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In-Depth Information
Classical concept of the synchronization of two oscillators is described as an
active adjustment of their rhythmicity that manifests in phase locking between the
synchronized oscillators. Specifically, given two signals
X
1
(
t
)
and
X
2
(
t
)
, and their
corresponding instantaneous phases
φ
1
(
t
)
and
φ
2
(
t
)
, the basic definition of the
phase
locking
states that
n
φ
1
(
t
)
−
m
φ
2
(
t
)=
C
≡
const
,
(18.23)
where integers
n
and
m
specify the phase locking ratio.
When investigating phase synchrony in neurophysiological signals, one must as-
sume that the constant phase locking ratio is valid within a limited time interval
T
, which usually means a few hundreds of milliseconds. As noted in [31], as a
consequence of volume conduction effects in brain tissues, the activity of a single
neuronal population can be recorded by two distant electrodes, which results in spu-
rious phase locking between their signals. Furthermore, in noninvasive EEG, the
true synchronies are hidden in a significant background noise. Hence, in the syn-
chronous state, the phase shifts back and forth around some constant value, and so
the signals can be viewed as
synchronous
or
not synchronous
only in a statistical
sense. Therefore, the condition (18.23) must be adjusted to account for the noise as
follows:
−
ε
≤
n
φ
(
t
)
−
m
φ
(
t
)
≤
C
+
ε
,
(18.24)
C
1
2
where
n
denotes a small positive constant.
To investigate phase synchrony, first the instantaneous phases need to be
extracted from the data, and then statistical approaches are applied to evaluate the
degree of phase synchronization. The following two methods for estimating the
phases applied to neuronal signals have recently been considered in the literature.
Tass and colleagues [40] extracted the instantaneous phases from original signals
by means of the
Hilbert transform
, and then applied to magnetoencephalographic
(MEG) motor data in patients affected by Parkinson's disease [40]. On the other
hand, Lachaux et al. [16] estimated the phases from the original signals by means
of convolution with a complex wavelet, and then applied it to EEG and intracranial
data recorded during cognitive tasks [32, 15].
The first step in quantifying phase synchronization between two time series
X
and
Y
is the determination of their instantaneous phases
,
m
,
C
are constants from (18.23), and
ε
.Thisis
achieved either via the Hilbert transform or via the wavelet transform. Next, we
present phase estimation approach based on Hilbert transform.
φ
X
(
t
)
and
φ
Y
(
t
)
18.4 Phase Estimation Using Hilbert Transform
The first method used to extract the instantaneous phase from the time series is based
on the
analytic signal approach
, which was first introduced by D. Gabor [11] and
later extended for model systems and experimental data [35].
The
Hilbert transform
of a given real-valued function
f
(
t
)
with domain
T
is
defined as a real-valued function
f
(
t
)
on
T
as follows:
