Biomedical Engineering Reference
In-Depth Information
local maximum over a range of
K
values. In other words, the
i
th point in signal
x
would be an event if
x
i
>
x
i
±
k
,
k
represents the time within
which events from
x
and
y
must occur in order to be considered synchronized, and it
must be less than half of the minimum interevent distance; otherwise, a single event
in one signal could be considered to be synchronized with two different events in the
other signal.
Finally, the number of events in
x
that appear “shortly” (within
=
1, …,
K
. The term
τ
τ
) after an event
in
y
is counted as shown in (6.15) when
J
ij
τ
is defined as in (6.16):
m
m
y
x
()
=
∑
1
τ
τ
cxy
=
(6.15)
ij
i
1
j
=
x
y
⎧
⎪
⎪
10
12
0
if
if
else
<−
=
t
t
i
j
y
J
τ
=
t
x
t
(6.16)
ij
i
j
Similarly, the number of events in
y
that appear shortly after an event in
x
can
also be defined in an analogous way. This would be denoted
c
τ
(
y|x
). With these two
values, the synchronization measure
Q
τ
can be calculated. This measure is shown in
(6.17):
() ()
τ
τ
cxy cyx
+
Q
=
(6.17)
τ
m
xy
1and
Q
τ
is 1 if and only if
x
and
y
are
fully synchronized (i.e., always have corresponding events within
The metric is normalized so that 0
≤
Q
τ
≤
τ
).
6.5.2 Lag Synchronization
When two different systems are identical with the exception of a shift by some time
lag
, they are said to be lag synchronized [38]. This characteristic was tested by
Mormann et al. [39] when applied to EEG channels in the interictal and preictal
stage. To calculate the similarity of two signals they used a normalized cross-corre-
lation function (6.18) as follows:
τ
(
)( )
corr s
,
s
τ
(
)( )
a
b
Cs s
,
τ
=
(6.18)
a
b
(
)() (
)()
corr s
,
s
0
⋅
corr s
,
s
τ
a
a
bb
where corr(
s
a
,
s
b
)(
) represents the linear cross-correlation function between the two
time series
s
a
(
t
) and
s
b
(
t
)computed at lag time
τ
τ
as defined here:
∞
(
)( )
(
) ( )
corr s
,
s
τ
=
∫
s
t
+
τ
s
t dt
(6.19)
a
b
a
b
−∞
The normalized cross-correlation function yields a value between 0 and 1,
which indicates how similar the two signals (
s
a
and
s
b
) are. If the normalized
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