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healthy individuals, whereas Figure
5.2
(b) depicts the average exponent and singular-
ity spectrum for fourteen measurements of eight migraineurs. The convex form of the
singularity spectrum indicates that the CBF is multifractal. However, the degree of vari-
ability, as measured by the width of the singularity distribution, is vastly restricted for
the migraineurs compared with healthy individuals. Note that the partition function was
calculated by West
et al
.[
32
] using wavelet transforms.
The fractional multiplicative Langevin equation from the previous subsection pro-
vides one possible explanation of this effect. Consider the relation between the two
scaling exponents
ζ(
q
)
=
2
−
τ(
q
),
(5.95)
so that, since from (
5.87
)
ζ(
0
)
=
H
,wehave
τ(
0
)
=
2
−
H
,
(5.96)
resulting in the well-known relation between the fractal dimension and the global Hurst
exponent
D
(
0
)
=
2
−
H
.
(5.97)
When the multiplicative stochastic parameter in the fractional Langevin equation has
Gaussian statistics we obtain, by inserting
ζ(
q
)
into (
5.93
), the singularity spectrum
2
)
−
(
h
−
H
)
f
(
h
)
=
f
(
H
,
(5.98)
2
σ
for which the mode of the spectrum is located by
(
)
=
−
f
H
2
H
(5.99)
and the quadratic term arises from
q
The solid curves in Figure
5.2
are the best least-square fits of (
5.98
) to the empirical singularity spectrum.
A significant change in the multifractal properties of the MCAfv time series of the
migraineurs is apparent. Namely, the interval for the singularity distribution
f
=
(
h
−
H
)/(
2
σ).
on
the local Hurst exponent
h
is vastly constricted. This is reflected by the small value of
the width of the singularity spectrum squared,
(
h
)
013 for the migraineurs, which is
almost three times smaller than the value for the control group
σ
=
0
.
σ
=
0
.
038. In both cases
the distribution is centered at
H
were determined
via the least-square fit to (
5.98
). This collapse of the singularity spectrum suggests that
the underlying process has lost its flexibility. The biological advantage of multifractal
processes is that they are highly adaptive, so that in this case the brain of a healthy
individual adapts to the multifractality of the interbeat interval time series. Here again
we see that disease, in this case migraine, may be associated with the loss of complexity
and consequently the loss of adaptability, thereby suppressing the normal multifractality
of blood-flow time series.
=
0
.
81. The parameters
H
and
σ
