ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Navrotskaya E. V., Smirnov D. A., Bezruchko B. P. The reconstruction of the couplings structure in the ensemble of oscillators according to the time series via phase dynamics modeling. Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, vol. 27, iss. 1, pp. 41-52. DOI: 10.18500/0869-6632-2019-27-1-41-52

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The reconstruction of the couplings structure in the ensemble of oscillators according to the time series via phase dynamics modeling

Navrotskaya Elena Vladimirovna, Saratov State University
Smirnov Dmitrij Alekseevich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Bezruchko Boris Petrovich, Saratov State University

Topic. In this paper, we investigate the applicability of the known method for detecting the couplings between two oscillators, based on experimental phase dynamics modeling by phases time series in cases where there are several peaks in the signal power spectrum. The problem of revealing the structure of couplings in ensembles (presence, directions and intensity of interactions between elements) by experimental recordings of their oscillations is relevant for systems of different nature and applications. In its solution for the system of two oscillators, the method of reconstruction of phase dynamics models by time series has shown its efficiency provided that there is a single peak in the signal power spectrum. Aim. The investigation of the conditions for the applicability of this method in less favorable cases when there are several peaks in the power spectrum and the width of these peaks is significant. Methods. An ensemble of three Van der Pol oscillators is considered as a test system in the numerical experiment: two coupled oscillators were exposed to the simultaneous influence from the third one. Using the investigated method, the presence of a couplings between the oscillators was estimated from the time series of the oscillator phases. The numerical experiment was carried out at different values of oscillator parameters. Results. The possibility of false conclusions about the structure of couplings in such system is demonstrated. The diagnostic criterion of possible errors based on the estimation of the autocorrelation function of the phase dynamics model residues is proposed. To obtain reliable estimates of interactions in problem situations, pre-filtering of signals was tested.    

  1. Hung Y.C., Hu C.K. Chaotic communication via temporal transfer entropy // Physical Review Letters. 2008. Vol. 101. P. 244102.
  2. Smirnov D.A., Barnikol U.B., Barnikol T.T., Bezruchko B.P., Hauptmann C., Buhrle C., Maarouf M., Sturm V., Freund H.-J., Tass P.A. The generation of parkinsonian tremor as revealed by directional coupling analysis // Europhysics Letters. 2008. Vol. 83. P. 20003.
  3. Mosekilde E., Maistrenko Yu., Postnov D. Chaotic Synchronization. Applications to Living Systems. Singapore: World Scientific, 2002.
  4. Tass P.A. Phase Resetting in Medicine and Biology: Stochastic Modeling and Data Analysis. Berlin, Heidelberg: Springer–Verlag, 1999. 329 p.
  5. Pereda E., Quian Quiroga R., Bhattacharya J. Nonlinear multivariate analysis of neurophysiological signals // Progr. Neurobiol. 2005. Vol. 77. P. 1–37.
  6. Tass P., Smirnov D., Karavaev A., Barnikol U., Barnikol T., Adamchic I., Hauptmann C., Pawelcyzk N., Maarouf M., Sturm V., Freund H.-J., Bezruchko B. The causal relationship between subcortical local field potential oscillations and parkinsonian resting tremor // J. Neural Engineering. 2010. Vol. 7. P. 016009.
  7. Karavaev A.S., Prokhorov M.D., Ponomarenko V.I., Kiselev A.R., Gridnev V.I., Ruban E.I., Bezruchko B.P. Synchronization of low-frequency oscillations in the human cardiovascular system // Chaos. 2009. Vol. 19. P. 033112.
  8. Kazantsev V.B., Nekorkin V.I., Makarenko V.I., Llinas R.R. Olivo-cerebellar clusterbased universal control system // Proc. Natl. Acad. Sci. USA. 2003. Vol. 100, No 22. Pp. 13064.
  9. Lacaux J.-P., Rodriguez E., Le Van Quyen M., Lutz A. Studying single-trials of phase synchronous activity in the brain // Int. J. Bif. Chaos. 2000. Vol. 10. Pp. 2429–2455.
  10. Mokhov I.I., Smirnov D.A. News of RAS. Atmospheric and ocean physics, 2008, no. 44, pp. 283–293.
  11. Barnston A.G., Livezey R.E. Classification, seasonality and persistence of low frequency atmopheric circulation patterns // Mon. Wea. Rev. 1987. Vol. 115. P. 1083.
  12. Mokhov I.I., Smirnov D.A. El Nino Southern Oscillation drives North Atlantic Oscillation as revealed with nonlinear techniques from climatic indices // Geophys. Res. Lett. 2006. Vol. 33. L03708.
  13. Mokhov I.I. SpB: Hydrometeoizdat, 1993. 271 p. (in Russian).
  14. Rosenblum M.G., Pikovsky A.S. Detecting direction of coupling in interacting oscillators // Phys. Rev. E. 2001. Vol. 64. P. 045202(R).
  15. Smirnov D.A., Bezruchko B.P. Detection of coupling in ensembles of stochastic oscillators // Physical Review E. 2009. Vol. 79. P. 046204.
  16. Smirnov D.A., Bezruchko B.P. Estimation of interaction strength and direction from short and noisy time series // Phys. Rev. E. 2003. Vol. 68. P. 046209.
  17. Weinstein L.A., Vakman D.E. Razdelenie chastot v teorii kolebanij i voln. M.: Nauka, 1983, 288 p. (in Russian).
  18. Torrence C., Compo G.P. A practical guide to wavelet analysis. Bull. Am. Meteorol. Soc., 1998, vol. 79, pp. 61–78.
  19. Koronovskij A.A., Hramov A.E. Nepreryvnyj Vejvletnyj Analiz i Ego Prilozheniya. М.: Fizmatlit, 2003, 176 p. (in Russian).
  20. Rosenblum M.G., Pikovsky A., Kurths J., Schafer C. Phase synchronization: From theory to data analysis // Neuro-informatics. Handbook of Biological Physics. 2000. Vol. 4. P. 279.
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