1. 1 Time Series Analysis of fMRI II: Noise, Inference, and Model Error Douglas N. Greve greve@nmr.mgh.harvard.edu

2. 2 fMRI Analysis Overview Higher Level GLM First Level GLM Analysis First Level GLM Analysis Subject 3 First Level GLM Analysis Subject 4 First Level GLM Analysis Subject 1 Subject 2 CX CX CX CX Preprocessing MC, STC, B0 Smoothing Normalization Preprocessing MC, STC, B0 Smoothing Normalization Preprocessing MC, STC, B0 Smoothing Normalization Preprocessing MC, STC, B0 Smoothing Normalization Raw Data CX

3. 3 Hemodynamic Response Function (HRF) Model Modeling the Entire BOLD Signal Contrasts Noise Propagation Inference Design Efficiency HRF Model Errors Overview

4. 4 Review: Neuro-vascular Coupling Delay Dispersion – spreading out Amplitude Stimulus Delay Dispersion Amplitude

5. 5 Review: Fundamental Concept Amplitude of neural firing is monotonically related to the HRF amplitude. Interpretation: if the HRF amplitude for one stimulus is greater than that of another, then the neural firing to that stimulus was greater. Emphasis is on estimating the HRF amplitude by constructing and fitting models of the HRF and the entire BOLD signal.

6. 6 Parametric Models of the HRF Dale and Buckner, 1997, HBM 5:329-340.  =2.25s,  =1.25s Gamma Model: Parameters:  Delay  Dispersion  Amplitude Model converts a neural “impulse” into a BOLD signal (convolution)

7. 7 Review: Estimating the Amplitude  m =Slope=Amp Forward Model: y=x h*  m 16 Equations 1 Unknown xhxh y y = x h = 16x1 h(t1, ,  ) h(t2, ,  ) h(t3, ,  ) … mm

8. 8 Review: Contrast Matrix Two Conditions (E1 and E2) X = [x hE1 x hE2 ],  E1  E2   Hypothesis: Response to E1 and E2 are different Null Hypothesis (Ho):  E1 –  E2 = 0 Statistic:  C*  c 1 c 2  E1  E2     c 1 *  E1 + c 2  E2 =0 c 1 = +1, c 2 = -1 C = [+1 -1] 1x2  Means E 1 > E 2  Means E 2 > E 1

9. 9 Noise

10. 10 Observed Never Matches Ideal

11. 11 Sources of Temporal Noise Thermal/Background – Gaussian; reduce by temporal averaging and/or spatial smoothing Scanner Instability – drift, instability in electronics Physiological Noise Motion – motion correction, motion regressors Heart Beat – aliasing, external monitor, nuisance regressors (RETROICOR) Respiration, CO 2 – aliasing, external monitor, nuisance regressors (RETROICOR) Endogenous (non-task related) Neural Activation Model Errors Behavioral/Cognitive Variability Wrong assumed shape ?????

12. 12 Noise Composition Physiological Noise contributes the most in cortex/gray matter. First level (time series) noise generally less than intersubject

13. 13 fMRI Noise Spectrum Voxel-wise, no smoothing fMRI Noise gets much worse at low frequencies. Thermal Noise Floor Block Period=60s Longer Shorter

14. 14 Full Model with Noise y – observable s – signal = X  n – noise, Model: Gaussian, 0-mean, stddev  n,  =I for white noise nn s=Signal=X  n=Noise y=Observable

15. 15 Noise Propagation Assumed shape is the same Actual shape is the same Noise is different Amplitude estimates (slopes) are different (  ) Measurement noise creates uncertainty in estimates Experiment 1Experiment 2

16. 16 Full Model with Noise

17. 17 Contrasts and the Full Model

18. 18 Propagation of Noise Noise in the observable gets transferred to the contrast through (X T X) -1. The properties of (X T X) are important! Singularity Invertibility Efficiency Condition

19. 19 M*A = I, then A= M -1 Complicated in general Simple for a 2x2 Review: Matrix Inverse m 11 2x2 M = m 12 m 21 m 22  = m 11 * m 22 - m 12 * m 21 m 22 2x2 M -1 = -m 12 -m 21 m 11 

20. 20 Review: Invertibility 1.0 2x2 M = 2.0 0.51.0  = 1.0*1.0 - 2.0*0.5 = 1-1 = 0 1.0 2x2 M -1 = -2.0 -0.51.0  IMPORTANT!!! Not all matrices are invertible  =0 “Singular”

21. 21 Review: Singularity and “Ill-Conditioned” 1.0 2x2 M = 2.0 0.51.0  = 1.0*1.0 - 2.0*0.5 = 1-1 = 0 Column 2 = twice Column 1 Linear Dependence Ill-Conditioned:  is “close” to 0 Relates to efficiency of a GLM.

22. 22 Review: GLM Solution Intercept: b Slope: m Age x1x2 y2 y1 y = X*   =X -1 *y y1 y2 1 x1 1 x2 bmbm =* 1 X = x1 1x2 X -1 = -x1 1 11  = x2-x1 Non-invertible if x1=x2 Ill-conditioned if x1 near x2 Sensitive to noise

23. 23 Noise Propagation through X T X

24. 24 X T X for Orthogonal Design A BBA x hA x hB

25. 25 Orthogonal Design (Twice as long) A BBA x hA x hB A BBA

26. 26 X T X for Fully Co-linear Design A+V x hA x hB Auditory and Visual Presented Simultaneously Auditory Regressor Visual Regressor Singular! Does not work! Note: DOF is OK

27. 27 X T X for Partially Co-linear Design x hA x hB Working Memory: Encode and Probe Encode Regressor Probe Regressor Note: textbook (HSM) suggests orthogonalzing. This is not a good idea. This design requires a lot of overlap which reduces the determinant, but this does not mean that it is a bad design.

28. 28 Co-linearity and Invertibility What causes co-linearity? Synchronized presentations/responses Many regressors –Derivatives –Basis sets (eg, FIR) –Lots of nuisance regressors Noise tends to be low-frequency Task tends to be low-frequency Only depends on X Note: more stimulus presentations generally improves things

29. 29 Number of Regressors y = N tp x1 How many regressors can you have? How many regressors should you have? X = N tp x10 x h1 x h2 111…111… 123…123… T x1 T x2 T x3 … T y1 T y2 T y3 … T z1 T z2 T z3 … R x1 R x2 R x3 … R y1 R y2 R y3 … R z1 R z2 R z3 … DOF = Rows(X)-Cols(X) > 0 #Equations > #Unknowns More regressors = more colinearity (less efficiency) X T X is 10x10

30. 30 Hemodynamic Response Function Model Error

31. 31 HRF Model Error Systematic deviation between the assumed shape and the actual shape Sources –Delay, Dispersion –Form (undershoot) –Duration (stimulus vs neural) –Non-linearity

32. 32 HRF Delay Error Delay error of 1 sec Loss of amplitude/slope (Bias), smaller t-values Larger “noise” – ie, residual error. –Cannot be fixed by more acquisitions (bias) Scaling still preserved xhxh y

33. 33 HRF Bias vs Duration and Delay Error As stimulus gets longer, bias gets less 2s 5s 10s

34. 34 HRF Bias vs Duration and Shape Error HSM Fig 9.10 As stimulus gets longer, overall shape gets similar even if individual shapes are very different

35. 35 Does Neural Activation Match Stimulus? May be shorter – eg, due to habituation, not needing as much time to process information May be longer – eg, emotional stimuli May not be constant 2 5 10

36. 36 Does Model Error Matter? More False Negatives (Type II Errors) –Loss of amplitude –More noise (usually trivial compared to other sources) Scaling still preserved Real Problem: Systematic Errors across … –Subjects, Brain Areas, Time …

37. 37 When Does Model Error Matter? Actual Estimation Difference Group 1Group 2 Groups have same true amplitude but different delays Estimated amplitudes are systematically different False Positives (or False Negatives) Groups could be from different: populations (Normals vs Schizophrenics), times (longitudinal), brain regions

38. 38 Adding Derivatives

39. 39 Still have a Problem Fit to observed waveform is better (residual variance less) But now have two Betas x h and derivative are orthgonal, so  m does not change (bias is not removed). No good solution to the problem (maybe Calhoun 2004)

40. 40 End of Presentation

41. 41 Noise Propagation Centered at 2.0 (true amplitude) Variance depends on –noise variance, number of samples, a few other things Uncertainty Type II Errors (false negatives) Type I Errors (false positives)   true (Unknown)

42. 42 “Student’s” t-Test t-value: t DOF DOF = Rows(X)-Cols(X) = TimePoints-Parameters Significance (p-value): area under the curve to the right   true Null Hypothesis  ˆ Probability “under the null” (  True =0) False Positive Rate (Type I Error) Null Hypothesis How to get   

43. 43 Optimal Rapid Event-related design Lots of overlap in jittered design Should push apart so no overlap? No overlap means fewer stimuli (fixed scanning time) How to balance? How to schedule? Periodic Design (N=3) Jittered Design (N=21) x hA x hB x hA x hB

44. 44 Optimal Experimental Design Efficiency  only depends on X and C X depends on stimulus onset times and number of presentations Choose stimulus onset times to max  Can be done before collecting data! Can interpret  as a variance reduction factor

45. 45 Finite Impulse Response (FIR) Model

46. 46 Finite Impulse Response (FIR) Model 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0  y = X*  First Stimulus Onset First Stimulus Onset + 1TR Second Stimulus Onset Average at Stimulus Onset Average Delayed by 1TR Average Delayed by 2TR Average Delayed by 3TR Average Delayed by 4TR First Stimulus Onset + 2TR Second Stimulus + 1TR