Image Processing Reference
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of the BOLD signal, whereas complex parametric models that rely heavily on a
prior knowledge of nuisance parameters (due to biophysical details), almost never
do not have a reliable and straightforward means of estimation. This fact makes
it unlikely that such comprehensive models will be used as reliable generative
models of the BOLD signal. In the following subsections we describe modeling
issues in greater detail to further underline the limited applicability of many of
the multimodal analysis methods covered in Section 8.4.
8.4.2.1
Convolutional Model of BOLD Signal
Various experimenters had originally focused on simple contrast designs such as
block design paradigms in order to exploit the presumed linearity between their
design parameters and the HR. This assumption depends critically on the ability
of the block design to amplify the SNR, and the implicit belief that the HR
possesses more temporal resolution than indicated by the TR.
In order to account for the present autocorrelation of the HR caused by its
temporal dispersive nature, Friston et al. [120] suggested to model HR with a
LTIS. To describe the output of such a system, a convolution of an input (joint
intrinsic and evoked neuronal activity
q ()
) with a hemodynamic response func-
tion (HRF)
h ()
is used to model the HR
bt
() [
=⊗
h
q t
]()
(8.16)
Localized neuronal activity itself is not readily available via means of non-
invasive imaging, and therefore it is more appropriate to verify LTIS modeling
on real data as a function of parameters of the presented stimuli (i.e., duration,
contrast).
The convolutional model was used on real data to demonstrate linearity
between the BOLD response and the parameters of presented stimuli [121, 122].
In fact, many experimenters have shown apparent agreement between LTIS mod-
eling and real data. Specifically, it has been possible to model responses to longer
stimuli durations by constructing them using the responses to shorter-duration
stimuli, which is consistent with LTIS modeling. Because of the predictive suc-
cess, its relative simplicity of application, and the resulting ignorance of biophys-
ical details, this modeling approach became widely accepted. Unfortunately, LTIS
as a modeling constraint is very weak, therefore allowing an arbitrary choice of
parametric HRF based only on preference and familiarity.
Over the years, multiple models for the HRF have been suggested. The most
popular and widely used up until now is a single probability density function
(PDF) of gamma distribution by [123]. It was elaborated by [124] to perform the
deconvolution of the HR signal, and the nuisance parameters (
ntn t a
11 2 2 2
,, ,,
) of
the next HRF were estimated for motor and auditory areas
n
= %
(
*
i
1
a
c
e
nt
2
ht
()
=
te
n
−/
t t
te
n
−/
t t
where
c
=
max
tte
n
−/
t t
(8.17)
1
1
2
2
i
i
'
i
c
t
1
2
ii
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