Phase dynamics of coupled oscillators reconstructed from data

Bjrn Kralemann, Laura Cimponeriu, Michael Rosenblum, and Arkady PikovskyDepartment of Physics and Astronomy, University of Potsdam, Karl-Liebknecht Strasse 24-25, D-14476 Potsdam, Germany

Ralf MrowkaInstitut fr Physiologie, AG Systems Biology-Computational Physiology, Charit-Universittsmedizin Berlin, Berlin, Germany

Received 29 January 2008; published 9 June 2008

We systematically develop a technique for reconstructing the phase dynamics equations for coupled oscil-lators from data. For autonomous oscillators and for two interacting oscillators we demonstrate how phaseestimates obtained from general scalar observables can be transformed to genuine phases. This allows us toobtain an invariant description of the phase dynamics in terms of the genuine, observable-independent phases.We discuss the importance of this transformation for characterization of strength and directionality of interac-tion from bivariate data. Moreover, we demonstrate that natural autonomous frequencies of oscillators can berecovered if several observations of coupled systems at different, yet unknown coupling strengths are available.We illustrate our method by several numerical examples and apply it to a human electrocardiogram and to aphysical experiment with coupled metronomes.

DOI: 10.1103/PhysRevE.77.066205 PACS numbers: 05.45.Tp, 05.45.Xt, 02.50.Sk, 87.19.Hh

I. INTRODUCTION

Inferring the laws of interaction between different oscil-lating objects from observations is an important theoreticalproblem with many experimental applications. Identificationand characterization of coupled dynamics from data has beenlargely approached within the framework of nonlinear dy-namics, using techniques developed for bivariate or, moregenerally, multivariate data analysis 1. In the presentwork, we concentrate on a revealing phase dynamics descrip-tion of the interaction of oscillatory systems, a highly rel-evant problem in numerous fields of research: e.g., in thestudy of coupled lasers 2, electronic systems 3, chemicalreactions 4,5, cardiorespiratory interaction 6,7, neuronalsystems 8, and functional brain connectivity 911, tomention only a few. The major goal of this paper is to showthat, under certain assumptions about the intrinsic dynamicsand the coupling, it is possible to reconstruct in anobservable-independent, invariant way, the equations of thephase dynamics from the observed oscillatory data.

Any technique for reconstructing the phase dynamicsequations from data requires as a first step the computationof phase estimates from the observed scalar signals. Thiscomputation, based, e.g., on the Hilbert transform, complexwavelet transform, or on the marker event method see, e.g.,1214 for details, became a popular tool in the context ofa description of the phase synchronization of chaotic oscilla-tors and quantification of synchronization in experiments. Assoon as the phases of, say, two interacting oscillators areobtained, one can estimate phase derivatives instantaneousfrequencies and obtain the desired equations, fitting the de-pendence of each instantaneous frequency on both phase es-timates 15,16. Reconstructed in this way, the equationscapture many important properties of the interaction. For ex-ample, these equations can be used for quantification of di-rectionality and delay in coupling 6,1618, as well as forrecovery of the phase resetting curve 8 see also later pub-lications 5,19. However, this approach has a fundamentaldrawback: the phase estimates and reconstructed equations

depend on the observables used and, generally, differ fromthe phases and the equations used in the theoretical treatmentof interacting oscillatory systems. To overcome this problem,we propose an approach which bridges the gap betweentheory and data analysis.

In our approach, we treat the phase estimates, obtained,e.g., via the Hilbert transform, as preliminary variables. Toemphasize this issue, we hereafter call these variables pro-tophases. Next, we transform the protophases to obtain theobservable-independent, genuine phases, which correspondto those used in a theoretical description. In this way, weobtain an observable-invariant description of the interactionof coupled oscillators.

We mention two other important features of our approach.First, as an intermediate result we derive a transformationfrom a protophase to the phase for an autonomous oscillator.This transformation can be used as a preprocessing step forany technique, dealing with phases, e.g., for quantification ofthe phase diffusion. Second, we develop a technique for re-covery of the autonomous frequencies of systems from ob-servations of coupled dynamics. This is possible if two ormore observations of the coupled systems for different,though unknown values of the coupling are available.

In this paper we systematically analyze and further de-velop the approach suggested in our Rapid Communication20. The paper is organized as follows. Our approach forreconstructing the phase dynamics is based on the theoreticalframework of coupled self-sustained oscillators 12,21,which is briefly outlined in Sec. II. Further, we contrast therethe concept of phase used in the theoretical description ofautonomous oscillators with that of an angular variable, orprotophase, estimated from data. Next, we introduce our ap-proach for the reconstruction of invariant phase dynamicsfrom arbitrary protophases for one oscillator Sec. III andfor bivariate observations Sec. IV. In Sec. V, numericalsimulation examples are used to validate the efficacy of theproposed methodology. Furthermore, we verify our theory byan experiment with two coupled metronomesa variant ofthe classical Huygens study of pendulum clocks Sec. VI. In

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the end Sec. VII, we discuss the implications of our analy-sis for characterization and quantification of interaction fromdata.

II. PRELIMINARIES AND GOALS OF THE WORK

A. Theoretical framework: Coupled oscillatorsin the phase approximation

Let us recall basic notions of the phase dynamics. Math-ematically, an autonomous oscillator is described by a vectorof state variables x and a differential equation of motion, x=Fx. When such a system has a stable limit-cycle solutionx0t=x0t+T0, then the motion on the limit cycle and in itsvicinity can be characterized by a phase x, which growslinearly with time according to

= 0, 1

where 0=2T01 is the natural autonomous frequency

12,21. A perturbation say, periodic or noisy to the pureautonomous equation leads to a deviation of the phase from the linear growth 1, resulting in such effects as phaselocking or phase diffusion.

Of particular interest is the description of the interactionof two oscillators in terms of their phases. As the first step ofthe theoretical treatment, one introduces phases 1,2 for twouncoupled autonomous systems; these phases grow withnatural frequencies 1,2. Next, the coupling terms in the gov-erning equations should be taken into account. In a morecomplex setting, the coupling may include additional dy-namical variables requiring additional equations. Physically,the coupling appears, e.g., due to an overlap of radiationfields of two lasers or due to synaptic and/or gap junctioncurrents between interacting neurons, etc. For the followingit is convenient to distinguish between weak, moderate, andstrong coupling.

If the coupling is weak, the dynamics of the coupled sys-tem is confined to a torus in the phase space; hence, it caneffectively be described by means of two phases only. Thisdescription can be obtained explicitly with the help of a per-turbation technique, where one neglects the dynamics of theamplitudes, because they are robust. The robustness of theamplitude follows from the fact that the amplitude corre-sponds to the direction in the phase space, transversal to thelimit cycle, and this direction is stable. Its stability is char-acterized by the negative Lyapunov exponent

of the dy-namical system. Hence, the amplitude can be considered asenslaved if the magnitude of forcing or coupling is much lessthan

. Contrary to the amplitude, the phase corresponds tothe direction along the limit cycle and, correspondingly, tothe zero Lyapunov exponent. Thus, the phase is a marginallystable variable and is therefore affected even by a weak ex-ternal perturbation, which can be due to coupling or due tonoise; see 12,21 for more details. The result of the applica-tion of the perturbation technique is the following system ofequations:

1 = 1 + q11,2 ,

2 = 2 + q22,1 , 2

where q1,2 are coupling functions which are 2 periodicwith respect to their arguments. Due to coupling, the aver-aged velocities of phase rotation,

1 = 1t, 2 = 2t , 3

called observed frequencies, generally differ from 1,2. So-lutions of the system 2 can be quasiperiodic, with incom-mensurate 1,2, or periodic. The latter case is designated assynchronization.

In the case of moderate coupling, the approximation usedin the perturbation theory is not valid anymore, so that Eqs.2 cannot be explicitly derived from the equations in statevariables. However, the equations in this form still exist. In-deed, as long as the full dynamics remains confined to thetorus, it still can be described by two phase variables. Fi-nally, a strong coupling means that the torus is destroyed,e.g., due to the emergence of chaos and Eqs. 2 are not validanymore.

Real-world systems are unavoidably noisy. However,when the noise is weak, it can be accounted for by incorpo-rating random terms into Eqs. 2, so that Eqs. 2 becomeLangevin-type equations. These equations have been alsosuggested for describing the dynamics of weakly coupledchaotic oscillators 12.

For the rest of the paper we restrict our consideration tothe case of weakly and moderately coupled noisy-chaoticoscillatory systems in the asynchronous regimei.e., to thecase when the amplitudes are enslaved and the phase portraitof the individual system resembles a smeared limit cycle. Inthis case, the coupled system can be adequately described bythe phase variables.

B. Phases vs protophases

In the theoretical consideration of a particular periodicoscillatory system, an analytical expression for the phasex obeying Eq. 1 can be found only in exceptional cases.As the first step, one typically introduces an angle variableon the cycle, or protophase, =x0, which grows mono-tonically within one oscillation period T0 and obeys (x0t+T0)=(x0t)+2. This can be easily done numericallyonce the dynamical equations are known. For example, inmany cases can be identified with an angle =arctan y2 /y1 in a projection of the limit cycle onto someplane y1 ,y2. Obviously, the choice of the protophase isnot unique: different projections generally yield differentprotophases. However, it does not constitute a problem, sinceany protophase (x0t) can be easily transformed to theunique phase (x0t). Indeed, from

ddt

=

dd

ddt

= 0,

we obtain

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dd

=

0ddt

= 0dtd

, 4

which further reveals the desired transformation

= 00

dtd

d. 5

We impose an additional condition =0=0.The crucial issue raised by our study is that the above

discussed distinction between protophases and phases hasnever been exploited in the context of data analysis. Indeed,all known methods of phase estimation from data see1214 and detailed discussion belowe.g., via the Hilberttransformactually provide protophases which heavily de-pend on the scalar observables available and on the analysistechnique. If the goal of the analysis is the quantification ofthe frequency locking, then the difference between phasesand protophases is not relevant, since they provide the sameaverage frequencies, = . However, if the goal of theanalysis is to get more insight into the interaction and toreconstruct the equations of the phase dynamics, this differ-ence becomes decisive. Indeed, in terms of protophases Eqs.2 read

1 = f 11,2 ,

2 = f 22,1 , 6where the functions f 1,2 depend on the choice of pro-tophases 1,2. Equations 6 can be easily reconstructed fromthe data see 16 and Sec. IV A below, but they do notprovide a complete, invariant description of interaction, sincethey are observable dependent.

The main goal of our study is to develop a technique fortransformation of protophases to true phases, 1,21,2, andfor a subsequent transformation of observable-dependentfunctions f 1,2 to observable-invariant coupling functionsq1,2. Next, we propose a solution for the problem of thedetermination of autonomous frequencies 1,2 from observa-tions of coupled systems. Note that the latter task is highlynontrivial: the functions f 1,2 can be represented as a Fourierseries, but their constant terms do not equal 1,2. Beforeaddressing the problem of two coupled oscillators, we firstdiscuss how the transformation 5 can be extended to treatthe case of a noisy or a weakly chaotic autonomous oscilla-tor.

III. ONE OSCILLATOR: FROM TIME SERIES TO PHASE

In this section we describe our approach for recoveringthe phase of an autonomous oscillator from a single oscillat-ing scalar observable. This case, being interesting by itself, isa necessary first step for the subsequent consideration ofmultivariate data.

A. From time series to protophase

The first step of data analysis is to obtain a protophaset from an oscillatory scalar time series Yt; the latter is

assumed to be generated by a noisy or a weakly chaoticself-sustained oscillator. In the following we denote the timeseries of corresponding observables by capital letters. Notethat observable yt is generally a nonlinear function of statevariables x; we require it to be nondegenerate.

There are different ways to accomplish this task. The mostpopular approach is to construct the analytic signal st=Yt+ iY t, where Y t is the Hilbert transform of Yt 22;see also 12 for details of practical implementation. For anoscillatory signal Yt, the Hilbert transform provides a two-dimensional embedding of the smeared limit cycle orstrange attractor. A protophase can be computed by takingthe argument of st,

t = arctanY t Y 0Yt Y0

, 7and then unwrapping it so that it is defined on a real line.Here Y0 ,Y 0 is an offset point, chosen in such a way that itis evolved by the trajectory. Alternatively, one can exploit thecomplex wavelet transform 13,14, which is equivalent to asubsequent application of a bandpass filter and the Hilberttransform 31. Obviously, the protophase obtained accord-ing to 7 depends not only on the observable Y, but also onthe offset Y0 ,Y 0.

As already mentioned, to obtain the phase by means ofa transformation we need a protophase that growsmonotonically within one oscillatory cycle. However, if theprotophase is chosen according to Eq. 7, monotonicity isnot ensured. For example, is not monotonic if the trajectoryin the embedding y , y intersects itself cf. Figs 1a and1d below. However, in this case, we can define a mono-tonic protophase in a following way. First, we choose aPoincar surface of section and denote the moments of timewhen the trajectory intersects this surface as ti. Next, wecompute the length Lt along the trajectory, starting fromthe first intersection. The protophase is then defined as

t = 2Lt Lti

Lti+1 Lti+ 2i, ti t ti+1. 8

This definition can be considered as a generalization of themarker event method 12. Note that using the marker eventprotophasei.e., linearly interpolating phase between twointersections with the Poincar surface of sectionwe losethe information about the fast of the order of the oscillationperiod component of the phase dynamics and retain onlyinformation about the slow variation of the phases. Hence,although this technique is useful, e.g., for the computation ofobserved frequencies, is of a limited use if a complete recon-struction of the phase dynamics is required. This issue, aswell as a determination of protophases from observables ofdifferent complexity, will be discussed elsewhere 23.

Note that in the following we alternatively, depending onthe context, use both wrapped defined on 0,2 interval andunwrapped defined on the real line phases and protophases.

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B. From protophase to phase

Once a monotonic protophase t has been obtained froma time series, we can look for a transformation tt.For a noise-free limit-cycle oscillator this transformation isgiven by Eq. 5. Our next goal is to extend this transforma-tion to the case of a noisy system. The sought transformationshall satisfy two requirements: i it should be 2 periodic in and ii it should minimize the deviations of from thelinear growthi.e., provide a maximally uniform instanta-neous frequency .

In the presence of weak noise and/or chaos, the state tra-jectory does not repeat itself. However, for small perturba-tions, the trajectory of the stable oscillator is expected toremain close to the unperturbed one. In this case, the trans-formation can be performed on average, requiring asufficiently long observation of the oscillatory time series.Given an estimated protophase t over a time interval T,the oscillation frequency can be computed from

0 =T 0

T.

To find the desired transformation, we average the right-handside rhs of 4 and write

dd

= 0 dtd = , 9where, in , the average is taken over all the oscillationcycles. Alternatively, due to ergodicity, the average over can be replaced by time integration over the trajectory:

= 1T0

T

dt . 10

The function 21 is nothing else but the probabilitydistribution density of , as it is inversely proportional to thevelocity of the point on the trajectory. The probability den-sity can be written as

21 = t , 11where the average is taken in the sense of Eq. 10.

Let us represent as a Fourier series

= n

Snein, 12

with the coefficients

Sn =1

202

eind . 13

The term S0=1 ensures the normalization condition for theprobability density. Substituting 11 into 13, we obtain

Sn =1T0

T

dt0

2

d eint = 1T0

T

eintdt .

14

For a time series given as Np points tj sampled with timestep T /Np, where tj = jTNp1, we replace the integration bysummation and obtain

Sn =1

Np

j=1

Np

eintj. 15

Finally, using Eqs. 9 and 12, we obtain the desired trans-formation in the form

= 0

d = n

Sn0

eind = + n0

Snin

ein 1 .

16

Equations 1416 solve the problem of finding the trans-formation from a protophase to the phase. Transformation16 gives the perfectly linearly growing phase for periodicnoiseless oscillations. For a time series with noise it gives thebest possible approximation to the linearly growing phase,where all nonuniformities, 2 periodic in the protophase, areeliminated and deviations of t from a strictly lineargrowth account only for external influences or fluctuations ofthe parameters of the underlying dynamical system.

C. Numerical example

We illustrate our theoretical consideration by simulationof a noisy van der Pol oscillator described by

x 1 x2x + 2x = 0.05t , 17

where =0.5, =1.11, and t denotes -correlated Gauss-ian noise. We solve the system numerically using the Eulermethod and estimate the protophase using four different ob-servables.

-2 -1 0 1 2

-2

-1

0

1

2

-2 -1 0 1 2

-2

-1

0

1

2

-2 0 2 4 6

-4

-2

0

2

4

-6 -3 0 3 6

-3

0

3

(a) (b)

(c) (d)

Y3 Y4

Y1 Y2

Y3

Y4

Y1

Y2

FIG. 1. Color online Embedding of data from the van der Poloscillator Eq. 17 in coordinates Yi and Y i, where Y i is the Hilberttransform of Yi, for four different observables see text. In a, b,and c the protophases have been obtained according to Eq. 7, ascorresponding angles. In d the protophase has been obtained viathe length of the trajectory according to Eq. 8. The bold red line ind marks the Poincar section used for this computation.

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i The first observable is just the solution of Eq. 17i.e., Y1t=xtand the first protophase 1 is computed ac-cording to Eq. 7 with Y0=Y 0=0.

ii The second protophase 2 is obtained from Y2t=xt with Y0=0, Y 0=0.8.

iii The third observable is obtained via a monotonic non-linear transformation of xt: Y3t=expxt2.2; Y0=Y 0=0.

iv Finally, the fourth observable is obtained via a non-monotonic transformation: Y4t= x2t1.7xt; Y0=Y 0=0.

The phase portraits of the system in coordinates Yi ,Y i areshown in Fig. 1. Obviously, in cases i, ii, and iii Figs.1a1c, rotation of the phase point is monotonic, so thatwe can estimate protophases 1,2,3t according to Eq. 7.For estimation of 4t Fig. 1d we employed the ap-proach based on the length of the trajectory Eq. 8; thePoincar section used for this procedure is shown by the boldred line in Fig. 1d. The deviations of the obtained phaseestimates i, i=1, . . . ,4 from the linear growth 0t areshown in Fig. 2a.

Next, we compute from these estimates the phases i,according to Eqs. 15 and 16 using 48 Fourier terms inexpansion 16; the results are shown in Fig. 2b. One cansee that for all observables the deviation from a linear growthis drastically reduced by the transformation . However,

as we illustrate in Fig. 3, our transformation preserves long-time-scale features of the phase dynamics: e.g., the phasediffusion. Moreover, we stress that the transformation (16) isfully invertible and does not mean any filtering of the data.Another way to illustrate the effect of the transformation is tocompare the distributions of wrapped protophases and corre-spondent phases Fig. 4. Summarizing this example, weconclude that our method efficiently works with simulateddata of the noisy self-sustained oscillator considered here.

D. Experimental data: Three ECG lead measurements

For the second example we estimate the protophases and phases from three-channel electrocardiogram ECGmeasurements from a healthy male. The data are taken fromthe PhysioNet database 24 subject mgh001 of databaseMGH/MF. Hence, we have at our disposal three differentobservations of the same oscillatory system. The samplingfrequency used by the recording system was 1 kHz. TheHilbert plane representations of these channels are presentedin Fig. 5. It can be seen that a monotonic protophase cannotbe obtained according to Eq. 7 due to the fact that thetrajectories are folded and display small loops, smeared by

0

0.5

1

1.5

2

2.5

0 5 10 15 20

0

0.5

1

1.5

2

2.5

(a)

(b)

time

i

0t

i

0t

FIG. 2. Color online Deviation of the angle variables i aand phases i b from linear growth for four different observablesof the van der Pol system see text are shown by solid black line,red bold line, dashed blue line, and dotted magenta line, respec-tively the lines are vertically shifted for clarity of presentation. Itis clearly seen that the transformation makes the growth ofthe corresponding variable almost uniform. Note that the bold redand solid black lines in b practically coincide; it means that ourtransformation completely removes the effect of the linearshift of the reference point for the angle calculations cf. Figs 1aand 1b.

0 1000 2000 3000 4000-1

-0.5

0

0.5

1

1.5

2

time

1

0t,

1

0t

FIG. 3. Color online The phase diffusion in the noisy van derPol oscillator 17 is preserved by the transformation 16, as illus-trated by the protophase 1 and phase 1, obtained from the firstobservable see text. The curve for the phase is shifted up forpresentation purposes.

0 20

0.01

0.02

0.03

0.04

0 2 0 2 0 21,1 2,2 3,3 4,4

FIG. 4. Color online The probability density of protophasesimod 2 bold curves and of the phases imod 2 solidregions, obtained from different observables of the van der Poloscillator. While the distributions of the phase estimates it arenot uniform and depend on the observables, the transformed, genu-ine phases of all observables are practically uniformly distributed.

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noise. Hence, we estimate the angular variables with the helpof Eq. 8.

The protophases and the phases obtained with the help oftransformation 16 for all channels are shown in Figs. 6aand 6b, respectively 200 Fourier terms have been used inthe transformation 15 and 16. Noteworthy, the phases

1,2,3t computed from three different observables nearlycoincide and exhibit a similar slow deviation from a lineargrowth. These deviations can be attributed to external pertur-bations to the cardiovascular system, most likely due torespiratory-related influences. Figure 7 illustrates the effectof the transformation ii on the distributions ofimod 2 and imod 2. As expected, the distributionsof the phases are nearly uniform.

Figure 6c shows the evolution of the phase during alonger time interval. Here the deviations from a uniformgrowth become apparent, and one recognizes two character-istic time scales of these deviations: i a shorter time scale,

of about 4 s, corresponding presumably to respiratory in-fluences, and ii slower variations with time scale 1 min.Hence, cleaning away inhomogeneities at the time scale ofone oscillatory period 1 s, the transformation to thephase preserves the manifestation of external perturbations tothe heart rhythm, thus providing an opportunity for an effi-cient analysis of these perturbations. Important is that theresulting time series of the phase is practically independentof the observable used i.e., of the ECG channel.

At the end of this discussion we would like to mentionthat using the phases obtained from different channels withuncorrelated measurement noise one can reduce this noise byaveraging over observables; all noisy components which arecommon for different measurements remain, of course, pre-served.

IV. FROM BIVARIATE TIME SERIES TO PHASEDYNAMICS EQUATIONS

In this section we describe how the phases 1,2, as well asthe equations for their dynamics, Eqs. 2, can be recon-structed from the observations of two scalar observables y1,2.A generalization of our approach to the case of a larger num-ber of interacting oscillators will be discussed elsewhere23. The first stepobtaining the protophases 1,2t fromthe time series Y1,2twas described in Sec. III A above;thus, now we presume that time series of protophases1t , 2t, 0 tT, are already available.

A. From time series to equations for protophases

We assume that the protophases stem from coupled oscil-lators satisfying Eq. 6, and we try to reconstruct these equa-tions. The first observation is that this is possible only in thenonsynchronous case when the oscillations are quasiperiodicor nearly quasiperiodic as we presume that the data arenoisy, the exact notion of quasiperiodicity does not apply.Indeed, in the case of synchrony for simplicity we can con-sider 1:1 locking one observes a periodic motion with acertain relation between the phases; i.e., the observables 1,2are functionally related. Therefore it is impossible to recon-struct functions of two variables in 6, because these func-tions are observed not on the full torus 01,22, butonly on a line. On the other hand, in the absence of syn-chrony, when the observed regime is nonperiodic, there is norelation between the observed protophases 1,2 and one mayhope to reconstruct the functions of two variables in 6. It isclear, that this reconstruction will work better if the angles

-3.2 0 3.2-3.2

0

3.2

-1.2 0 1.2-1.5

0

1.5

-1.4 0 1.4-1.4

0

1.4

Y1

Y1

Y2Y2

Y3

Y3

FIG. 5. Three channels of a multiple-lead ECG measurement,represented in Yi ,Y i coordinates, where Y i is the Hilbert transformof Yi.

-2

0

2

4

6

0 1 2 3 4

-2

0

2

4

6

0 20 40 60 80 100

-10

-5

0

5

(a)

(b)

(c)

time [s]

time [s]

1

0t

i

0t

i

0t

FIG. 6. Color online Protophases a i and phases b i ofthree ECG channels. The curves are vertically shifted for presen-tation purposes. Panel c shows 1 on a large time scale. One cansee that our transformation preserves all the low-frequency features,including the phase diffusion. In particular, one can see a regularmodulation of the cardiac phase with period 4 s, which is mostlikely due to influence of respiration.

0 20

0.04

0.08

0 2 0 21,1 2,2 3,3

FIG. 7. Color online The distributions of phase estimatesimod 2 bold lines and of the phases imod 2 solid re-gions, obtained from three ECG channels.

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nearly uniformly cover the torus 01,22.The next step is the estimation of the rhs of 6. For sim-

plicity of presentation we consider only the first equation tobe reconstructed:

11,2 = f 11,2 = n,m

Fn,m1 ein1+im2. 18

We find the coupling function by minimizing the error ofapproximation in 18 according to

1 n,m

Fn,m1 ein1+im22 =

0

20

2

d1d21,2

1 n,m

Fn,m1 ein1+im22

=!min, 19

where 1 ,2 is the probability density. The minimizationcondition leads to a linear system

0

20

2

d1d21,2ein1+im2

= k,l

Fkl1

0

20

2

d1d21,2ein+k1+im+l2.

20

Replacing the integration over 1 ,2 through the integrationover the time according to

= 0

20

2

1,2d1d2 1T0T

dt21

this expression is analogous to Eq. 10, we can rewrite thissystem as

k,l

An+k,m+lFkl1

= Bn,m, 22

where

An+k,m+l =1T0

T

dt ein+k1t+im+l2t, 23

Bn,m =1T0

1T

d1ein1+im2. 24

Similar to Eq. 15, we can also write Eqs. 23 and 24 assums over the discrete time series:

An+k,m+l =1

Np

j=1

Np

ein+k1tj+im+l2tj, 25

Bn,m =1

Np 2

j=2

Np1 1tj+1 1tj12

ein1tj+im2tj.

26

For the second protophase 2, the corresponding equationsread

k,l

An+k,m+lFkl2

= Cn,m, 27

where

Cn,m =1T0

2T

d2ein1+im2. 28

These formulas solve the problem of finding the couplingfunctions f 1,2 in Eq. 6 from the linear equations 22 and27.

B. From protophases to phases

Now we want to find a transformation from the pro-tophases satisfying Eq. 6 to phases 1,2, which satisfy Eq.2. We immediately see that this condition is not well de-fined, as in system 2 we have a clear separation betweenterms of the coupling q1,2 and the natural frequencies 1,2only if the latter are known. Otherwise, this separation isambiguous, because coupling functions q1,2 generally con-tain constant terms. Therefore, here we have to make anadditional assumption: we assume that the coupling termsoscillatory depend on the forcing phase. This means thatthe following conditions are valid:

0

2

q11,2d2 = 0

2

q22,1d1 = 0. 29

This assumption is required because we cannot recover thecomponents of the phase dynamics which are independent ofthe forcing phase. We will discuss this condition in moredetails below, after obtaining equations of the dynamics.

Let us write the transformations 1,21,2 in a formsimilar to Eq. 9:

d1d1

= 11,d2d2

= 22 , 30

with the normalization condition

0

2

1,2d = 2 . 31

Then Eqs. 6 can be rewritten as the equations for 1,2:

d1dt

= 11f 11,2,d2dt

= 22f 22,1 .

32

Comparing the latter with Eqs. 2 we obtain

1 + q11,2 = 11f 11,2 , 33

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2 + q22,1 = 22f 22,1 .Integrating 33 and using the conditions 29, we finallyobtain

1 =11

2 02

d2f 11,2 , 34

2 =22

2 02

d1f 22,1 . 35

Changing the integration variables to 1,2 we obtain a closedsystem of equations for unknown functions 1,2 and con-stants 1,2, provided the functions f 1,2 are given:

1 =11

2 02

d222f 11,2 , 36

2 =22

2 02

d111f 21,2 . 37

This is a nonlinear system that can be easily solved numeri-cally two methods are described in the Appendix. Aftersolving this system we obtain 1,2 and the transformations11 and 22, which can be represented as Fourierseries:

11 = n

Sn1ein1, 22 =

n

Sn2ein2. 38

With these results we compute the phases 1 and 2 from theprotophases 1 and 2 by integration of Eqs. 30:

11 = 1 + n0

Sn1

inein1 1 , 39

22 = 2 + n0

Sn2

inein2 1 .

From 1 and 2 and their derivatives, which are alreadygiven by Eqs. 32, we finally obtain by substitution the ex-pressions for the phase dynamics:

d1dt

= q11,2 = n,m

Qn,m1 ein1+m2, 40

d2dt

= q22,1 = n,m

Qn,m2 ein2+m1.

After we have derived the equations for the phases, wewould like to discuss again the assumption made at theirderivation. For simplicity, we will speak about the equationfor 1 only and drop for the moment the upper index. Let usrepresent the equation for 1 as

1 = 1 + q1,2 = 1 + q0 + q1 + q1,2 , 41

where the separation of the term q1 can be made unam-biguously if the condition 0

2q1 ,2d2=0 is imposed.The terms on the rhs of Eq. 41 can be interpreted as fol-

lows: 1 is the natural frequency, q0 is a constant shift of thisfrequency due to the coupling, the term q1 is thecoupling-induced permanent in the sense that it does notdepend on the driving phase 2 inhomogeneity of the phaserotation, and q is the part of the coupling that explicitly de-pends on the driving phase. In our derivation of the phasesfrom protophases we have assumed that the term q is notpresent; if it is present, we make here an approximation byneglecting it. This appears to be unavoidable: indeed, themain reason for transforming from protophases to phases isin the obtaining a maximally homogeneously rotating phase.Thus during the transformation we cleanse all inhomoge-neities, those due to the protophases, but also those that arecoupling induced, the latter being described by the term q.Since we cannot separate protophase-induced and coupling-induced inhomogeneities, we prefer to cleanse both of them.

As a result, in the obtained system 40 the term q1=nQn,0ein1 is absent, as according to the constructionQn,0=0, if n0. Still, we have an uncertainty in the separa-tion of the terms 1 and q0; we discuss this problem in thenext subsection.

C. Restoring natural frequencies

The constant terms in the reconstructed equations 40 areQ0,01,2=1,2+q01,2. They yield the natural frequencies only ifthe constant terms q0

1,2 in the coupling vanish. In the ex-amples we studied this was not the case. Nevertheless, if datasets from at least two observations of the same oscillatorswith different, though unknown, coupling strengths are avail-able, q0

1,2 can be computed under some assumptions, mak-ing the estimation of natural frequencies more precise.

We shall now discuss this issue, for simplicity, for the firstoscillator only, again dropping the upper index. The mainassumption is that the coupling functions for observationswith different coupling strengths we denote those by a andb have the same form and differ by their amplitudes only.In other words, the constant terms q0 are proportional to thenorms of the phase-dependent terms:

q0a

q0b =

NaNb , N

a,b= Qn,ma,b21/2, 42

where the summation is over all terms except for Q0,0. Fromthis assumption we can easily obtain the natural frequencyby making an extrapolation to the case of vanishing cou-pling:

1 = Q0,0b Q0,0a Q0,0bNa NbN

b. 43

A similar equation can be written for the frequency of thesecond oscillator.

D. Invariance of recovered coupling functions

Finally, we demonstrate that the coupling functions, re-covered with the help of Eqs. 3437, are observable in-dependent; this is valid at least for a wide class of nondegen-erate observables preserving the main frequency. Suppose

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that we start the reconstruction procedure with a differentpair of observables 1,2, related to old ones 1,2, via a trans-formation

d1,2d1,2

= 1,2.

Then,

d1,2d1,2

=

d1,2d1,2

d1,2d1,2

= 1,2,

with 1,2=1,21,2. Next,

1,2 =

1,2d1,2d1,2

=

f 1,21,2

= g1,2.

Similar to Eqs. 36 and 37 we write the equations for thetransformation 1,21,2:

1 =11

2 02

d222g11,2 ,

2 =22

2 02

d111g22,1 . 44

Substituting here 1,2, g1,2, and d1,2, one can easily seethat Eqs. 44 are equivalent to Eqs. 36 and 37.

V. NUMERICAL EXAMPLES

A. Data from a phase model

As the first example, we take coupled phase oscillatorsi.e., a system of coupled differential equations for phase dy-namics in the form of Eqs. 2:

1 = 1 + 1 sin2 1 1 ,

2 = 2 + 2 sin2 1 2 . 45

The parameters are 1=1, 2= 51 /2, 1= /4, 2= /3, 1=0.05, and 2=0.03. As the protophases we tookdistorted phases, calculated according to 1,2=1,2+ 12sin1,2+

110cos21,2. A reconstruction of the equations

of motion 6, in terms of protophases 1,2, leads to the func-tions f1,2, depicted in Fig. 8a and 8b. Clearly, thesestrongly differ from the actual coupling functions in Eqs.45. The absolute values of Fourier coefficients of the re-constructed functions F1,1

1 =0.050 43 and F1,12 =0.0302are very close to the true values F1,1

1,2=1,2; however othercoefficients do not vanish and are not small e.g., F1,2

1,2=0.011 78. Application of our method, described in Sec.IV B, provides coupling functions in terms of genuine phases1,2; these reconstructed functions are shown in Figs. 8cand 8d. They are in full agreement with the rhs of theoriginal equations 45: the absolute values of the Fouriercoefficients Q1,11 =0.049 94 and Q1,12 =0.2994; all othercoefficients are smaller by at least two orders of the magni-tude.

B. Data from noisy van der Pol oscillatorsIn this subsection we test the described method using a

system of two coupled noisy van der Pol oscillators

x1 + 1 x12x1 + 1

2x1 = x2 x1 + d1t ,

x2 + 1 x22x2 + 2

2x2 = x1 x2 + d2t . 46

Here 1=1, 2= 51 /2, and noise is Gaussian with cor-relations 1t2t=1,2t t. For this model the equa-tions for the phases are not known analytically, so we justcheck the reliability of the method and its robustness towarddynamical and toward noise due to the measurement. Thelatter was implemented by calculating the protophases fromthe embedding:

1,2 = arctan x1,2 0.2 + 1x1,2 0.2 + 2

, 47where 1,2 are random variables uniformly distributed fromd to d. The results for a moderate coupling =0.1 and theintensity of both noise sources d=0.05 are presented in Fig.9. The upper row shows the reconstructed functions f 1,2describing the dynamics in terms of the protophases 1,2.One can see a strong variation of these functions with theown variable i.e., f 1 strongly varies with 1, etc.. Next,we applied one-dimensional transformations to each of theprotophases separately, as described in Sec. III. In these new,improved protophases 1 and 2 the dependence on theown variables is almost cleansed, as one can see from themiddle row of Fig. 9. Finally, we applied the full two-dimensional transformation as described in Sec. IV to theimproved protophases and obtain the final system for thegenuine phases with functions presented in the bottom rowof Fig. 9. One can see only a small difference between the

(a) (b)

0.61

1.41.8

0

2

2

1

2

f (1)f (1)

0.2

0.6

1

0

0

22

1

2

f (2)f (2)

(c) (d)

0.80.91

1.11.2

0

2

2

1

2

q(1)q(1)

0.5

0.6

0.7

0.8

0

0

22

1

2

q(2)q(2)

FIG. 8. Color online Reconstruction of phase dynamics ofcoupled oscillators 45 from a time series of protophases. Top pan-els: functions f 1,2 governing the dynamics of the protophases ac-cording to Eq. 6. Bottom panels: functions q1,2 governing thedynamics of genuine phases, obtained from functions f 1,2 with thehelp of the transformation of Sec. IV B. These functions agree verywell with the true coupling functions in the original equations 45.

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improved protophases and genuine phases. Thus, in caseswhere a high accuracy is not needed or the data sets arerather limited, performing a one-dimensional transformationmight be sufficient for practical purposes. In any case wesuggest to perform such a transformation prior to the two-dimensional one, as it is rather fast and significantly reducesthe computational efforts.

For a quantitative comparison of the reconstructed func-tions we again can compare their Fourier coefficients. Wepresent here the results for the main interaction termsei12 and ei12. The absolute values of the Fouriercoefficients of the reconstructed functions for protophasesare F1,1

1 =0.034 51 and F1,12 =0.227 79. After a one-dimensional transformation we obtain F 1,1

1 =0.021 30 andF 1,1

2 =0.033 51; the two-dimensional transformation yieldsF 1,1

1 =0.020 77 and F 1,12 =0.034 18. We see that for this

example the coupling functions for protophases not only pos-sess additional terms which should be eliminated, but alsothe amplitudes of the true modes are significantly distorted.In summary, this example again demonstrates that a transfor-mation to true phases is a necessary step in model recon-struction.

Finally, we demonstrate the robustness of our approachwith respect to noise by plotting the largest Fourier coeffi-cients of reconstructed coupling functions versus the noiseintensity. The results, shown in Fig. 10, indicate that therecovered coupling functions are rather insensitive to thenoise level.

VI. PHYSICAL EXPERIMENT WITH COUPLEDMETRONOMES

Apart from the numerical examples above, we searchedfor experimental evidence supporting the efficacy of our the-oretical approach. For this purpose, we performed a versionof the classical Huygens experiment, examining two me-

0.51

1.52

2.5

0

0

2

2

1

2

f (1)(1, 2)f (1)(1, 2)

0.51

1.52

0

0

2

2

1

2

f (2)(2, 1)f (2)(2, 1)

00.51

1.5

0

0

2

2

1

2

f (1)(1, 2)f (1)(1, 2)

00.5

11.5

0

0

2

2

1

2

f (2)(2, 1)f (2)(2, 1)

00.51

1.5

0

0

2

2

1

2

q(1)(1, 2)q(1)(1, 2)

00.5

11.5

0

0

2

2

1

2

q(2)(2, 1)q(2)(2, 1)

FIG. 9. Color online Recon-struction of the phase dynamicsfor coupled van der Pol oscillatorswith intrinsic and measurementalnoises. Upper row: functions f 1,2for the protophases, calculatedfrom the embedding given by Eq.47. Intensity of the intrinsicnoise is d=0.05; the measuremen-tal noise is modeled by randomvariables 1,2, which are uni-formly distributed from d to d.Middle row: functions f1,2 forthe improved protophases 1,2 ob-tained by applying one-dimensional transformations to1,2. Bottom row: functions q1,2governing the dynamics of genu-ine phases.

0.025 0.05 0.075 0.10.01

0.02

0.03

0.04

0.05

d

|Q(1

,2)

n,m

|

0

FIG. 10. Color online Robustness of reconstructed couplingfunctions q1,2 with respect to noise. Absolute values of three larg-est Fourier coefficients Qn,m1,2 of q1,2 are plotted against the inten-sity of the both intrinsic and measuremental noise d. It is seen thatthe variation of coefficients is rather weak. Bold black lines andsolid red lines correspond to the first and second systems, respec-tively. Circles, squares, and diamonds represent Q1,11,2, Q1,11,2, andQ3,11,2, respectively.

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chanical metronomes Cherub WSM-330, placed on a rigidbase; see Fig. 11a cf. 25. The oscillation of the metro-nomes pendulums was observed for three different experi-mental conditions: i metronomes are uncoupled, ii pendu-lums of the metronomes are linked by a rubber band, andiii pendulums of the metronomes are linked by two rubberbands.

The first measurement was used to determine the autono-mous frequencies. Measurements ii and iii were used toreconstruct the coupling functions for two different couplingstrengths and to recover the autonomous frequencies fromobservations of coupled systems by virtue of the methoddescribed in Sec. IV C. The experiment was performed twicefor two different settings of autonomous frequencies.

The rubber bands were customized by longitudinal dissec-tion to have a spring constant of about k=0.08 N /m; withinthe range of operation, k is nearly constant. The diameter ofthe rubber band was about 60 m. The motion of the met-ronomes pendulums was filmed by a JVC GR-D245E digitalvideo camera cf. 26, with an exposure time of 1/1000 s,resolution 720576 pixels interlaced, and a rate of 25frames/s, thus providing a time resolution of 0.04 s. Theduration of the recording was about 140 s.

The camera was placed on a stable tripod in front of themetronomes at a distance of 4 m. A piece of a circularreflecting material appearing as a dot on the camera imagewas pasted to the top of each pendulum. The metronomeswere positioned in a dark room. The reflecting material con-sisted of microprismatic reflecting film having the propertyto return the light to its source. Therefore, the light sourcewas placed directly on the camera and directed at the metro-nomes in order to have the light source as close as possibleto the optical system of the camera. This setup provided anexcellent contrast ratio between the reflecting dots on thependulums and the remaining structures which were effec-tively invisible on the captured images.

After the video capture the data was transferred to a PCvia the digital fire wire interface. The movies were split intosingle-image frames. No blurring or any other image en-

hancement algorithm was allowed in the capturing soft-ware. The frames were de-interlaced and split vertically intotwo image parts, one for the left and one for the right pen-dulum.

A motion tracking algorithm was used for off-line pro-cessing and recovering of the horizontal and vertical coordi-nates of motion in time for each metronome on a frame-by-frame basis. The algorithm was implemented using the opensource WXDEV-C development package 33. Image pro-cessing was greatly facilitated by using the open source CIMGpackage 34. The position detection algorithm uses essen-tially the computation of the cross-correlation function for agiven frame with a picture of metronomes at rest. The shift,corresponding to a maximum of the cross correlation of eachframe with the rest image, provided the position of the mov-ing light dot.

As the scalar observable characterizing the state of met-ronomes, we have chosen the x coordinates of the pendulumtops of the two metronomes since the horizontal resolution of

Light

Reflectors

Metronomes

Light Source

Video - Camera

distance approx. 4 m

Dark room

FIG. 11. Color online Twometronomes, placed on a rigidsupport upper panel. The imageof the rubber band is artificiallyenlarged for visibility. Scheme ofthe experimental setup lowerpanel. The motion of the metro-nomes pendulums has beenfilmed by a mini-DV camera forthree experimental conditions: nocoupling and coupling via one asshown here or two rubberbands.

2 4 6 8 10

-50

50

100

150

200

0

0

X1,2(t

)[p

ixel

s]

time [s]

FIG. 12. Color online Time series of horizontal displacementfor the case of metronomes, coupled via one rubber band. Thesesignals are provided by the motion-tracking algorithm applied tovideo frames. Only 10 s out of 140 s are shown. The time seriesX1 is shifted upwards for better visualization.

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the camera is higher than the vertical one. In Fig. 12 shortsegments of the time records of these observables X1 and X2are presented. From these observables, the protophases wereobtained by means of the Hilbert transform with a zero off-set.

Next, we used our approach to reconstruct the couplingfunctions f 1,2 in terms of the protophases 1,2 and to trans-form them to observable-independent coupling functionsq1,2; the results are shown in Figs. 13 and 14. Comparingthe top and middle rows in Fig. 14, one can see that thecoupling functions for two different interaction strengthshave quite a similar form; only the amplitude is rescaled.Thus, the assumption used for the recovery of autonomous

frequencies is justified. Indeed, using the analysis describedin Sec. IV C, we were able to recover them with a goodprecision; see Fig. 15.

VII. IMPLICATIONS FOR CHARACTERIZATIONOF COUPLING

Characterization of the strength, directionality, and delayin coupling from observed phases became recently a populartool of data analysis, with applications to brain activity9,11, cardiorespiratory interaction 7, electronic circuits27, etc. In the following, we discuss the importance of the

FIG. 13. Color online Cou-pling function for the protophasesof metronomes. Top row: couplingwith one rubber band. Middlerow: coupling with two bands.Bottom row: uncoupled systems.Points show the original data; thesurfaces depict the result of fitting.Notice the difference in the verti-cal scales of the graphs. The cou-pling function for the protophasesof uncoupled systems does notvanish, but demonstrates a depen-dence on the own protophase.

FIG. 14. Color online Cou-pling functions for the phases, ob-tained after application of ourmethod to functions in Fig. 13.Top row: coupling with one rub-ber band. Middle row: couplingwith two bands. Bottom row: un-coupled systems. The verticalscales are the same as in Fig. 13,so that one can clearly see the re-duction of the phase dependenceof the coupling function in thebottom row.

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above-proposed transformation for the precision ofthis analysis.

A. Implications for the synchronization analysis

Here we consider the drastic effect, first discussed byIzhikevich and Chen 28, of the difference between the pro-tophase and the phase on quantification of interrelation be-tween the phases. Typically, after calculating the protophases1 and 2 from the bivariate data, one computes the so-calledsynchronization index 29,30

n,m = ein1m2 . 48

More precisely, the index n,m quantifies an interrelation be-tween the protophases and cannot distinguish between syn-chronization and other types of interaction: e.g., modulationsee discussion in 12. For the case of a synchronizing in-teraction, n,m1 indicates n :m locking.

Below, we analytically study two examples and demon-strate that the value of the synchronization index 48, com-puted from protophases, can be significantly biased with re-spect to the true value, computed from genuine phases. Weemphasize that this bias is solely due to the difference be-tween phases and protophases and not due to numerical arti-facts, not discussed here.

For this goal we first suppose that the synchronizationindex is calculated for independent oscillators; certainly, in

this case one expects it to vanish. However, it is easy to seethat Eq. 48 yields

n,m = ein1eim2 = Sn1Sm

2 , 49

where Sn is defined according to Eq. 14. Thus, generally,the computation of index 48 for independent oscillatorsfrom protophases provides a spurious, nonzero, value, andone obtains vanishing synchronization indices for genuinephases only, for which Sn=n,0. An alternative approach,recently suggested by Daido 32, is to compute the synchro-nization index as n,m= ein1 ein1eim2 eim2.This measure yields correct value n,m=0 for uncoupled os-cillators. For illustration, consider a particular relation be-tween phases and protophases see Eq. 30, taking func-tions 1,2=1+a1,2 cos1,2, where a1,2=const, a1,21.Computation of the 1:1 index from protophases yields 1,1=

a1a24 ; i.e., 1,1 can reach 0.25.Consider now the other limit case, when two oscillators

are perfectly 1:1 locked, so that 21=, where =const.Computation of the synchronization index 1,1 from genuinephases certainly provides

1,1 = ei12 = ei = 1.

Computation of the synchronization index from protophasesprovides

0.02 0.04 0.06 0.081.42

1.44

1.46

1.48

1.5

1.52

1.54

0.02 0.04 0.06 0.080.9

1

1.1

1.2

1.3

1.4

0.02 0.04 0.06 0.081.28

1.3

1.32

1.34

1.36

1.38

1.4

0.02 0.04 0.06 0.080.7

0.8

0.9

1

1.1

0

0

1/2

[Hz]

1/2

[Hz]

2/2

[Hz]

2/2

[Hz]

N (1)/2 [Hz] N (2)/2 [Hz]

0

0

FIG. 15. Recovery of autonomous frequencies of two metronomes from observations of motion of coupled systems. Top panels, firstexperiment: autonomous frequencies are 1 / 2=1.434 Hz and 2 / 2=0.929 Hz. Bottom panels, second experiment: autonomousfrequencies are 1 / 2=1.305 Hz and 2 / 2=0.797 Hz. Constant terms of the coupling function 1,2 are plotted vs the norm of itsoscillatory component N1,2. The recovered natural frequencies are 1.4370.004 Hz and 0.920.02 Hz first setting and1.2950.006 Hz and 0.7950.01 Hz second setting; they are in good agreement with the frequencies of uncoupled systems. Errorbounds for linear regression, estimated from an analysis of disjoint segments of the full records, are shown as gray stripes.

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1,1 = ei12 = eir11r22 = eir11r21+ ,

where the functions 1,2=r1,21,2 are functions inverse tothose defined by Eqs. 40. Performing the average, we write

1,1 = 1202

d eir1r2+ .Considering a particular case r1,21,2=1,2+a1,2 cos1,21,2, with constants a1,2 and 1,2, a1,21, we obtain afterintegration

1,1 = J0a12 + a22 2a1a2 cos1 2 + ,where J0 is the Bessel function. Its argument is smaller than2, which implies that the spurious value of the synchroniza-tion index can be as low as 0.22i.e., 4 times smaller thanthe true value.

To summarize this section, we emphasize that the syn-chronization index computed from protophases can be bothover- and underestimated; this bias can be essential. Thetransformation from the protophases to phases prior to thecalculation of the index 48 allows one to obtain the latter ina reliable and observable-independent way.

B. Implications for directionality analysis

Directionality of the coupling of two oscillators may beinferred from the reconstructed phase dynamics. Severalmethods of quantification and a discussion can be found in6,16,18. The main idea is to quantify the mutual influenceof the first second phase on the time derivative of the sec-ond first one. In particular, this influence can be quantifiedby means of the indices

c1,22

=

1420

20

2 q1,22,1

2d1d2, 50calculated from the functions q1,2 in Eqs. 2. Furthermore,the directionality of coupling can be quantified by a singleindex

d =c2 c1

c2 + c1, 51

which varies from d=1, if system 2 drives system 1, and tod=1, if the unidirectional driving is in the other direction;the values 1d1 correspond to a bidirectional coupling.

In the case when protophases are used instead of thegenuine phases, the indices c1,2 are calculated in a similarway, but in terms of the reconstructed functions f 1,2:

c1,22

=

1420

20

2 f 1,22,1

2d1d2. 52In order to find a relation between c1,2 and c1,2, we introduce,similarly to Eqs. 30, the transformation functions s1,2:

d1d1

= s11,d2d2

= s22 , 53

with an obvious relation s1,21,2=1 /1,21,2. Comput-ing the derivative of Eq. 33 we obtain

q1,21,22,1

= 1,21,2 f 1,22,1

d2,1d2,1

,

which yields

s11s22

q1,21,22,1

=

f 1,21,22,1

.

Substituting into Eq. 52 and changing integration variables,we obtain

c1,22

=

1420

20

2 q1,22,1

2s1,23s2,11d1d2.54

This expression differs from 50 by an observable-dependent factor s1,23s2,11 in the integral. Thus, theindices calculated for protophases generally differ from thosecalculated for genuine phases. In the case of a unidirectionalcoupling, say, from system 1 to system 2, both f

1

2and q

1

2vanish, and, hence, the directionality can be correctly char-acterized both from phases and from protophases. However,in the case of a bidirectional coupling, the error due to usingprotophases instead of genuine phases can be highly signifi-cant.

A natural way to achieve quantification of directionality inan invariant fashion is as follows. First, we reconstruct Eqs.2, as discussed in details above. For the given equations,the strength of the action of one system on the other is un-ambiguously determined by the coupling functions q1,2 andcan be quantified by their norms N1,2 cf. Fig. 14 and Eq.42. A relative dimensionless measure of this action isgiven by

C1,2 =N1,2Q0,01,2

. 55

Defined in this way, the coefficients C1,2 provide, togetherwith Eq. 51, an observable-independent measure of direc-tionality. Note that computation of the directionality indexvia Eq. 55 does not require computation of derivatives cf.Eq. 54, which essentially improves the numerical perfor-mance.

We emphasize, again, that here we do not bring into dis-cussion the effects of the finite-data-size effects and noise onthe estimation of indices used for quantifying the couplingcharacteristics synchronization, coupling coefficients, anddirectionality. The effects we consider are solely due to thedifference between phases and protophases and can be elimi-nated by the above proposed transformation.

VIII. CONCLUSION

In this paper we have described a method that bridges thegap between the theoretical description of coupled oscilla-tory systems and the data analysis. The key idea of the ap-proach is to distinguish between the protophases, which arecomputed from scalar signals with the help of the Hilberttransform or equivalent embedding methods, and genuinephases, used in the theoretical treatment of coupled oscilla-

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tors. Contrary to phases, protophases are observable depen-dent and nonuniversal. Therefore, an invariant description ofthe phase dynamics can be obtained only after a transforma-tion from protophases to phases. We provided explicit rela-tions for the corresponding transformation for the cases of anisolated and two interacting oscillators, and illustrated themby several numerical examples. We thoroughly discussed thereconstruction of invariant phase dynamics equations frombivariable observations of oscillatory time series. The tech-nique we developed is applicable to weakly and moderatelycoupled oscillatorsi.e., when the amplitudes are enslavedand the full dynamics is represented by two phases. Note thatthe validity of our technique goes far beyond the validity ofthe first-order approximation in the coupling strength used inthe perturbation theory. The approach excludes the synchro-nous regime when the two phases become functionally re-lated. We emphasize that our transformation from pro-tophases to phases is not a filtering or an interpolation, but aninvertible transformation that preserves all the relevant infor-mation and, at the same time, cleanses all observable-dependent features.

Furthermore we discussed the implications of our ap-proach for characterization of the interaction from data. Asfirst pointed out by Izhikevich and Chen 28, the synchro-nization index, computed without such a cleansing, can besignificantly overestimated. Our results show that the esti-mates can be biased both upwards and downwards. After thetransformation to genuine phases the calculation of the syn-chronization index becomes reliable. Moreover, we showedthat the coupling strength and directionality can be effi-ciently quantified from the reconstructed equations. In thisway, one significantly improves the performance of previ-ously used approaches that exploit the dynamical equationsin terms of protophases see, e.g., 6,1618.

We have also shown, both theoretically and experimen-tally, that autonomous frequencies of self-sustained oscilla-tors can be recovered from at least two observations ofcoupled dynamics if these observations correspond to differ-ent, though unknown values of coupling. Next, we havedemonstrated that our techniques can be straightforwardlyapplied to real noisy data obtained in both physical andphysiological experiments. The reconstruction of the equa-tion for phases in the form used in the theoretical analysissignificantly contributes to a better comparison of experi-ments and theory.

ACKNOWLEDGMENTS

We thank E. Izhikevich, H. Kantz, and E.Ott for usefuldiscussions. We especially thank Professor J. Schaefer IIftCKiel for fruitful discussions and constant support. The workhas been supported by the EU Project BRACCIA, DFGSFB 555, and the Ursula Merz Stiftung, Berlin.

APPENDIX: SOLVING EQUATIONS FOR THE PHASETRANSFORMATIONS

The system 3437 can be solved in two different ways,described below. In the first method we employ iterations,starting, e.g., from the uniform functions 1,2=1. Then wecalculate the frequencies 1,2 from the normalization condi-tions

11

= 0

2 d1

0

2

22f 11,2d2, A1

12

= 0

2 d2

0

2

11f 22,1d1.

Substituting this into 36 and 37, we get the next iterationfor the functions :

11 =21

0

2

22f 11,2d2, A2

22 =22

0

2

11f 22,1d1.

These iterations converge rather fast. In the practical imple-mentation we used the functions f and defined on a grid.

In the second method these functions are represented viaFourier series:

1,21,2 = n

Sn1,2ein1,2, f 1 =

l,kFl,k

1eil1+ik2,

A3

f 2 = l,k

Fl,k2eil2+ik1.

Substitution of these expressions into 36 and 37 yieldsafter integration a nonlinear set of equations for unknownFourier coefficients Sn

1,2:

n,m

Sn1Sm

2Fjn,m1

= 1 j,0, n,m

Sn2Sm

1Fjn,m2

= 2 j,0.

A4

This nonlinear system of coefficients can be solved numeri-cally using a standard routine for finding a root of a nonlin-ear system, e.g., the MATLAB algorithm FSOLVE. Notice thatfrom the normalization follows S0

1,2=1.

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