Nonparametric Inference of Hemodynamic Response for

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Tingting Zhang
University of Virginia
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Joint work with Fan Li
Data from Duke Department of Psychology
Lab of Ahmad Hariri
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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
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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.
Three stimuli:
Fearful face matching, Angry face matching
and neutral shape matching
 Block Design
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For every TR of 2s, a 3d brain image of
dimension
is acquired
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The total experiment time is 390 s, so there
are195 times points for each voxel
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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.
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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
Extract important quantitative characteristics of
individual HRF estimate to be regressed with
individual NEO scores
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With K different stimuli, the fMRI is modeled as
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Here, we would be interested in estimating
Parametric approaches
usually assume parametric forms of HRF with
only one free parameter measuring the
amplitude (Worsley and Friston, 1995)
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Linear Fit: only magnitude is the free parameter
Nonlinear Fit, using Gauss-Newton algorithm
to estimate six free parameters by minimizing
MSE
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Inverse Logit Regression Model (Lindquist &
Wager 2007)
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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)
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can be rewritten as
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Usually, the
least square
estimate
has an
unnatural
high-frequency
noise
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The smoothing parameters vary for different
HRFs.
For easy computations, we consider do kernel
smoothing on the least square estimate: use
Nadaraya-Watson estimator
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For each stimulus k, we choose the optimal h that minimizes
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We
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and
. Then the kernel
estimate is linear of the least square estimate:
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Because for least square estimate, we have
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Then
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The variance can be estimated by plugging
the OLS estimate of
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In many situations, the matrix
is ill-conditioned.
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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.
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We consider add Tikhonov regularization
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We select the parameters
that minimize
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The bias and variance of the estimate can be easily
estimated
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With large
, a large bias is incurred, so bias
correction is necessary.
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Because
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The new estimator is defined as
10
15
Theoretically
Optimal
Lambda
0
10
5
Frequency
20
30
Theoretically
Optimal
Bandwidth
0
Frequency
40
Histogram of SelectLamda2
0
1
2
3
4
SelectLamda2
5
6
7
2
4
6
h
8
10
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Due to the large individual variance, all the
existing nonparametric methods are
only feasible for magnitude estimation.
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Represent
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Connecting HRF and other subject covariates
with response variables
Interpretation
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