The method of twin surrogates has been introduced to test for phase synchronization of complex systems in the case of passive experiments. In this paper we derive new analytical expressions for the number of twins depending on the size of the neighborhood, as well as on the length of the trajectory. This allows us to determine the optimal parameters for the generation of twin surrogates. Furthermore, we determine the quality of the twin surrogates with respect to several linear and nonlinear statistics depending on the parameters of the method. In the second part of the paper we perform a hypothesis test for phase synchronization in the case of experimental data from fixational eye movements. These miniature eye movements have been shown to play a central role in neural information processing underlying the perception of static visual scenes. The high number of data sets (21 subjects and 30 trials per person) allows us to compare the generated twin surrogates with the “natural” surrogates that correspond to the different trials. We show that the generated twin surrogates reproduce very well all linear and nonlinear characteristics of the underlying experimental system. The synchronizationanalysis of fixational eye movements by means of twin surrogates reveals that the synchronization between the left and right eye is significant, indicating that either the centers in the brain stem generating fixational eye movements are closely linked, or, alternatively that there is only one center controlling both eyes.
In a typical laboratory experiment, in which phase synchronization of two systems is studied, the coupling strength between both systems is systematically increased, until both systems adapt their rhythms, and hence, become phase synchronized. In the case of passive experiments, it is not possible to systematically vary the coupling strength. This is the case in many natural systems, such as, the synchronization among the electrical activity of different brain areas. There, we have only access to one single value of the coupling strength. Computing the phase synchronization index in these cases is not enough to assess the statistical significance of the result. The method of twin surrogates has been proposed to overcome this problem, allowing the performance of a hypothesis test that assess the significance of the result. In this paper, we revisit the method of twin surrogates and derive new analytical expressions for the number of twins depending on the size of the recurrence neighborhood and the number of points of the trajectory. These results allow us to determine the optimal parameters for the generation of twin surrogates, which is a very relevant problem in the case of experimental data. Moreover, we validate the method of twin surrogates comparing the generated surrogates to “natural” surrogates in an experimental system consisting of fixational eye movements, and show that the phase synchronization of the left and right fixational eye movements is statistically significant.
We thank Norbert Marwan for fruitful discussions. M. C. R. would like to acknowledge the Scottish Universities Life Science Alliance (SULSA) for the financial support. M. T. would like to acknowledge the RCUK academic fellowship from EPSRC. J. K. and R. E. acknowledge the Research Group of Computational Modeling of Behavioral and Cognitive Dynamics, funded by DFG.
II. ALGORITHM FOR THE GENERATION OF TWIN SURROGATES
III. NUMBER OF TWINS
IV. COMPARISON OF TWIN SURROGATES WITH ORIGINAL TRAJECTORIES
A. Dependence on
B. Dependence on embedding parameters
C. Dependence on the number of data points
V. APPLICATION TO EXPERIMENTAL DATA FROM A PASSIVE EXPERIMENT: SYNCHRONIZATION OF FIXATIONAL EYE MOVEMENTS
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