Measuring synchronization in coupled model systems: A comparison of different approaches

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Abstract

The investigation of synchronization phenomena on measured experimental data such as biological time series has recently become an increasing focus of interest. Different approaches for measuring synchronization have been proposed that rely on certain characteristic features of the dynamical system under investigation. For experimental data the underlying dynamics are usually not completely known, therefore it is difficult to decide a priori which synchronization measure is most suitable for an analysis. In this study we use three different coupled model systems to create a ‘controlled’ setting for a comparison of six different measures of synchronization. All measures are compared to each other with respect to their ability to distinguish between different levels of coupling and their robustness against noise. Results show that the measure to be applied to a certain task can not be chosen according to a fixed criterion but rather pragmatically as the measure which most reliably yields plausible information in test applications, although certain dynamical features of a system under investigation (e.g., power spectra, dimension) may render certain measures more suitable than others.

Introduction

Synchronization between dynamical systems has been an active field of research in many scientific and technical disciplines since the first description of this phenomenon in the seventeenth century [1]. Starting in the 1980s and following the development of the theory of deterministic chaos, the notion of synchronization was extended to the case of interacting chaotic oscillators [2], [3], [4], [5]. Since the definition of chaos implies the rapid decorrelation of nearby orbits due to their high sensitivity on initial conditions, synchronization of two coupled chaotic systems is a rather counter-intuitive phenomenon. Thus the study of synchronization phenomena in chaotic systems has been a topic of increasing interest since the early 1990s (for an overview cf. Ref. [6]). A great deal of attention has been paid to identifying certain regimes of synchronization in coupled identical or non-identical systems with varying parameters (e.g., different frequency mismatches or an increasing coupling strength).

Since synchronization phenomena can manifest themselves in many different ways, various concepts for their description have been proposed. The simplest case of complete synchronization can be attained if identical systems are coupled sufficiently strongly so that their states coincide after transients have died out [2], [3]. Phase synchronization, first described for chaotic oscillators [7], [8], [9], is defined as the global entrainment of phases, whereas the exact phase differences and the amplitudes remain chaotic and, in general, weakly correlated. The concept of generalized synchronization, introduced for uni-directionally coupled systems [4], [10], [11], denotes the presence of some functional relation between the states of responder and driver. Since this function does not have to be the identity, generalized synchronization is already a rather weak criterion, but it is still surpassed by the notion of interdependence [12] where the mapping of local neighborhoods in the first system onto local neighborhoods in the second system is exploited as the quantifying criterion. Various methods exist for the detection of the different types of synchronization. Complete synchronization can be recognized by plotting a component of the driver versus the respective component of the responder while phase synchronization can be identified by a vanishing mean frequency difference [7]. To verify the existence of generalized synchronization in unidirectionally driven systems, the negativeness of the largest Lyapunov exponent of the responder is usually employed [5].

Corresponding to this variety of concepts and complementing the respective methods of synchronization detection many different approaches have been proposed aiming at a quantification of the degree of synchronization between two systems on a continuous scale. These approaches comprise linear ones like the cross correlation or the coherence function as well as essentially non-linear measures like mutual information [13] and transfer entropy [14]. Furthermore, different indices of phase synchronization have been introduced [15], [16], [17], [18]. Here the instantaneous phases are extracted from the time series by using e.g. the Hilbert transform [7] or the wavelet transform [19]. Topological approaches to quantify generalized synchronization include the method of mutual false nearest neighbors [10] and the index based on non-linear mutual predictions [20] as well as the non-linear interdependencies [12], the synchronization likelihood [21] and the predictability improvement [22]. Finally, event synchronization [23] quantifies the overall level of synchronicity from the number of quasi-simultaneous appearances of certain predefined events. While most of these measures are designed to estimate only the degree of synchronization between two systems, some of these measures are intended to also reveal possible directionalities between them and thus to detect driver–responder relationships. This aspect of directionality has already been addressed in Refs. [14], [24], [25], [26], [27], a comparison of different directionality approaches is not part of the present study, but can be found in Ref. [28].

Investigations on model systems are typically carried out by analyzing long and noise-free time series (typically of the order of 105 data points or more). Only rarely has the dependence between coupled model systems been evaluated by applying bivariate measures to short time series (of the order of 103 data points) that are typically found in field applications. If at all, almost exclusively a single approach to measure synchronization has been used. Examples of investigations on phase synchronization include Refs. [15], [16], [17], [18], [29], [30], while approaches to characterize generalized synchronization have been studied in Refs. [10], [20], [21], [24], [31], [32]. A comprehensive comparison of different approaches applied to different coupled model systems in a ‘controlled’ setting is still missing and is thus the aim of the present study. Note that it is beyond the scope of this study to give a comprehensive characterization of the coupled model system in terms of their dynamic regimes of synchronization or their bifurcations and transitions. Rather we are aiming to use their controlled setting (due to the given equations of motion we are able to calculate the dependence of the largest Lyapunov exponent on the coupling strength) as a validation for judging the performance of the different measures.

In our analysis six measures of synchronization are applied to time series (length N=4096) of three coupled model systems with different individual properties (e.g., power spectra, dimension). However, all show a rather monotonic dependence of the Lyapunov exponent on the coupling strength. Note that this is not necessarily the case for other systems but we deliberately chose systems with such monotonic behavior in order to provide a controlled setting. Model systems comprise coupled Hénon maps as well as coupled Rössler and coupled Lorenz systems. Applied measures are composed of symmetric ones like the maximum linear cross correlation Cmax, the mutual information I, and two different indices of phase synchronization γH and γW (where the phase of the time series is extracted by using the Hilbert or the wavelet transform, respectively) as well as symmetrized variants of the asymmetric measures non-linear interdependence S and event synchronization Q. Using the coupling strength as the first control parameter, it is tested to what extent the different measures are able to distinguish different degrees of coupling. This is essential in many field applications since rarely is the absolute value of synchronization of interest, but rather the change of synchronization between different states, times, or recording sites. The signal-to-noise ratio is used as a second parameter to investigate whether the results of the different measures are robust against contaminations with noise. This is another important aspect for choosing the most suitable measure for an application to field data (e.g. of medical origin). Finally, to assess the perspective of a combined use of different approaches to quantify synchronization, the degree of redundance between the different measures is evaluated by means of a correlation analysis.

The remainder of the paper is organized as follows: The statistical evaluation designed to compare these measures is described in Section 2.1, the methodology for the correlation analysis is outlined in Section 2.2. Results on the measures’ capability to reflect the strength of coupling, their robustness against noise and their mutual correlations are presented in Sections 3.1, 3.2, 3.3, respectively. Conclusions are drawn in Section 4. Finally, in Appendix A the coupling schemes and the underlying model systems are introduced, while the measures of synchronization and their implementation are described in Appendix B.

Section snippets

Criterion for comparing different measures

To compare the different measures of synchronization in terms of their capability to distinguish between different degrees of coupling, an indicator was designed to yield maximum values for those measures for which higher values are obtained for higher coupling strengths. For every model system each measure of synchronization s was estimated at r=81 monotonously increasing coupling strengths, resulting in values si,i=1,,r. If s depends monotonically on the coupling strength, then sisj for ij

Dependence on coupling strength

In Fig. 1 the dependence of six representative measures of synchronization on the coupling strength C between two identical Hénon systems (cf. Appendix A.1) is shown along with the maximum Lyapunov exponent of the responder system (cf. Ref. [24]). The latter attains negative values for couplings larger than 0.7, when generalized synchronization between the systems takes place finally leading into identical synchronization (cf.  Fig. 13). In the regime 0.47<C<0.52 it is also slightly negative

Discussion

The aim of this study was to perform a comparison of different measures of synchronization on model systems to help to decide which measure may be most suitable for an application to field data. To this end we compared different measures regarding their capability to reflect different degrees of coupling between two model systems both in the presence and absence of noise. The dependence on the coupling strength was evaluated using the degree of monotonicity M.

It should be noted that there are

Acknowledgments

Many thanks to R. Quian Quiroga for useful discussions. T.K. acknowledges support from the EU Marie Curie Human Resources and Mobility Activity, R.G.A. from the Alexander von Humboldt Foundation, F.M. intramural funding from BONFOR. T.K., F.M., K.L. and P.G. acknowledge support from the Deutsche Forschungsgemeinschaft [SFB TR3].

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