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

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

fMRI Analysis Overview Subject 1 Preprocessing MC, STC, B0 Smoothing Normalization First Level GLM Analysis Raw Data C X Subject 2 Preprocessing MC, STC, B0 Smoothing Normalization First Level GLM Analysis Raw Data C X Experimental Design Higher Level GLM Subject 3 Preprocessing MC, STC, B0 Smoothing Normalization CG XG First Level GLM Analysis Raw Data C X Subject 4 Correction for Multiple Comparisons Preprocessing MC, STC, B0 Smoothing Normalization First Level GLM Analysis Raw Data C X

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

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.

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

Review: Estimating the Amplitude y = xh = 16x1 h(t1,D,t) h(t2,D,t) h(t3,D,t) … bm y xh Forward Model: y=xh*bm 16 Equations 1 Unknown bm=Slope=Amp

Review: Contrast Matrix Two Conditions (E1 and E2) X = [xhE1 xhE2], b=[bE1 bE2]T Hypothesis: Response to E1 and E2 are different Null Hypothesis (Ho): bE1 –bE2= 0 Statistic: g = C*b = [c1 c2]*[bE1 bE2]T = 0 g = c1* bE1 + c2*bE2=0 c1= +1, c2 = -1 C = [+1 -1] g>0 Means E1 > E2 g<0 Means E2 > E1 1x2

Noise and Inference

Observed Never Matches Ideal Noise creates uncertainty – need to quantify

Full Model with Noise y – observable s – signal = Xb n – noise, Model: Gaussian, 0-mean, stddev sn, S=I for white noise s=Signal=Xb n=Noise sn y=Observable

Noise Propagation Assumed shape is the same Actual shape is the same Experiment 1 Experiment 2 Assumed shape is the same Actual shape is the same Noise is different Amplitude estimates (slopes) are different (b1, b2) Measurement noise creates uncertainty in estimates

Full Model with Noise

Contrasts and the Full Model

Student’s t-Test and the Null Hypothesis t-value: tDOF DOF = Rows(X)-Cols(X) = TimePoints-Parameters Significance (p-value): area under the curve to the right Null Hypothesis Probability “under the null” (bTrue=0) False Positive Rate (Type I Error) sb btrue b ˆ How to get sb?

Inference Null Hypothesis (H0) = Nothing happening p-value = probability H0 is “true” If p is “small”, then nothing is not happening, then infer that something is happening For publication, p usually must be <.05 AFTER correction for multiple comparisons sb btrue Null Hypothesis b ˆ

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, CO2– aliasing, external monitor, nuisance regressors (RETROICOR), Changes in B0 Endogenous (non-task related) Neural Activation Model Errors Behavioral/Cognitive Variability Wrong assumed shape ?????

Noise Composition Physiological Noise contributes the most in cortex/gray matter. First level (time series) noise generally less than intersubject Greve, et al , 2009, A Novel Method for Quantifying Scanner Instability in fMRI. MRM.

Physiological Origins of the BOLD HRF Artery Vein Capillary 5-10um BOLD HRF: Deoxy Concentration Blood Volume Blood Flow Muscle controls diameter of artery, regulates blood flow

Voxel-wise, no smoothing fMRI Noise Spectrum Voxel-wise, no smoothing Thermal Noise Floor Block Period=60s Longer Shorter fMRI Noise gets much worse at low frequencies. Thermal noise is white (broad spectrum) Physio noise is low frequency

Noise Propagation

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

1 D D = m11* m22 - m12* m21 Review: Matrix Inverse m11 m12 m22 -m12 M*A = I, then A=M-1 Complicated in general Simple for a 2x2 m11 m12 m22 -m12 1 D M = M-1 = m21 m22 -m21 m11 2x2 2x2 D = m11* m22 - m12* m21

1 Review: Invertibility D = 1.0*1.0 - 2.0*0.5 = 1-1 = 0 IMPORTANT!!! Not all matrices are invertible D=0 “Singular” D = 1.0*1.0 - 2.0*0.5 = 1-1 = 0 1.0 2x2 M = 2.0 0.5 1.0 2x2 M-1 = -2.0 -0.5 1

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

Review: GLM Solution y = X*b b=X-1*y = x2-x1 Non-invertible if x1=x2 Intercept: b Slope: m Age x1 x2 y2 y1 y = X*b b=X-1*y 1 x1 x2 -x1 1 D X = X-1 = 1 x2 -1 1 = x2-x1 Non-invertible if x1=x2 Ill-conditioned if x1 near x2 Sensitive to noise y1 y2 1 x1 1 x2 b m = * 26

Noise Propagation through XTX

XTX for Orthogonal Design B xhA xhB

Orthogonal Design (Twice as long) B A B A B A B xhA xhB

XTX for Fully Co-linear Design Auditory and Visual Presented Simultaneously A+V A+V A+V A+V Auditory Regressor xhA xhB Visual Regressor Singular! Does not work! Note: DOF is OK

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

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

Number of Regressors How many regressors can you have? How many regressors should you have? X = Ntpx10 xh1 xh2 1 … 2 3 Tx1 Tx2 Tx3 Ty1 Ty2 Ty3 Tz1 Tz2 Tz3 Rx1 Rx2 Rx3 Ry1 Ry2 Ry3 Rz1 Rz2 Rz3 y = Ntpx1 XTX is 10x10 DOF = Rows(X)-Cols(X) > 0 #Equations > #Unknowns More regressors = more colinearity (less efficiency)

Hemodynamic Response Function Model Error

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

HRF Delay Error Delay error of 1 sec xh y 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

HRF Bias vs Duration and Delay Error As stimulus gets longer, bias gets less

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

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

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 …

Summary Used at higher level for random and mixed effects group analysis Signal Size HRF Errors Timing Errors Acquisition Used at higher level for mixed effects group analysis Measurement Noise Acquisition Parameters Motion Cor Smoothing Nuisance Regressors Stimulus schedule, Variance reduction, Efficiency of Experimental Design (Noise Propagation, Co-linearity), # Regressors

fMRI Analysis Overview Subject 1 Preprocessing MC, STC, B0 Smoothing Normalization First Level GLM Analysis Raw Data C X Subject 2 Preprocessing MC, STC, B0 Smoothing Normalization First Level GLM Analysis Raw Data C X Experimental Design Higher Level GLM Subject 3 Preprocessing MC, STC, B0 Smoothing Normalization CG XG First Level GLM Analysis Raw Data C X Subject 4 Correction for Multiple Comparisons Preprocessing MC, STC, B0 Smoothing Normalization First Level GLM Analysis Raw Data C X

End of Presentation

When Does Model Error Matter? Group 1 Group 2 Actual Estimation Difference 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

Adding Derivatives

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

Noise Propagation Centered at 2.0 (true amplitude) Variance depends on sb btrue (Unknown) 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)

Optimal Rapid Event-related design xhA xhB Periodic Design (N=3) xhA xhB Jittered Design (N=21) 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?

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

Finite Impulse Response (FIR) Model

Finite Impulse Response (FIR) Model y = X*b 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 b1 b2 b3 b4 b5 First 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 + 1TR First Stimulus Onset + 2TR Second Stimulus Onset Second Stimulus + 1TR