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where
i
,
j
5 are the standard parameters used
for the oscillators to be in the chaotic regime, while we introduce a mismatch in their
parameter
=
1, 2, and
α
i
=
0
.
38,
β
i
=
0
.
3,
γ
i
=
4
.
ω
(i.e.,
ω
1
=
1 and
ω
2
=
0
.
9) to make them nonidentical,
ε
ji
denotes
the strength of the diffusive coupling from oscillator
j
to oscillator
i
;
ε
ii
denotes
self-coupling in the
i
th oscillator (it is taken to be 0 in this example). Also, in this
example,
0 (unidirectional coupling) so that the direction of information flow
is from oscillator 2
ε
12
=
1 (see Fig. 15.1a for the coupling configuration). The data
were generated using an integration step of 0.01 and a fourth-order Runge-Kutta
integration method. The coupling strength
→
ε
21
is progressively increased in steps
of 0.01 from a value of 0 (where the two systems are uncoupled) to a value of
0.25 (where the systems become highly synchronized). Per value of
ε
21
, a total
of 10,000 points from the
x
time series of each oscillator were considered for the
estimation of each value of the TE after downsampling the data produced by Runge-
Kutta by a factor of 10 (common practice to speed up calculations after making
sure the integration of the differential equations involved is made at a high enough
precision). The last data point generated at one value of
ε
21
was used as the initial
condition to generate data at a higher value of
21
. Results from the application of
the TE method to this system, with and without our improvements, are shown next.
Figure 15.1b shows the time-delayed mutual information MI of oscillator 1
(driven oscillator) at one value of
ε
ε
21
(
ε
=
0
.
05) and for different values of
k
.The
21
first minimum of MI occurs at
k
16 (see the downward arrow in Fig. 15.1b). The
state spaces were reconstructed from the
x
time series of each oscillator with em-
bedding dimension
p
=
=
k
+
l
+
1. Figure 15.1c shows the ln
C
2
→
1
(
r
)
vs. ln
r
(dotted
line), and ln
C
1
→
2
(
vs. ln
r
(solid line), estimated according to Equation (15.5). TE
was then estimated according to Equation (15.4) at this value of
r
)
ε
21
. The same pro-
cedure was followed for the estimation of TE at the other values of
ε
21
in the range
[0, 0.25]. Figure 15.1d shows the lags of the autocorrelation function AF of oscilla-
tor 1 (driven oscillator - see Figure 15.1a) at one value of
ε
21
(that is,
ε
21
=
0
.
05)
and for different values of
k
.Thevalueof
k
at which AF drops to 1
e
of its max-
imum value was found equal to 14 (see the downward arrow in Fig. 15.1b), that is
close to 16 that MI provides us with. Thus, it appears that AF could be used instead
of MI in the estimation of
k
, an approximation that can speed up calculations, as
well as end up with an accurate estimate for the direction of information flow.
/
15.3.1 Statistical Significance of
TE
and
NTE
A total of 50 surrogate data series for each original data series at each
21
coupling
value were produced. The null hypothesis that the obtained values of TE
o
are not
statistically significant was then tested at
ε
α
=
.
0
005 for each value of
ε
21
. For every
ε
21
,iftheTE
o
values were greater than 2.68 standard deviations from the mean
of the TE
s
values, the null hypothesis was rejected (one-tailed
t
-test;
005).
Figure 15.2a depicts the TE
o
and the corresponding mean of 50 surrogate TE
s
values
along with 99% confidence interval error bars in the directions 1
α
=
0
.
→
2 and 2
→
1(using
