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


Cite this article as:

Smirnov D. A. Revealing nonlinear couplings between stochastic oscillators from time series. Izvestiya VUZ. Applied Nonlinear Dynamics, 2010, vol. 18, iss. 2, pp. 16-38. DOI: https://doi.org/10.18500/0869-6632-2010-18-2-16-38

Language: 
Russian

Revealing nonlinear couplings between stochastic oscillators from time series

Abstract: 

The problem of detection and quantitative characterization of nonlinear directional couplings between stochastic oscillators is considered. Coupling characteristics and a technique for their estimation from time series are suggested. An analytic expression for a statistical significance level of the conclusion about coupling presence is derived that allows a reliable inference from relatively short signals. Performance of the approach is demonstrated in numerical experiments with diverse individual properties of oscillators and different kinds of coupling functions.

DOI: 
10.18500/0869-6632-2010-18-2-16-38
References: 

1. Пиковский А.С., Розеблюм М.Г., Куртс Ю. Синхронизация: фундаментальное нелинейное явление. М.: Техносфера. 2003. 2. Tass P.A. Phase resetting in medicine and biology – stochastic modelling and data analysis. Berlin: Springer, 1999. 3. Boccaletti S., Kurths J., Osipov G., Valladares D., Zhou C. The synchronization of chaotic systems // Phys. Rep. 2002. Vol. 366. P. 1. 4. Mosekilde E., Maistrenko Yu., Postnov D. Chaotic Synchronization. Applications to Living Systems. Singapore: World Scientific, 2002. 5. Balanov A., Janson N., Postnov D., Sosnovtseva O. Synchronization: From Simple to Complex. Berlin: Springer-Verlag, 2008. 6. Rosenblum M.G., Pikovsky A.S. Detecting direction of coupling in interacting oscillators // Phys. Rev. E. 2001. Vol. 64, 045202(R). 7. Smirnov D.A., Bezruchko B.P. Estimation of interaction strength and direction from short and noisy time series // Phys. Rev. E. 2003. Vol. 68, 046209. 8. Palus M., Stefanovska A. Direction of coupling from phases of interacting oscillators: An information-theoretic approach // Phys. Rev. E. 2003. Vol. 67, 055201. 9. Kralemann B., Cimponeriu L., Rosenblum M., Pikovsky A., Mrowka R. Uncovering interaction of coupled oscillators from data // Phys. Rev. E. 2007. Vol. 76, 055201. 10. Kralemann B., Cimponeriu L., Rosenblum M., Pikovsky A., Mrowka R. Phase dynamics of coupled oscillators reconstructed from data // Phys. Rev. E. 2008. Vol. 77, 066205. 11. Baccala L.A., Sameshima K. Partial directed coherence: a new concept in neural structure determination // Biological Cybernetics, 2001. Vol. 84. P. 463. 12. Ancona N., Marinazzo D., Stramaglia S. Radial basis function approach to nonlinear Granger causality of time series // Phys. Rev. E. 2004. Vol. 70, 056221. 13. Schreiber T. Measuring information transfer // Phys. Rev. Lett. 2000. Vol. 85. P. 461. 14. Verdes P.F. Assessing causality from multivariate time series // Phys. Rev. E. 2005. Vol. 72, 026222. 15. Hlavackova-Schindler K., Palus M., Vejmelka M., Bhattacharya J. Causality detection based on information-theoretic approaches in time series analysis // Phys. Rep. 2007. Vol. 441. P. 1. 16. Vejmelka M., Palus M. Inferring the directionality of coupling with conditional mutual information // Phys. Rev. E. 2008. Vol. 77, 026214. 17. Rosenblum M.G., Cimponeriu L., Bezerianos A., Patzak A., Mrowka R. Identification of coupling direction: Application to cardiorespiratory interaction // Phys. Rev. E. 2002. Vol. 65, 041909. 18. Prokhorov M.D., Ponomarenko V.I., Gridnev V.I., Bodrov M.B., Bespyatov A.B. Synchronization between main rhythmic processes in the human cardiovascular system // Phys. Rev. E. 2003. Vol. 68, 041913. 19. Luchinsky D.G., Millonas M.M., Smelyanskiy V.N., Pershakova A., Stefanovska A., McClintock P.V. Nonlinear statistical modeling and model discovery for cardiorespiratory data // Phys. Rev. E. 2005. Vol. 72, 021905. 20. Hramov A.E., Koronovskii A.A., Ponomarenko V.I., Prokhorov M.D. Detection of synchronization from univariate data using wavelet transform // Phys. Rev. E. 2007. Vol. 75, 056207. 21. Sosnovtseva O.V., Pavlolv A.N., Mosekilde E., Holstein-Rathlou N.H., Marsh D.J. Double-wavelet approach to study frequency and amplitude modulation in renal autoregulation // Phys. Rev. E. 2004. Vol. 70, 031915. 22. Pavlov A.N., Sosnovtseva O.V., Pavlova O.N., Mosekilde E., Holstein-RathlouN.-H. Characterizing multimode interaction in renal autoregulation // Physiological Measurements, 2008. Vol. 29. P. 945. 23. Eguia M.C., Dawson S.P., Mindlin G.B. Information transmission and recovery in neural communications channels // Phys. Rev. E. 2000. Vol.62. P. 7111. 24. Kiemel T., Gormley K., Guan L., Williams T., Cohen A. Estimating the strength and direction of functional coupling in the lamprey spinal cord // J. Computational Neuroscience. 2003. Vol. 15. P. 233. 25. Blinowska K.J., Kus R., Kaminski M. Granger causality and information flow in multivariate processes // Phys. Rev. E. 2004. Vol. 70, 050902(R). 26. Pereda E., Quian Quiroga R., Bhattacharya J. Nonlinear multivariate analysis of neurophysiological signals // Progress in Neurobiology. 2005. Vol. 77. P. 1. 27. Brea J., Russell D.F., Neiman A.B. Measuring direction in the coupling of biological oscillators: A case study for electroreceptors of paddlefish // Chaos. 2006. Vol. 16, 026111. 28. Schelter B., Winterhalder M., Eichler M., Peifer M., Hellwig B., Guschlbauer B., Luecking C., Dahlhaus R., Timmer J. Testing for directed influences among neural signals using partial directed coherence // J. Neurosci. Methods. 2005. Vol. 152. P. 210. 29. Wang S., Chen Y., Ding M., Feng J., Stein J.F., Aziz T.Z., Liu X.J. Revealing the dynamic causal interdependence between neural and muscular signals in Parkinsonian tremor // J. Franklin Institute. 2007. Vol. 344. P. 180. 30. Osterhage H., Mormann F., Wagner T., Lehnertz K. Measuring the directionality of coupling: Phase versus state space dynamics and application to EEG time series // Int. J. Neural Syst. 2007. Vol. 17. P. 139. 31. Smirnov D.A., Barnikol U.B., Barnikol T.T., Bezruchko B.P., Hauptmann C., Buehrle C., Maarouf M., Sturm V., Freund H.-J., Tass P.A. The generation of Parkinsonian tremor as revealed by directional coupling analysis // Europhys. Lett. 2008. Vol. 83, 20003. 32. Mokhov I.I., Smirnov D.A. El Nino Southern Oscillation drives North Atlantic Oscillation as revealed with nonlinear techniques from climatic indices // Geophysical Research Letters. 2006. Vol. 33, 024557. 33. Smirnov D.A., Bezruchko B.P. Detection of couplings in ensembles of stochastic oscillators // Phys. Rev. E. 2009. Vol. 79, 046204. 34. Kori H., Kuramoto Y. Slow switching in globally coupled oscillators: robustness and occurrence through delayed coupling // Phys. Rev. E. 2001. Vol. 63, 046214. 35. Kuramoto Y. Chemical Oscillations, Waves and Turbulence. Berlin: Springer-Verlag, 1984. 36. Pikovsky A.S., Rosenblum M.G., Kurths J. Phase synchronization in regular and chaotic systems // Int. J. Bifurc. Chaos. 2000. Vol.10, No 10. P. 2291. 37. Gabor D. Theory of communication // J. Inst. Elect. Eng. (London). 1946. Vol. 93. P. 429. 38. Rosenblum M.G., Pikovsky A.S., Kurths J., Schaefer C., Tass P.A. Phase synchronization: from theory to data analysis // Neuro-informatics. Handbook of Biological Physics. Edited by F. Moss and S. Gielen. New York: Elsevier Science, 2001. P. 279. 39. Kendall M.C., Stuart A. The advanced theory of statistics. New York: Hafner, 1979. 40. Никитин Н.Н., Разевиг В.Д. Методы цифрового моделирования стохастических дифференциальных уравнений и оценка их погрешностей // Журнал выч. математики и мат. физики. 1978. Т. 18, No 1. С. 106. 41. Анищенко В.С., Вадивасова Т.Е., Окрокверцхов Г.А., Стрелкова Г.И. Статистические свойства динамического хаоса // УФН. 2005. Т. 175. С. 163. 42. Thompson J.M.T. and Stewart H.B. Nonlinear Dynamics and Chaos. New York: Wiley, 1987. 43. Morris C., Lecar H. Voltage oscillations in the barnacle giant muscle fiber // Biophys. J. 1981. Vol. 35. P. 193. 44. Izhikevich E.M. Neural excitability, spiking and bursting // Int. J. Bifurc. Chaos. 2000. Vol. 10. P. 1171. 45. Ermentrout G.B., Kopell N. Oscillator death in systems of coupled neural oscillators // SIAM J. Appl. Math. 1990. Vol. 50. P. 125. 46. Smirnov D., Schelter B., Winterhalder M., Timmer J. Revealing direction of coupling between neuronal oscillators from time series: Phase dynamics modeling versus partial directed coherence // Chaos. 2007. Vol. 17, 013111.

Short text (in English):
(downloads: 6)
Full text:
(downloads: 8)