^{1,2,a)}, Marco Thiel

^{1}, Jürgen Kurths

^{3}, Konstantin Mergenthaler

^{4}and Ralf Engbert

^{4}

### Abstract

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.

I. INTRODUCTION

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

VI. CONCLUSIONS

### Key Topics

- Time series analysis
- 45.0
- Synchronization
- 28.0
- Eyes
- 9.0
- Data analysis
- 7.0
- Brain
- 6.0

## Figures

Recurrence plot of a trajectory from the Lorenz system [Eq. (A2)].

Recurrence plot of a trajectory from the Lorenz system [Eq. (A2)].

A: Two neighbors in the one-dimensional space with distance . The nonoverlapping regions (NOR) have the length . B: Two neighbors in the two-dimensional space with distance . The nonoverlapping regions (NOR) depend on and on the radius that defines the neighborhoods.

A: Two neighbors in the one-dimensional space with distance . The nonoverlapping regions (NOR) have the length . B: Two neighbors in the two-dimensional space with distance . The nonoverlapping regions (NOR) depend on and on the radius that defines the neighborhoods.

Computation of the average number of twins depending on the threshold and the number of points of the time series in the one-dimensional case: A, B [analytical, Eq. (3)], C, D (random uniformly distributed time series), and E, F (chaotic Bernoulli map). The inset in A presents a zoom to show that for the analytical expression for the average number of twins is equal to zero, in accordance with the numerical simulations.

Computation of the average number of twins depending on the threshold and the number of points of the time series in the one-dimensional case: A, B [analytical, Eq. (3)], C, D (random uniformly distributed time series), and E, F (chaotic Bernoulli map). The inset in A presents a zoom to show that for the analytical expression for the average number of twins is equal to zero, in accordance with the numerical simulations.

A: Distribution of distances of the bivariate random uniformly distributed time series (points: numerical simulations; solid: curve fitted). B: Distribution of distances of the Lorenz system [Eq. (A2)] (points: numerical simulations; solid: curve fitted). The curve fitted in both cases has the form .

A: Distribution of distances of the bivariate random uniformly distributed time series (points: numerical simulations; solid: curve fitted). B: Distribution of distances of the Lorenz system [Eq. (A2)] (points: numerical simulations; solid: curve fitted). The curve fitted in both cases has the form .

Computation of the average number of twins depending on the threshold and the number of points of the time series in the two-dimensional case: A, B [analytical estimation, Eq. (4)], C, D (bivariate random uniformly distributed time series), and E, F [chaotic Lorenz system, Eq. (A2)]. The length of the time series used for the left panels was (A, C, E) and the values for the thresholds were (B), (D) and (F).

Computation of the average number of twins depending on the threshold and the number of points of the time series in the two-dimensional case: A, B [analytical estimation, Eq. (4)], C, D (bivariate random uniformly distributed time series), and E, F [chaotic Lorenz system, Eq. (A2)]. The length of the time series used for the left panels was (A, C, E) and the values for the thresholds were (B), (D) and (F).

Comparison between the twin surrogates and the “real” trajectories for several statistics for the logistic map [Eq. (A1)] depending on the threshold of the recurrence matrix. A: Mean value of the ACF. B: Standard deviation of the ACF. C: Mean value of the MI. D: Standard deviation of the MI. E: MDL, and F: MVL.

Comparison between the twin surrogates and the “real” trajectories for several statistics for the logistic map [Eq. (A1)] depending on the threshold of the recurrence matrix. A: Mean value of the ACF. B: Standard deviation of the ACF. C: Mean value of the MI. D: Standard deviation of the MI. E: MDL, and F: MVL.

Comparison between the twin surrogates and the “real” trajectories for several statistics for the Lorenz system [Eq. (A2)] depending on the threshold of the recurrence matrix. A: Mean value of the ACF. B: Standard deviation of the ACF. C: Mean value of the MI. D: Standard deviation of the MI. E: MDL, and F: MVL.

Comparison between the twin surrogates and the “real” trajectories for several statistics for the Lorenz system [Eq. (A2)] depending on the threshold of the recurrence matrix. A: Mean value of the ACF. B: Standard deviation of the ACF. C: Mean value of the MI. D: Standard deviation of the MI. E: MDL, and F: MVL.

Comparison between the twin surrogates and the “real” trajectories for several statistics for the AR-model [Eq. (A3)] depending on the threshold of the recurrence matrix. A: Mean value of the ACF. B: Standard deviation of the ACF. C: Mean value of the MI. D: Standard deviation of the MI. E: MDL, and F: MVL.

Comparison between the twin surrogates and the “real” trajectories for several statistics for the AR-model [Eq. (A3)] depending on the threshold of the recurrence matrix. A: Mean value of the ACF. B: Standard deviation of the ACF. C: Mean value of the MI. D: Standard deviation of the MI. E: MDL, and F: MVL.

Errors of the twin surrogates depending on the embedding delay for the Lorenz system. A: Mean value of the ACF. B: Standard deviation of the ACF. C: Mean value of the MI. D: Standard deviation of the MI. E: MDL, and F: MVL. The curves have been computed for the following values of the embedding dimension: (+ solid), (× long-dashed), (∗ short-dashed), and (◻ dotted).

Errors of the twin surrogates depending on the embedding delay for the Lorenz system. A: Mean value of the ACF. B: Standard deviation of the ACF. C: Mean value of the MI. D: Standard deviation of the MI. E: MDL, and F: MVL. The curves have been computed for the following values of the embedding dimension: (+ solid), (× long-dashed), (∗ short-dashed), and (◻ dotted).

Errors of the twin surrogates depending on the length of the time series for the logistic map. A: Mean value of the ACF. B: Standard deviation of the ACF. C: Mean value of the MI. D: Standard deviation of the MI. E: MDL, and F: MVL.

Errors of the twin surrogates depending on the length of the time series for the logistic map. A: Mean value of the ACF. B: Standard deviation of the ACF. C: Mean value of the MI. D: Standard deviation of the MI. E: MDL, and F: MVL.

Errors of the twin surrogates depending on the length of the time series for the Lorenz system. A: Mean value of the ACF. B: Standard deviation of the ACF. C: Mean value of the MI. D: Standard deviation of the MI. E: MDL, and F: MVL.

Errors of the twin surrogates depending on the length of the time series for the Lorenz system. A: Mean value of the ACF. B: Standard deviation of the ACF. C: Mean value of the MI. D: Standard deviation of the MI. E: MDL, and F: MVL.

Errors of the twin surrogates depending on the length of the time series for the AR-model. A: Mean value of the ACF. B: Standard deviation of the ACF. C: Mean value of the MI. D: Standard deviation of the MI. E: MDL, and F: MVL.

Errors of the twin surrogates depending on the length of the time series for the AR-model. A: Mean value of the ACF. B: Standard deviation of the ACF. C: Mean value of the MI. D: Standard deviation of the MI. E: MDL, and F: MVL.

Simultaneous recording of left and right fixational eye movements. A) Horizontal component of the left (red, solid line) and right (blue, dashed line) eye. B) Detrended data.

Simultaneous recording of left and right fixational eye movements. A) Horizontal component of the left (red, solid line) and right (blue, dashed line) eye. B) Detrended data.

Comparison between the twin surrogates and the “real” trajectories for several statistics for the fixational eye movements from one subject depending on the threshold of the recurrence matrix. A: Mean value of the ACF. B: Standard deviation of the ACF. C: Mean value of the MI. D: Standard deviation of the MI. E: MDL, and F: MVL.

Comparison between the twin surrogates and the “real” trajectories for several statistics for the fixational eye movements from one subject depending on the threshold of the recurrence matrix. A: Mean value of the ACF. B: Standard deviation of the ACF. C: Mean value of the MI. D: Standard deviation of the MI. E: MDL, and F: MVL.

A) Twin surrogate of the left (red, solid line) and right (blue, dashed line) filtered fixational eye movements, horizontal component. B) Segment of another filtered trial of the same subject. The twin surrogates reproduce the structure of the measured time series very well.

A) Twin surrogate of the left (red, solid line) and right (blue, dashed line) filtered fixational eye movements, horizontal component. B) Segment of another filtered trial of the same subject. The twin surrogates reproduce the structure of the measured time series very well.

Histogram of the values obtained for with (bars). The dashed vertical line indicates the value obtained for for the original data. Hence, in this case the null hypothesis is rejected, which indicates that there is PS between the left and right fixational eye movements. This test was performed with the data from subject 2, trial 10.

Histogram of the values obtained for with (bars). The dashed vertical line indicates the value obtained for for the original data. Hence, in this case the null hypothesis is rejected, which indicates that there is PS between the left and right fixational eye movements. This test was performed with the data from subject 2, trial 10.

## Tables

Results for the test for PS between the trajectories of the left and right fixational eye movements performed for 30 trials for 21 subjects. Trials in which the participants blinked, were discarded. 100 twin surrogates were used for the test.

Results for the test for PS between the trajectories of the left and right fixational eye movements performed for 30 trials for 21 subjects. Trials in which the participants blinked, were discarded. 100 twin surrogates were used for the test.

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