Ch. 5 Bayesian Treatment of
Neuroimaging Data
Will Penny and Karl Friston
18 Dec. 2008
summarized by Soo-Jin Kim
Contents
1.
2.
3.
Summary
The General Linear Model (GLM)
Parameter Estimation
1.
2.
4.
5.
ML for GLM
Bayes Rule for GLM
Posterior Probability Mapping (PPM)
Dynamic Causal Modeling (DCM)
© 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/
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Summary

A general issue in the analysis of brain imaging data
 the relationship between the neurobiological hypothesis one
posits and the statistical models adopted to test that
hypothesis

fMRI: functional maps showing which regions are
specialized for specific functions
 General linear model
 Analysis of functional specialization

Posterior probability maps (PPM)
 Analysis of functional integration

Dynamic causal modeling (DCM)
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Design matrix

Design matrix for study of
attention to visual motion
Time
point
 four columns: Photic stimulation,
Motion, Attention and a constant
 rows : time points in the imaging
time series (360 images)

The relative contribution of each
of these variables can be assessed
using standard least squares or
Bayesian estimation.
Designed effect
 Classical inferences : T or F statistics
 Bayesian inferences: posterior or
conditional probability
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General Linear Model

The general linear model is an equation that
expresses an observed response variable y in terms of
a linear combination of explanatory variables X
where y is a T × 1 vector comprising responses at, e.g. T time points, X is
a T × K design matrix, β is a vector of regression coefficients, and e is a
T × 1 error vector

GLMs are fitted to each voxel using ML or OLS estimators.
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Temporal Basis Functions (1/2)
Functional MRI using blood oxygen level dependent
(BOLD) contrast provides an index of neuronal
activity indirectly via changes in blood oxygenation
levels
 Temporal basis functions are important because they
enable a graceful transition between conventional
multilinear regression models with one stimulus
function per condition and FIR models with a
parameter for each time point following the onset of
a condition.

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Temporal Basis Functions (2/2)

The convolution model for fMRI responses takes a stimulus function
encoding the supposed neuronal responses and convolves it with an
HRF to give a regressor that enters into the design matrix
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Parameter Estimation: ML

Maximum Likelihood
 N(x; m, Σ) specifies a multivariate Gaussian distribution over
x with mean m and covariance Σ. The likelihood specified by
the GLM described in the previous section is given by
p(y|β) = N(y;X β,Ce), where Ce is an error covariance
matrix.
 The maximum-likelihood (ML) estimator is given by [25]:
βML=(XTCe-1X)-1XTCe-1y
 For other modalities, Ce=σ2I often suffices. The ML estimate
then reduce to the ordinary least squares (OLS) estimator
βOLS=(XTX)-1XTy
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Parameter Estimation: Bayesian

Bayes rule for GLMs
 The Bayesian framework allows us to incorporate our prior
beliefs about parameter values.
 If our beliefs can be specified using the Gaussian distribution
 P(β )=N(β; μp, Cp), where μp is the prior mean and Cp is the prior
covariance, then the posterior distribution is [18]
p(β |Y ) = N(β; μ,C), Where
C-1=XTCe-1X+Cp-1 (posterior precision)
μ=C(XTCe-1y+Cp-1 μp (posterior mean)
 In the abscence of prior information, i.e. Cp-1= 0, the above
estimate reduces to the ML estimate.
 If the priors are correct, then Bayesian estimation is more
accurate than ML estimation.
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Bayesian estimation: Nonlinear case
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Posterior Probability Mapping (PPM)

Posterior probability map ( p(β > γ|Y ) > α ): images of
the probability that an activation exceeds some specified
threshold, given the data
 Posterior distribution (p(β |Y ) ): probability of getting an effect,
given the data
Activation threshold 
Probability 
Here, we used an
effect-size threshold of
0.7% of whole-brain
mean, and a probability
threshold
of 0.95.
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SPM vs. PPM

Maximum intensity projections (MIPs) of SPM (left)
and PPM (right) for the fMRI study of attention to
visual
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SPM vs. PPM

The difference bwt SPM and PPM
 The SPM identifies a smaller number of voxels than the PPM

The SPM appears to have missed a critical and bilaterally
represented part of the V5 complex
 The SPM is more conservative because the correction for
multiple comparisons
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Dynamic causal modeling (DCM)
Dynamic causal modeling (DCM) is used to make
inferences about functional integration and has
been formulated for the analysis of fMRI time series
in Friston et al.
 Current DCMs for fMRI comprise a bilinear model
for the neurodynamics and an extended balloon
model for the hemodynamics
z

λ
 The modelled neuronal dynamics (z) is transformed into
area-specific BOLD signals (y) by a hemodynamic forward
model (λ)
y
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DCM: bilinear state equation

The neurodynamics are described by the multivariate
differential equation
 This is known as a bilinear model because the dependent variable,
ż, is linearly dependent on the product of uj and z.
m
z  ( A   u j B j ) z  Cu
j 1
state
changes
intrinsic
connectivity
 z1    a11
   
  
 zn    an1

modulation of
connectivity
a1n  m b11j
 u 
j 
 
j 1
bnj1
ann 

b1jn  


bnnj  
system
state
direct
inputs
 z1   c11
 
  
 zn  cn1
© 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/
m external
inputs
(known)
c1m   u1 
 
 
cnm  um 
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DCM: balloon model
activity



In brief, for the ith region, neuronal
activity zi causes an increase in the
vasodilator signal si that is subject
to autoregulatory feedback. Inflow
fi responds in proportion to this
signal with concomitant changes in
blood volume vi and
deoxyhemoglobin content qi.
This process converts neuronal
activity in the ith region zi to the
hemodynamic response.
The parameters θ = {A,B,C, h} are
updated until convergence.
z(t )
vasodilatory signal
s  z  s  γ( f  1)
flow induction
f  s
changes in volume
changes in dHb
τv  f  v 1 /α
τq  f E ( f, ) q  v1/α q/v
BOLD signal
y (t )
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DCM for the fMRI study of attention
to visual motion

The interesting aspects of
this connectivity involve the
role of motion and attention
in exerting bilinear effects.
 The effect of motion in the
visual field was modeled
as a bilinear modulation of
the V1 to V5 connectivity
 The effect of attention was
allowed to modulate the
backward connections
from IFG and SPC.
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