Image Processing Reference
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which can be described as the sum of two unscaled PDFs of the gamma distribu-
tion. The first term captures the positive BOLD HR, and the second term is to
capture the overshoot often observed in the BOLD signal. Many other simple, as
well as more sophisticated, models of HRF were suggested: Poisson PDF [120],
Gaussians [125], Bayesian derivations [126-128], and others. The particular choice
of any of them was primarily dictated by some motivation other than biophysics:
easy Fourier transformation, presence of postresponse dip, or best-fit properties.
Since the suggestion of the convolutional model describing the BOLD
response, different aspects of HR linearity became an actively debated question.
If HR is linear, then with what features of the stimulus (e.g., duration, intensity)
or neuronal activation (e.g., firing frequency, field potentials, frequency power)
does it vary linearly? As a first approximation, it is important to define the ranges
of the above-mentioned parameters in which HR was found to behave linearly.
For example, early linearity tests [124] showed the difficulty in predicting long-
duration stimuli based on an estimated HR from shorter-duration stimuli. Soltysik
et al. [129] reviewed existing papers describing different aspects of nonlinearity
in BOLD HR and attempted to determine the ranges of linearity in respect to
stimuli duration in three cortical areas: motor, visual, and auditory complex. The
results of these analyses have shown that although there is a strong nonlinearity
observed on small stimuli durations, long stimuli durations show a higher degree
of linearity.
It appears that a simple convolutional model generally is not capable of
describing the BOLD responses in terms of the experimental design parameters
if these vary over a wide range during the experiment. Nevertheless, LTIS might
be more appropriate to model the BOLD response in terms of neuronal activation
if most of the nonlinearity in the experimental design can be explained by the
nonlinearity of the neuronal activation itself.
8.4.2.2
Neurophysiologic Constraints
In the previous subsection we explored the subject of linearity between the
experimental design parameters and the observed BOLD signal. For the purpose
of this chapter it may be more interesting to explore the relation between neuronal
activity and HR.
It is a well-known fact that E/MEG signals are produced by the large-scale
synchrony of neuronal activity, whereas the nature of the BOLD signal is not
clearly understood. The BOLD signal does not even seem to correspond to the
most energy-consuming neural activity [130], as early researchers believed. Fur-
thermore, the transformation between the electrophysiological indicators of neu-
ronal activity and BOLD signal cannot be linear for a whole dynamic range of
signals, under all experimental conditions, and across all the brain areas. Gener-
ally, a transformation function cannot be linear as the BOLD signal is driven by
a number of nuisance physiologic processes such as cerebral metabolic oxygen
consumption (CMRO
2
), cerebral blood flow (CBF), and cerebral blood volume
(CBV), as suggested by the Balloon model [118], which are not generally linear.
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