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Fractal-based linear model of resting state hemodynamic response in fMRI
Wonsang You
0
Sophie Achard
1
Jrg Stadler
0
0
Special Lab Non-invasive Brain Imaging, Leibniz Institute for Neurobiology
,
Magdeburg
,
Germany
1
GIPSA-lab
,
CNRS, UMR 5216, Grenoble
,
France
From Twenty First Annual Computational Neuroscience Meeting: CNS*2012
Decatur, GA, USA. 21-26 July 2012
Understanding endogenous dynamics of the brain has
been one of crucial issues in neuroscience since it will
disclose a huge default-mode functional network hidden
behind resting state signals. Recent studies have shown
that a resting state fMRI time series tend to exhibit the
fractal property such as 1/f-type spectral density which
is characterized by a fractal exponent [1]. There exist
indirect evidences supporting that the fractal exponent
is associated with neurophysiological activities [2],
however the relevance of fractal behavior with functional
connectivity has been still unclear.
Recently it was observed that the spontaneous
fluctuation of cerebral blood volume also exhibits fractal
properties [3]. Since fMRI is a non-invasive measurement of
hemodynamic activities, it leads us to consider
attributing the fractal behavior of BOLD signals to cerebral
Figure 1 The plots of classical HRF (without delay) and rs-HRFs with two memory parameters. While the classical HRF may have undershoot, the
rs-HRF has no undershoot but longer tail.
hemodynamics as well as neuronal activities. Motivated
by this idea, we propose a fractal-based linear model of
resting state hemodynamic response function (rs-HRF)
whose behavior is summarized by a fractal exponent
(See Figure 1). It comprises the infinite or large number
of density functions with slowly decaying coefficients
according to a fractal exponent while the classical HRF
consists of just two density functions. We showed that
the special condition of rs-HRF causes long memory in
BOLD signals. The rs-HRF model also implies that a
resting state BOLD signal can be well approximated as a
fractionally integrated process (FIP) rather than the
typical fractional Gaussian noise (FGN) [4].
The simulation studies based on the rs-HRF model
enables us to figure out the influence of hemodynamic
fractal behavior on functional connectivity of resting
state BOLD signals. First, we simulated a multivariate
autoregressive process as an approximation of stationary
resting state neuronal activity, and obtained its
corresponding BOLD signals by convolving them with
rsHRF. Then, we observed the dissimilarity of wavelet
correlation matrices between neuronal activities and BOLD
signals on the basis of the symmetric Kullback-Leibler
divergence which can be utilized as a measure of
difference between two distributions. The dissimilarity of
wavelet correlation increases in high frequency scales as
the variance of fractal exponents increases while the
dissimilarity approaches to zero in low frequency scales.
These results suggest that the difference of fractal
exponents between brain regions may cause apparent
discrepancy of functional connectivity between neuronal
activities and BOLD signals in high frequency scales.
Conclusion
All of these results may give us insight into the
dependence of functional connectivity on fractal behavior, and
direct us to the default-mode functional network
dominated by neuronal activities beyond fractal behavior of
BOLD signals. While the linear relationship between
neuronal activities and BOLD signals in evoked state
has been well represented by HRF, it is questionable
that the classical HRF is appropriate even for resting
state since the model does not well reflect the fractal
properties of cerebral hemodynamics as well as BOLD
signals. In the future work, it will be valuable to
constitute the nonlinear model of fractal behavior induced by
cerebral hemodynamics as an extension of the classical
Balloon model.
(...truncated)