICA) • Fuzzy Clustering • Others • Structural equation modeling

2. Matrix Notation of fMRI Data 1 voxel BOLD signal t=1 t=2 t=3 t=4 . . . Voxels X Data Matrix time Slice 1

3. Calculating level of Significance significance: ~ t statistic i/i  G X +  = fMRI Data + = Variability explained by the model total variability noise

4. SPM Nomenclature for Design Matrix G (interesting) H (non-interesting) Covariate Hc G1 Indicator variable Gc H1 Global activity Linear trends E.g. dose of drug subject Design matrix G

5. Some General Linear Model (GLM) Assumptions: • Design matrix known without error • the design matrix is the same everywhere in the brain • the ’s follow a Gaussian distribution • the residuals are well modeled by Gaussian noise • the voxels are temporally aligned • each time point is independent of the others (time courses of voxels are white) • each voxel is independent of the others

6. Regression Coefficients Hypothesis  < 0!!! Global signal Inclusion of Global Signal in Regression Global Signal Hypothesis Test voxel < 5 degrees difference between Global Signal & Hypothesis !

7. 2 X1 db2 X’1 1 db1 Inclusion of Global Covariate in Regression:Effect of non orthogonality b = (GTG)-1GTX 2 db2 X1 X’1 1 “Reference Function, R”

8. Consider an fMRI experiment with only 3 time points

9. Consider an fMRI experiment with only 3 time points

10. Analysis of Brain Systems Correlation viewed as a projection reference function R1 R2 R2 Corr(R2, ref) ref Corr(R1, ref) Although R1 and R2 both somewhat correlated with the reference function, they are uncorrelated with each other R1

11. Principal Component Analysis (PCA) PC1 Voxel 3 Voxel 1 Voxel 2 Voxel 1 Voxel 2 t Voxel 3 Eigenimage + time course

12. Independent Component Analysis (ICA) Without knowing position of microphones or what any person is saying, can you isolate each of the voices?

13. Independent Component Analysis (ICA) Assumption:each sound from speaker unrelated to others (independent)

14. g(C) : Some ICA assumptions • Position of microphones and speakers is constant (mixing matrix constant) • Sources Ergodic • The propagation of the signal from the source to the microphone is instantaneous • Sources sum linearly • Number of microphones equals the number of speakers • In Bell-Sejnowski algorithm, the non-linearity approximates the cdf of the sources

15. ?M S = X Mixing Independent Sources = Data (individuals’ speech) matrix time = C W X Data = Independent Components Weight matrix time Goal of ICA: Find W, so that Kullback-Leibler divergence between f1(C) and f2(S) is minimized ? ‘InfoMax’ algorithm: Iteratively estimate W, so that: g(C) : Key point: maximizing H(y) implies that rows of C are maximally independent Independent Component Analysis (ICA) Goal of ICA: given Data (X), can we recover the sources (S), without knowing M?

16. Non task-related activations (e.g. Arousal) Measured Signal Pulsations Machine Noise Independent Component Analysis (ICA) Task Assumption: spatial pattern from sources of variability unrelated (independent)

17. The fMRI data at each time point is considered a mixture of activations from each component map Mixing COMPONENT MAPS MEASURED fMRI SIGNAL #1 t = 1 S time #2 ‘mixing matrix’, M S t = 2 S n t = n S

18. Selected Components: Transiently task-related Abrupt head movement Quasi-periodic Slow head movement Activated Suppressed Consistently task-related Slowly-varying

19. Comparison of Three Linear Models PCA (2nd order) 4th order ICA (all orders) r = 0.46 r = 0.85 r = 0.92 Increasing spatial independence between components

20. ? 0.4, 1.2, 4.3, -6.9, ... -2.1, 0.2 ... A B å = ICA (all orders) p q 0 A B i i i å = 0 A B i i i 0.1, 1.2, 1.3, -1.9, ... -0.1, 4.2 ... PCA (2nd order) Are Two Maps Independent? Statistically Independent Identical 2nd-order statistics Higher- order statistics Comon’s 4th order Decorrelated

21. Derived Independent Components Histogram of voxel values for component map z > 1 0 ICA Component A component map specified by voxel values 0.4, 1.2, 4.3, -6.9, ... -2.1, 0.2 ... Component map after thresholding associated time course

22. Unexpected Frontal-cerebellar activation detected with ICA Self-paced movement Rest Movie

23. A Transiently task-related (TTR) component (active during first two trials) Martin J. McKeown, CNL, Salk Institute, martin@salk.edu

24. Single trial fMRI Trial 1 ICA component time course Aligned ICA component spatial distribution (a) (b)

25. Single trial fMRI (c) (d) (e) All p < 10-20 19-sec

26. Assessing Statistical Models PRESS Statistic: ^ -i +  G Eliminate 1 time point = Data • How well does G-i match data? • Gives some idea of the influence of the ith time point

27. Exp Exp Exp Exp Con Con Con Con Hybrid Techniques Hypothesis Driven Data Driven

28. HYBICA: L arm pronation/supination hypothesis Hybrid activation

29. S1 Use of HYBICA for Memory Load Hypothesis testing

30. Use of HYBICA for Memory Load Hypothesis testing Maintenance

31. Use of HYBICA for Memory Load Hypothesis testing S2