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

For citation:

Astakhova D. I., Sysoeva M. V., Sysoev I. V. Effect of nonlinearity on coupling estimations between oscillators using partial directed coherence approach. Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, vol. 27, iss. 6, pp. 8-24. DOI: 10.18500/0869-6632-2019-27-6-8-24

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Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
530.182
DOI: 
10.18500/0869-6632-2019-27-6-8-24

Effect of nonlinearity on coupling estimations between oscillators using partial directed coherence approach

Autors: 
Astakhova Daria Ivanovna, Saratov State University
Sysoeva Marina Vyacheslavovna, Saratov State University
Sysoev Ilya Vyacheslavovich, Saratov State University
Abstract: 

The purpose of this work is to determine the ability of the partial directed coherence method to identify directed interactions between nonlinear systems correctly in presence of nonlinear couplings between systems, as well as in the
case of measured signals generated by objects of high dimension. The other purpose is to determine the dependence of the coupling estimation results on the parameters: series length, sampling rate, model dimension and coupling architecture.

Methods. Ensembles composed of four differently coupled oscillators and dynamical mesoscale model of epilepsy are considered as test systems. Surrogate time series constructed by permutation of realization are used to determine the
significance of the results.

Results. Coupling architecture in ensembles of small-dimensional oscillators can be correctly identified for linear and nonlinear systems in both cases of linear and nonlinear coupling. For complex composite signals, when each measured time series is the sum of signals from many individual oscillators, the technique is not specific enough, revealing non-existent connections, and it is not sensitive enough, missing the existing ones.

Outcomes. The criteria for applying the partial directed coherence method to different signals are formulated. The measure does not show indirect couplings at sufficient series length, sampling rate and model dimension in contrast to the pairwise methods like Granger causality or transfer entropy. The measure works well for noisy time series. The method allows to study connectivity in an ensemble of arbitrary number of oscillators. The method allows to determine at what frequencies the interaction occurs. The partial directed coherence method gives acceptable results for series of length of 80 and more characteristic periods in comparison with the Granger causality method, for which the efficiency is declared already at 4–16 characteristic periods.

Key words: 
partial directed coherence
coupling
nonlinearity
Nonlinear systems
nonlinear coupling
high–dimensional system
Reference: 
  1. Gourevitch B., Le Bouquin-Jeannes R., Faucon G. Linear and nonlinear causality between signals: Methods, examples and neurophysiological applications // Biological Cybernetics. 2006. Vol. 95(4). P. 349–369.
  2. Sakkalis V. Review of advanced techniques for the estimation of brain connectivity measured with EEG/MEG // Computers in Biology and Medicine. 2011. Vol. 41, iss. 12. P. 1110–1117.
  3. Pereda E., QuianQuiroga R., Bhattacharya J. Nonlinear multivariate analysis of neurophysiological signals // Progress in Neurobiology. 2005. Vol. 77, iss. 1–2. P. 1–37.
  4. Sysoeva M.V., Sitnikova E., Sysoev I.V., Bezruchko B.P., van Luijtelaar G. Application of adaptive nonlinear Granger causality: Disclosing network changes before and after absence seizure onset in a genetic rat model // Journal of Neuroscience Methods. 2014. Vol. 226. P. 33–41.
  5. Kurgansky A.V. Study of cortico-cortical functional connectivity with vector autoregressive model of multichannel EEG. Zh Vyssh Nerv Deyat+, 2010. vol. 60(6), pp. 740–759 (in Russian).
  6. Kurgansky A.V. Quantitative measures of cortical functional connectivity: A state-of-the-art brief survey. Human Physiology, 2013, vol. 39, no. 4, pp. 432–440.
  7. Sakkalis V., Doru Giurcaneanu C., Xanthopoulos P., Zervakis M.E., Tsiaras V., Yang Y., Kara- konstantaki E., Micheloyannis S. Assessment of linear and nonlinear synchronization measures for analyzing EEG in a mild epileptic paradigm // IEEE Transactions on Information Technology in Biomedicine. 2009. Vol. 13, iss. 4. P. 433–441.
  8. Micheloyannis S., Sakkalis V., Vourkas M., Stam C.J., Simos P.G. Neural networks involved in mathematical thinking: Evidence from linear and non-linear analysis of electroencephalographic activity // Neuroscience Letters. 2005. Vol. 373, iss. 3. P. 212–217.
  9. Luttjohann A., van Luijtelaar G. The dynamics of cortico-thalamo-cortical interactions at the transition from pre-ictal to ictal LFPs in absence epilepsy // Neurobiology of Disease. 2012. Vol. 47. P. 47–60.
  10. Sysoeva M.V., Luttjohann A., van Luijtelaar G. and Sysoev I.V. Dynamics of directional coupling underlying spike-wave discharges // Neuroscience. 2016. Vol. 314. P. 75–89.
  11. Lehnertz K., Andrzejak R.G., Arnhold J., Kreuz T., Mormann F., Rieke C., Widman G., and Elger C.E. Nonlinear EEG analysis in epilepsy: Its possible use for interictal focus localization, seizure anticipation, and prevention // Journal of Clinical Neurophysiology. 2001. Vol. 18, iss. 3. P. 209–222.
  12. Tass P., Smirnov D., Karavaev A., Barnikol U., Barnikol T., Adamchic I., Hauptmann C., Pawelcyzk N., Maarouf M., Sturm V., Freund H.-J., and Bezruchko B. The causal relationship between subcortical local field potential oscillations and Parkinsonian resting tremor // Journal of Neural Engineering. 2010. Vol. 7. 016009.
  13. Smirnov Dmitry A. Quantifying causal couplings via dynamical effects: A unifying perspective // Phys. Rev. E. 2014. Vol. 90. 062921.
  14. Baccala L.A., Takahashi D.Y. Partial directed coherence: A new concept in neural structure determination // Biological Cybernetics. 2001. Vol.84. P.272–273.
  15. Sameshima K., Baccala L.A. Using partial directed coherence to describe neuronal ensemble interactions // Journal of Neuroscience Methods. 1999. Vol. 94. P. 93–103.
  16. Takahashi D.Y., Baccala L.A., Sameshima K. Connectivity inference between neural structures via partial directed coherence // Journal of Applied Statistics. 2007. Vol. 34(10). P. 1255–1269.
  17. Granger C.W.J. Investigating causal relations by econometric models and cross-spectral methods // Econometrica. 1969. 37(3). P. 424–438.
  18. Baccala L.A., Takahashi D.Y., Sameshima K. Directed transfer function: Unified asymptotic theory and some of its implications // IEEE Transactions on Biomedical Engineering. 2016. Vol. 63(12). P. 2450–2460.
  19. Milde T., Schwab K., Walther M., Eiselt M., Schelenz C., Voss A., Witte H. Time-variant partial directed coherence in analysis of the cardiovascular system: A methodological study // Physiological measurement. 2011. Vol. 32. P. 1787–1805.
  20. Schelter B., Timmer J., Eichler M. Assessing the strength of directed influences among neural signals using renormalized partial directed coherence // Journal of Neuroscience Methods. 2009. Vol.179. P.121–130.
  21. Chen Y., Rangarajan G., Feng J., Ding M. Analyzing multiple nonlinear time series with extended Granger causality // Phys. Lett. A. 2004. 324(1). P. 26–35.
  22. Kornilov M.V., and Sysoev I.V. Recovering the architecture of links in a chain of three unidirectionally coupled systems using the Granger-causality test. Technical Physics Letters, 2018, vol. 44, no. 5, pp. 445–449.
  23. Sato J.R., Takahashi D.Y., Arcuri S.M., Sameshima K., Morettin P.A., Baccala L.A. Frequency domain connectivity identification: An application of partial directed coherence in fMRI // Human Brain Mapping. 2009. Vol. 30. P. 452–461.
  24. Sommerlade L., Eichler M., Jachan M., Henschel K., Timmer J., Schelter B. Estimating causal dependencies in networks of nonlinear stochastic dynamical systems // Physical Review E. 2009. Vol. 80, iss. 5. 
  25. 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(11).
  26. Medvedeva T.M., Sysoeva M.V., Sysoev, I.V. Coupling analysis between thalamus and cortex in mesoscale model of spike-wave discharges from time series of summarized activity of model neurons // Proceedings of 2nd School on Dynamics of Complex Networks and their Application in Intellectual Robotics. 2018. 8589208. P. 137–138.
  27. Volnova A.B., Lenkov D.N. Absence epilepsy: Mechanisms of hypersyncronization of neuronal networks. Medical academic journal, 2012, vol. 12, no. 1, pp. 7–19.
  28. Tikhonov V.I., Mironov M.A. Markovian processes. Moscow: Soviet Radio, 1977, 488 p.
  29. Rabinovich M.I., Trubetskov D.I. Introduction to the Theory of Oscillations and Waves. «Regular and chaotic dynamics», 2000, 560 p. (in Russian).
  30. Sysoeva M.V., Medvedeva T.M. Optimization of Granger causation method parameters for the study of limbic epilepsy. Izvestia VUZ. Applied Nonlinear Dynamics, 2018, vol. 26, iss. 5, pp. 39–62 (in Russian). 
  31. Sysoeva M.V. , Kuznetsova G.D., Sysoev I.V. Modelling EEG signals from rats when analysing absence epilepsy in application to analysis of coupling between brain areas Biophysics, 2016, vol. 61(4), pp. 661–669.
  32. Medvedeva T.M., SysoevaM.V., van Luijtelaar G., Sysoev I.V. Modeling spike-wave discharges by a complex network of neuronal oscillators // Neural Networks. 2018. Vol. 98. P. 271–282.
Received: 
18.07.2019
Accepted: 
13.10.2019
Published: 
02.12.2019
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