Nonparametric Inference of Hemodynamic Response for fMRI data

1. Nonparametric Inference of Hemodynamic Response for fMRI data Tingting Zhang University of Virginia

2. Joint work with Fan Li • Data from Duke Department of Psychology • Lab of Ahmad Hariri

3. Real Problem • Data fMRI data under a specific experimental design Neuroticism-Extroversion-Openness (NEO) Inventory measurements: regarding individual personality, for example, Anxiety, Extraversion, Conscientiousness • Goal of the experiment Explore the differences of brain activities across subjects to emotional stimuli Understand the relationship between individual brain functions and their temperament and personality

4. Experiment • Participants completed a standardized protocol comprised of four blocks of a facial expression matching task interleaved with five blocks of a shape-matching control task.

5. Experiment • Three stimuli: Fearful face matching, Angry face matching and neutral shape matching • Block Design

6. The fMRI Data • For every TR of 2s, a 3d brain image of dimension is acquired • The total experiment time is 390 s, so there are195 times points for each voxel

7. fMRI model • Nonlinear ones such as the Balloon model (Buxon et al., 1998; Friston et al., 2000; Riera et al., 2004) in which differential equations are constructed to describe the brain hemodynamics • Linear Models: General Linear Model (GLM) (Friston et al., 1995a; Worsley and Friston, 1995; Goutte et al., 2000) in which fMRI time series are assumed to follow a linear regression of stimulus effects.

8. GLM • Let ,I i=1,…N and t=1,…,T beone fMRI time series for subject i • Let v(t) be the stimulus function, v(t)=1, if the stimulus is evoked at time t, otherwise it equals zero. • GLM: where m is some known constant, and is the hemodynamic response function of ith subject describing the underlying evoked brain activity due to the stimulus

9. HRF Extract important quantitative characteristics of individual HRF estimate to be regressed with individual NEO scores

11. More than one Stimulus • With K different stimuli, the fMRI is modeled as • Here, we would be interested in estimating

12. Existing Methods for Estimation HRF • Parametric approaches usually assume parametric forms of HRF with only one free parameter measuring the amplitude (Worsley and Friston, 1995)

13. Canonical HRF • Linear Fit: only magnitude is the free parameter • Nonlinear Fit, using Gauss-Newton algorithm to estimate six free parameters by minimizing MSE

14. Other Parametric Models • Inverse Logit Regression Model (Lindquist & Wager 2007)

15. Nonparametric • The most flexible approach is to treat HRF at every time point as a free parameter (Glover, 1999; Goutte et al., 2000; Ollinger et al., 2001)

16. The most flexible approach can be rewritten as

17. The least square estimate • Usually, the • least square • estimate • has an • unnatural • high-frequency • noise

18. Least square estimate of one voxel in ROI of one subject

19. Inhomogeneous Variance

20. Smoothing Finite Inverse Regression (SFIR) (Goutteet al. 2000)

22. Kernel Smoothing • The smoothing parameters vary for different HRFs. • For easy computations, we consider do kernel smoothing on the least square estimate: use Nadaraya-Watson estimator

23. Bandwidth Selection • For each stimulus k, we choose the optimal h that minimizes

24. Estimate Bias and Variance • We • and . Then the kernel estimate is linear of the least square estimate:

25. Estimate the MSE • Because for least square estimate, we have • Then • The variance can be estimated by plugging the OLS estimate of

26. HRF EstimationRidge Regression and Smoothing • In many situations, the matrix is ill-conditioned. • Even though is not ill-conditioned, due to the many parameters to be estimated, and the large variance involved, the kernel smoothing is not sufficient to reduce the estimation error. • We consider add Tikhonov regularization

27. We select the parameters that minimize • The bias and variance of the estimate can be easily estimated

28. Bias-Correction • With large , a large bias is incurred, so bias correction is necessary. • Because • The new estimator is defined as

31. Comments • Due to the large individual variance, all the existing nonparametric methods are only feasible for magnitude estimation.

32. B-spline • Represent

33. Boundary

38. Future Research • Connecting HRF and other subject covariates with response variables • Interpretation