Nonparametric Inference of Hemodynamic

Response for fMRI data

Tingting ZhangUniversity of Virginia

Joint work with Fan Li

Data from Duke Department of Psychology Lab of Ahmad Hariri

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

Real Problem

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.

Experiment

Three stimuli: Fearful face matching, Angry face matching

and neutral shape matching Block Design

Experiment

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

The fMRI Data

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.

fMRI model

Let ,I i=1,…N and t=1,…,T be one 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

GLM

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

HRF

With K different stimuli, the fMRI is modeled as

Here, we would be interested in estimating

More than one Stimulus

Parametric approaches usually assume parametric forms of HRF with

only one free parameter measuring the amplitude (Worsley and Friston, 1995)

Existing Methods for Estimation HRF

Linear Fit: only magnitude is the free parameter

Nonlinear Fit, using Gauss-Newton algorithm to estimate six free parameters by minimizing MSE

Canonical HRF

Inverse Logit Regression Model (Lindquist & Wager 2007)

Other Parametric Models

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)

Nonparametric

can be rewritten as

The most flexible approach

Usually, the least square estimate has an unnatural high-frequency noise

The least square estimate

Least square estimate of one voxel in ROI of one subject

Inhomogeneous Variance

Smoothing Finite Inverse Regression (SFIR) (Goutte et al. 2000)

The smoothing parameters vary for different HRFs.

For easy computations, we consider do kernel smoothing on the least square estimate: use Nadaraya-Watson estimator

Kernel Smoothing

For each stimulus k, we choose the optimal h that minimizes

Bandwidth Selection

We

and . Then the kernel

estimate is linear of the least square estimate:

Estimate Bias and Variance

Because for least square estimate, we have

Then

The variance can be estimated by plugging the OLS estimate of

Estimate the MSE

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

HRF EstimationRidge Regression and Smoothing

We select the parameters that minimize

The bias and variance of the estimate can be easily estimated

With large , a large bias is incurred, so bias correction is necessary.

Because

The new estimator is defined as

Bias-Correction

Histogram of SelectLamda2

SelectLamda2

Frequency

0 1 2 3 4 5 6 7

010

2030

40

TheoreticallyOptimalBandwidth

h

Frequency

2 4 6 8 100

510

15

TheoreticallyOptimalLambda

Due to the large individual variance, all the existing nonparametric methods are

only feasible for magnitude estimation.

Comments

Represent

B-spline

Boundary

Connecting HRF and other subject covariates with response variables

Interpretation

Future Research