A problem of revealing a coupling between two oscillatory systems by time series of their oscillations, i.e., a discrete sequence of values of the observed char acteristics is considered in radio physics  In this work, an alternative approach and the other characteristic of coupling r, i.e., the coefficient of cor relation between the phase increments are proposed. A law of the distribution of an estimate of this value for uncoupling systems with almost arbitrary properties of individual phase dynamics is analytically derived. On the basis of this law, a formula for the confidence probability of the difference of this estimate from zero is derived. The efficiency of this approach is shown using examples of reference oscillators with various coupling types and with phase nonlinearity. Condi tions for the superiority of this approach over the esti mate of phase coherence coefficient ρ in sensitivity are demonstrated. Let us consider the time series of the oscillation phases of two systems {φ 1 (t 1 ), …, φ 1 (t N )} and {φ 2 (t 1 ), …, φ 2 (t N )}, where t n = nΔt and Δt is the sampling interval. We do not consider methods for calculating the phase  here and hereinafter, the up arrow means an estimate obtained by a finite length time series. Let us designate the phase increments during time The coefficient of correlation between the phase increments is used as the characteristic of a coupling between the systems, where w 1, 2 = 〈Δφ 1, 2 〉 are the mathematical expectations of the phase increments; and are their standard deviations. For independent from each other systems r = 0. If a coupling exists, r can take Abstract-A method based on calculating the coefficient of correlation between the increments of oscillation phases is proposed for revealing a coupling between two oscillatory systems according to their time series. A distribution of the estimate of this characteristic for uncoupling systems is found; it was used to obtain a criterion for judging the availability of the coupling with a specified confidence probability. The proposed method is simpler than known methods and has a wider range of application, since it also includes oscillators with fairly strong phase nonlinearity. The efficiency of this method is illustrated by examples of reference sys tems in a numerical experiment

B P Bezruchko

D A Smirnov

E V Sidak

CiteSeerX

ISSN 10637850, Technical Physics Letters, 2013, Vol. 39, No. 7, pp. 601–605. © Pleiades Publishing, Ltd., 2013.
Original Russian Text © D.A. Smirnov, E.V. Sidak, B.P. Bezruchko, 2013, published in Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 2013, Vol. 39, No. 13, pp. 40–48.
601
A problem of revealing a coupling between two
oscillatory systems by time series of their oscillations,
i.e., a discrete sequence of values of the observed char
acteristics is considered in radio physics [1, 2], infor
mation transfer [3], biomedical applications [4–6],
climatology [7], and other fields. For nonlinear oscil
latory systems, approaches based on the introduction
and analysis of the oscillation phases φ1(t) and φ2(t)
turn out to be the most sensitive to a weak coupling
and, hence, practically efficient [1, 4, 5, 8, 9]. Various
phase synchronization indices are used [9]; among
them, phase coherence coefficient ρ is the best known.
This coefficient is the amplitude of the first Fourier
mode of the stationary phase difference distribution
[4]: ρ = ; here and hereinafter, angle
brackets mean mathematical expectation. The value
of ρ is equal to unity in the case of strict phase syn
chronization φ1(t) = φ2(t) and to zero in the case of
uncoupled oscillators (without phase nonlinearity)
[10]. On this basis, the availability of a coupling is fre
quently revealed from a significantly nonzero estimate
of ρ and statistical significance is checked using surro
gate data [11] or special analytical formulas [12].
However, in both cases, it is assumed that the oscilla
tors do not possess individual phase nonlinearity and
phase noises are white, which restricts the practical
application of this method.
In this work, an alternative approach and the other
characteristic of coupling r, i.e., the coefficient of cor
relation between the phase increments are proposed.
A law of the distribution of an estimate of this value for
uncoupling systems with almost arbitrary properties of
individual phase dynamics is analytically derived.
On the basis of this law, a formula for the confidence
e
i φ1 t( ) φ2 t( )–( )
〈 〉
probability of the difference of this estimate from zero
is derived. The efficiency of this approach is shown
using examples of reference oscillators with various
coupling types and with phase nonlinearity. Condi
tions for the superiority of this approach over the esti
mate of phase coherence coefficient ρ in sensitivity are
demonstrated.
Let us consider the time series of the oscillation
phases of two systems {φ1(t1), …, φ1(tN)} and {φ2(t1), …,
φ2(tN)}, where tn = nΔt and Δt is the sampling interval.
We do not consider methods for calculating the phase
[1, 9] and believe that it is correctly obtained for each
of the systems under study, for example, by introduc
ing an analytical signal. The empirical estimate of the
abovementioned characteristic ρ is as follows:
here and hereinafter, the uparrow means an estimate
obtained by a finite length time series.
Let us designate the phase increments during time
interval τ as Δφk(tn) = φk(tn + τ) – φk(tn); k = 1, 2; n =
1, …, N* = N – τ/Δt. The coefficient of correlation
between the phase increments
is used as the characteristic of a coupling between the
systems, where w1, 2 = 〈Δφ1, 2〉 are the mathematical
expectations of the phase increments;  and 
are their standard deviations. For independent from
each other systems r = 0. If a coupling exists, r can take
ρ̂
1
N
 e
i φ1 tn( ) φ2 tn( )–( )
n 1=
N
∑ ,=
r
Δφ1 w1–( ) Δφ2 w2–( )〈 〉
σΔφ1σΔφ2
=
σΔφ1 σΔφ2
A Method for Revealing Coupling between Oscillators 
with Analytical Assessment of Statistical Significance
D. A. Smirnov, E. V. Sidak, and B. P. Bezruchko
Saratov Branch, Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Saratov, Russia
email: smirnovda@yandex.ru
Received December 5, 2012
Abstract—A method based on calculating the coefficient of correlation between the increments of oscillation
phases is proposed for revealing a coupling between two oscillatory systems according to their time series.
A distribution of the estimate of this characteristic for uncoupling systems is found; it was used to obtain a
criterion for judging the availability of the coupling with a specified confidence probability. The proposed
method is simpler than known methods and has a wider range of application, since it also includes oscillators
with fairly strong phase nonlinearity. The efficiency of this method is illustrated by examples of reference sys
tems in a numerical experiment.
DOI: 10.1134/S1063785013070110
602
TECHNICAL PHYSICS LETTERS  Vol. 39  No. 7  2013
SMIRNOV et al.
nonzero values up to unity. Let us use the sample cor
relation coefficient
(1)
as an estimate of characteristic r; here,  and  are
the sample means and  and  are the sample
standard deviations.
To reliably reveal a coupling, it is necessary to
check if the estimate  significantly differs from zero;
this requires a law of its distribution in the case of a
zero coupling to be known. Let us derive this law as
follows. First, if the series is fairly long, i.e., N*Δt is
much more than the autocorrelation times τcorr of the
processes Δφ1(t) and Δφ2(t), the estimate  is the sum
of a large number N*Δt/τcorr of independent terms
and, in accordance with the central limit theorem, has
a close to Gaussian distribution [13]. Second, sample
moment  is an asymptotically unbiased estimate [13,
14], so that, for a long series, the mathematical expec
tation  is equal, with a high accuracy, to  and, for a
zero coupling, to zero. Provided that the phase incre
ments have a normal or close to normal distribution,
the dispersion of  is described by Bartlett’s formula
[14] that takes the following form for uncoupling
oscillators:
where (nΔt) and (nΔt) are the autocorrelation
functions of Δφ1(t) and Δφ2(t). We obtain an estimate
of the dispersion through the sample estimates 
and  as follows:
Thus, the 95% confidence interval for the characteris
tic r is  ± 1.96  and, during an analysis of the time
series, a conclusion is drawn on the availability of the
coupling, i.e., a positive conclusion is drawn when | |
> 1.96  with a confidence probability of 0.95, i.e., at
a significant level of 0.05. Below, we present an analysis
of the efficiency of this approach that includes the esti
mates of probabilities of positive conclusions without
a coupling (these are false conclusions the probability
of which should not exceed 0.05) and with a coupling
available (these are true conclusions the probability of
which governs the sensitivity of the method).
r̂
1
N
 Δφ1 ti( ) ŵ1–( ) Δφ2 ti( ) ŵ2–( )
i 1=
N*
∑
σ̂Δφ1σ̂Δφ2
=
ŵ1 ŵ2
σΔφ1 σΔφ2
r̂
r̂
r̂
r̂ r̂
r̂
σ r̂
2 1
N*
 cΔφ1 nΔt( )cΔφ2 nΔt( ),
n ∞–=
∞
∑=
cΔφ1 cΔφ2
cΔφ1
cΔφ2
σ̂ r̂
2 1
N*
 ĉΔφ1 nΔt( )ĉΔφ2 nΔt( ).
n N*/4–=
N*/4
∑=
r̂ σ̂ r̂
r̂
σ̂ r̂
We use the following phase oscillators as reference
systems:
(2)
where ω1 and ω2 are the angular frequencies; b is the
phase nonlinearity parameter; kd, 1 and kd, 2 are the
“difference” coupling coefficients; km is the modulat
ing coupling coefficient; the noises ξ1 and ξ2 are inde
pendent and have the autocorrelation functions
〈ξk(t)ξk(t')〉 = δ(t – t'); k = 1, 2; δ is the delta func
tion; and  and  are the intensities of the noises
(see, e.g., [1, 2, 9, 10]). We consider either only a dif
ference (km = 0) or only a modulating (kd, 1 = kd, 2 = 0)
coupling. The difference coupling is considered in the
symmetric (kd, 1 = kd, 2 = kd) and “antisymmetric”
(kd, 1 = –kd, 2 = kd) forms; in both cases, b = 0. In the
first case, the coupling is synchronizing (for zero noise
and a low frequency mismatch, the phase synchroni
zation mode 1 : 1 becomes stable at kd > |ω1 – ω2|/2)
and, in the second case, the coupling is nonsynchro
nizing. The modulating coupling is considered for
“linear” (b = 0) and “nonlinear” (b ≠ 0) oscillators.
For each set of the parameters, we analyzed an
ensemble of M = 100 pairs of time series obtained by
integrating Eqs. (2) with a step of 0.01 using the Euler
method. The sampling interval was 0.3, i.e., 20 points
during a characteristic perimeter; the series length was
N = 2000 or about 100 characteristic periods. The
results presented below were obtained for a value of τ
equal to two characteristic periods, but they are similar
at any value of τ that exceeds about onefourth of the
characteristic period. For each pair of the series, the
value of  was calculated and a conclusion on the
availability of a coupling was drawn or not drawn by
the criterion | | > 1.96 . We calculated the frequency
or the estimate of the probability of positive conclu
sions f, i.e., a fraction of the time series for which a
conclusion on the availability of a coupling was drawn.
Figure 1 illustrates values of f and ensemble aver
ages 〈 〉 for the set of the parameters ω1 = 1.1, ω2 =
0.9,  = 0.2, and  = 0.1, i.e., for nonidentical
oscillators with a moderate noise level. They show that
the proposed method is correct since the frequency of
false conclusions does not exceed 0.05 (see the dashed
lines at kd = 0 or km = 0). In addition, this method is
fairly sensitive to a difference synchronizing coupling,
i.e., frequency f is high even at low, as compared with
the value (ω1 – ω2)/2 = 0.1, values of kd and grows with
increasing kd (Fig. 1a). For an antisymmetric differ
dφ1/dt ω1 b φ1sin kd 1, φ2 φ1–( )sin+ +=
+ km φ2 ξ1 t( ),+sin
dφ2/dt ω2 b φ2sin kd 2, φ1 φ2–( )sin+ +=
+ km φ1 ξ2 t( ),+sin
σξk
2
σξ1
2
σξ2
2
r̂
r̂ σ̂ r̂
τ̂
σξ1 σξ2
TECHNICAL PHYSICS LETTERS  Vol. 39  No. 7  2013
A METHOD FOR REVEALING COUPLING 603
ence coupling, f increases still more rapidly (Fig. 1b).
Sensitivity to a modulating coupling is somewhat
lower, but also fairly high (Figs. 1c and 1d), especially
when phase nonlinearity exists (Fig. 1d).
Somewhat unexpected negative correlations of r
and the nonmonotonic pattern of the graphs presented
in Fig. 1a for the synchronizing difference coupling
are explained as follows. We note that the right side of
the first and second equations of system (2) contains
the same term ksin(φ2 – φ1), but the signs of this term
differs for these equations. The phase increments are
obtained by integrating system (2) over interval τ, so
that
is a component of both phase increments. At b = 0, we
have Δφ1(t) = ω1τ + ε1(t) + η(t) and Δφ2(t) = ω2τ +
ε2(t) – η(t), where ε1(t) and ε2(t) are integrals of ξ1(t)
and ξ2(t); i.e., independent Gaussian processes with
zero average value. In the case of a weak coupling, εk(t)
and η(t) can be assumed to be independent of another.
Since common additive component η(t) with different
signs is available, we obtain a negative correlation of r
(Fig. 1a, weak couplings). In the mode close to syn
chronization, the mutual independence of εk(t) and
η(t) is violated and the phase increments during one
and the same time interval are almost equal, so that the
η t( ) k φ2 t'( ) φ1 t'( )–( )sin t'd
t
t τ+
∫=
value of r is positive and close to unity (Fig. 1a, strong
couplings). Therefore, r changes its sign from minus to
plus as the mode approaches the synchronization
mode and f has a dip at intermediate values of kd when
the graph of r intersects the straight line r = 0.
In the case of the antisymmetric difference cou
pling, the total component of the phase differences
η(t) has one and the same sign and results in a positive
correlation and a monotonic increase in r and f
(Fig. 1b). A similar analysis carried out for the modu
lating coupling shows that the value of r is sensitive to
this coupling in the case of phase nonlinearity b ≠ 0
when the right sides of the first and second equations
of system (2) contain common terms. This is in fact
observed in Fig. 1d. However, it is interesting that this
method is also sensitive to the modulating coupling in
the case of b = 0, but only at sufficiently higher values
of km (Fig. 1c). An analogous analysis shows that, in
the case of a unidirectional coupling, all results are
similar, but the sensitivity of the method somewhat
decreases (the graphs are not shown); the method is
totally insensitive only in the case of a unidirectional
modulating coupling.
Figure 2 shows that the use of r provides some
advantages over the use of ρ in addition to greater sim
plicity and versatility during the assessment of signifi
cance. Namely, the value of r is sensitive to the modu
lating coupling (Figs. 2c and 2d), where ρ is totally
insensitive since the phase difference distribution
1.0
0.5
0
−0.5
(a)
0 0.1 0.2 0.3 0.4
kd
0.5
0
0 0.1 0.2 0.3 0.4
kd
1.0 (b)
1.0
0.5
0
−0.5
(c)
0 0.1 0.2 0.3 0.4
km
0.5
0
0 0.1 0.2 0.3
km
1.0 (d)
0.4
Fig. 1. Average values of 〈 〉 (solid lines) and frequencies of positive conclusions f (dashed lines) calculated for ensemble of
100 time series: (a) difference coupling with kd, 1 = kd, 2 = kd, (b) difference coupling with kd, 1 = –kd, 2 = kd, and (c, d) modu
lating coupling at (c) b = 0, (d) b = 0.7.
r̂
〈r〉, f 〈r〉, f
〈r〉, f 〈r〉, f
604
TECHNICAL PHYSICS LETTERS  Vol. 39  No. 7  2013
SMIRNOV et al.
remains almost uniform with increasing intensity of
the coupling. A similar situation arises in the case of
the antisymmetric difference coupling (Fig. 2b). For
the synchronizing difference coupling, the absolute
values of r and ρ grow with increasing k at an almost
identical rate (Fig. 2a). A similar analysis shows that ρ
has greater advantages in sensitivity only in the case of
the unidirectional synchronizing coupling (graphs are
not shown).
We emphasize that the conclusion on the existence
of a coupling drawn on the basis of a nonzero value of
the estimate ρ with a specified confidence probability
is only possible for oscillators without phase nonlin
earity [10]. Otherwise, a special analysis is needed; for
example, in Fig. 2d, ρ is high even for the uncoupling
oscillators and significantly differs from zero at b ≠ 0.
This is due to phase nonlinearity that leads to the non
uniformity of the phase difference distribution within
the section from 0 to 2π rather than to the existence of
the coupling. The alternative approach proposed in
this work is also efficient for nonlinear oscillators
without any changes.
In this work, a method for revealing a coupling
between oscillators is proposed that makes it possible
to draw conclusions with a specified confidence prob
ability. It is based on calculating correlations between
the increments of phases of oscillations. This method
is easy to implement and does not require a large com
putation volume, unlike known approaches that
involve the construction of surrogate data [11]. Unlike
the widely used estimate of the phase coherence coef
ficient, it is applicable for a broad range of cases for
various coupling types and with individual phase non
linearity in the dynamics of the systems under study
[12]. Therefore, this method seems to be a useful new
means for investigating the coupling between oscilla
tory systems of various natures according to their time
series.
Acknowledgments. This study was supported by the
Russian Foundation for Basic Research (projects
nos. 110200599 and 120200377) and the Russian
Academy of Sciences Program “Fundamental Sci
ences for Medicine.”
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1.0
0.5
0
−0.5
(a)
0 0.1 0.2 0.3 0.4
kd
0.5
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0 0.1 0.2 0.3 0.4
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0.4
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TECHNICAL PHYSICS LETTERS  Vol. 39  No. 7  2013
A METHOD FOR REVEALING COUPLING 605
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Translated by D. Tkachuk


A Method for Revealing Coupling between Oscillators with Analytical Assessment of Statistical Significance

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A Method for Revealing Coupling between Oscillators with Analytical Assessment of Statistical Significance

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