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Reconstruction of ensembles of nonlinear neurooscillators with sigmoid coupling function

  • Ilya V. Sysoev
  • Vladimir I. Ponomarenko
  • Mikhail D. Prokhorov
Original Paper
  • 29 Downloads

Abstract

Inferring information about interactions between oscillatory systems from their time series is a highly debated problem. However, many approaches for solving this problem consider either linear systems or linear couplings. We propose a method for the reconstruction of ensembles of nonlinearly coupled neurooscillators described by first-order nonlinear differential equations. The method is based on the minimization of a special target function for each oscillator in the ensemble separately. To find the solution of optimization problem the nonlinear least-squares routine is used. The method does not exploit any parameterization for approximation of nonlinear functions of individual nodes. In addition, an original two-step algorithm for the removal of spurious couplings is proposed based on the clusterization of coefficients of the reconstructed coupling functions and the analysis of their variation. The method efficiency is shown for periodic and chaotic vector time series for ensembles of different size that contain from 8 to 32 oscillators. These oscillators have a cubic nonlinearity and sigmoid is considered as a coupling function. The effect of measurement noise on the results of coupling architecture reconstruction is studied in detail and the method is shown to be effective for relatively high noise (signal to noise ratio equal to eight).

Keywords

Network reconstruction Time series Neurooscillators Nonlinear coupling 

Notes

Acknowledgements

This research was funded by the Russian Science Foundation, Grant No. 14-12-00291.

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Boccaletti, S., Latora, V., Morenod, Y., Chavez, M., Hwang, D.-U.: Complex networks: structure and dynamics. Phys. Rep. 424, 175–308 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Sporns, O., Chialvo, D.R., Kaiser, M., Hilgetag, C.C.: Organization, development and function of complex brain networks. Trends Cognit. Sci. 8(9), 418–425 (2004)CrossRefGoogle Scholar
  3. 3.
    Sompolinsky, H., Crisanti, A., Sommers, H.E.: Chaos in random neural networks. Phys. Rev. Lett. 61(3), 259–262 (1988)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Sysoev, I.V., Ponomarenko, V.I., Pikovsky, A.: Reconstruction of coupling architecture of neural field networks from vector time series. Commun. Nonlinear Sci. Numer. Simulat. 57, 342–351 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Shandilya, S.G., Timme, M.: Inferring network topology from complex dynamics. N. J. Phys. 13(1), 013004 (2011)CrossRefGoogle Scholar
  6. 6.
    Sysoev, I.V., Ponomarenko, V.I., Kulminsky, D.D., Prokhorov, M.D.: Recovery of couplings and parameters of elements in networks of time-delay systems from time series. Phys. Rev. E 94, 052207 (2016)CrossRefGoogle Scholar
  7. 7.
    Pikovsky, A.: Reconstruction of a neural network from a time series of firing rates. Phys. Rev. E 93, 062313 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Xu, Y., Zhou, W., Fang, J.: Topology identification of the modified complex dynamical network with non-delayed and delayed coupling. Nonlinear Dyn. 68(1–2), 195–205 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Yang, X., Wei, T.: Revealing network topology and dynamical parameters in delay-coupled complex network subjected to random noise. Nonlinear Dyn. 82, 319–332 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mokhov, I.I., Smirnov, D.A.: El Niño – southern oscillation drives north atlantic oscillation as revealed with nonlinear techniques from climatic indices. Geophys. Res. Lett. 33, L03708 (2006)CrossRefGoogle Scholar
  11. 11.
    Kaminski, M., Brzezicka, A., Kaminski, J., Blinowska, K.: Measures of coupling between neural populations based on Granger causality principle. Front. Comput. Neurosci. 10(OCT), 114 (2016)Google Scholar
  12. 12.
    Porta, A., Faes, L.: Wiener–Granger causality in network physiology with applications to cardiovascular control and neuroscience. Proceedings of the IEEE, 12 (2015)Google Scholar
  13. 13.
    Chen, Y., Rangarajan, G., Feng, J., Ding, M.: Analyzing multiple nonlinear time series with extended Granger causality. Phys. Lett. A 324(1), 26–35 (2004)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kornilov, M.V., Medvedeva, T.M., Bezruchko, B.P., Sysoev, I.V.: Choosing the optimal model parameters for Granger causality in application to time series with main timescale. Chaos, Solitons Fractal. 82, 11–21 (2016)CrossRefGoogle Scholar
  15. 15.
    Baccala, L., Sameshima, K.: Partial directed coherence: a new concept in neural structure determination. Biol. Cybern. 84, 463–474 (2001)CrossRefGoogle Scholar
  16. 16.
    Kamiński, M., Ding, M., Truccolo, W.A., Bressler, S.L.: Evaluating causal relations in neural systems: Granger causality, directed transfer function and statistical assessment of significance. Biol. Cybern. 85, 145–157 (2001)CrossRefGoogle Scholar
  17. 17.
    Rosenblum, M.G., Pikovsky, A.S.: Detecting direction of coupling in interacting oscillators. Phys. Rev. E 64, 045202(R) (2001)CrossRefGoogle Scholar
  18. 18.
    Tokuda, I.T., Jain, S., Kiss, I.Z., Hudson, J.L.: Inferring phase equations from multivariate time series. Phys. Rev. Lett. 99, 064101 (2007)CrossRefGoogle Scholar
  19. 19.
    Koutlis, C., Kugiumtzis, D.: Discrimination of coupling structures using causality networks from multivariate time series. Chaos 26, 093120 (2016)CrossRefGoogle Scholar
  20. 20.
    Kralemann, B., Pikovsky, A., Rosenblum, M.: Reconstructing phase dynamics of oscillator networks. Chaos 21, 025104 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wu, X., Wang, W., Zheng, W.X.: Inferring topologies of complex networks with hidden variables. Phys. Rev. E 86, 046106 (2012)CrossRefGoogle Scholar
  22. 22.
    Yang, G., Wang, L., Wang, X.: Reconstruction of complex directional networks with group lasso nonlinear conditional Granger causality. Sci. Rep. 7(1), 2991 (2017)CrossRefGoogle Scholar
  23. 23.
    Smirnov, D.A., Andrzejak, R.G.: Detection of weak directional coupling: phase-dynamics approach versus state-space approach. Phys. Rev. E 71, 036207 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    De Feo, O., Carmeli, C.: Estimating interdependences in networks of weakly coupled deterministic systems. Phys. Rev. E 77(2), 026711 (2008)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Wu, X., Sun, Z., Liang, F., Yu, C.: Online estimation of unknown delays and parameters in uncertain time delayed dynamical complex networks via adaptive observer. Nonlinear Dyn. 73(3), 1753–1768 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Shemyakin, V., Haario, H.: Online identification of large-scale chaotic system. Nonlinear Dyn. 93(2), 961–975 (2018)CrossRefGoogle Scholar
  27. 27.
    Wang, W., Yang, R., Lai, Y., Kovanis, V., Grebogi, C.: Predicting catastrophes in nonlinear dynamical systems by compressive sensing. Phys. Rev. Lett. 106, 154101 (2011)CrossRefGoogle Scholar
  28. 28.
    Han, X., Shen, Z., Wang, W.-X., Di, Z.: Robust reconstruction of complex networks from sparse data. Phys. Rev. Lett. 114, 28701 (2015)CrossRefGoogle Scholar
  29. 29.
    Brunton, S., Proctor, J., Kutz, J.: Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl. Acad. Sci. U. S. A. 113, 3932–7 (2016)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Mangan, N., Brunton, S., Proctor, J., Kutz, J.: Inferring biological networks by sparse identification of nonlinear dynamics. IEEE Trans. Mol. Biol. Multi-Scale Commun. 2, 52–63 (2016)CrossRefGoogle Scholar
  31. 31.
    Casadiego, J., Nitzan, M., Hallerberg, S., Timme, M.: Model-free inference of direct network interactions from nonlinear collective dynamics. Nat. Commun. 8, 2192 (2017)CrossRefGoogle Scholar
  32. 32.
    Gouesbet, G., Meunier-Guttin-Cluzel, G., Menard, O.: Chaos and its Reconstruction. Nova Science Publishers, New York (2003)Google Scholar
  33. 33.
    Bezruchko, B.P., Smirnov, Da: Extracting Knowledge From Time Series: (An Introduction to Nonlinear Empirical Modeling). Springer Series in Synergetics. Springer, New York (2010)CrossRefGoogle Scholar
  34. 34.
    Wang, W.-X., Lai, Y.-C., Grebogi, C.: Data based identification and prediction of nonlinear and complex dynamical systems. Phys. Rep. 644, 1–76 (2016)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Timme, M., Casadiego, J.: Revealing networks from dynamics: an introduction. J. Phys. A Math. Theor. 47, 343001 (2014)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Smirnov, D.A.: Quantifying causal couplings via dynamical effects: a unifying perspective. Phys. Rev. E 90, 062921 (2014)CrossRefGoogle Scholar
  37. 37.
    Richards, F.J.: A flexible growth function for empirical use. J. Exp. Bot. 10(2), 290–300 (1959)CrossRefGoogle Scholar
  38. 38.
    Levenberg, K.: A method for the solution of certain non-linear problems in least squares. Q. Appl. Math. 2, 164–168 (1944)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Marquardt, D.: An algorithm for least-squares estimation of nonlinear parameters. SIAM J. Appl. Math. 11(2), 431–441 (1963)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Coleman, T.F., Li, Y.: An interior trust region approach for nonlinear minimization subject to bounds. SIAM J. Opt. 6, 418–445 (1996)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Kera, H., Hasegawa, Y.: Noise-tolerant algebraic method for reconstruction of nonlinear dynamical systems. Nonlinear Dyn. 85(1), 675–692 (2016)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Upadhyay, R.K., Mondal, A., Paul, C.: A method for estimation of parameters in a neural model with noisy measurements. Nonlinear Dyn. 85(4), 2521–2533 (2016)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Savitzky, A., Golay, M.: Smoothing and differentiation of data by simplified least squares procedures. Anal. Chem. 38(8), 1627–1639 (1964)CrossRefGoogle Scholar
  44. 44.
    Moré, J.J., Sorensen, D.C., Hillstrom, K.E., Garbow, B.S.: The minpack project. In: Cowell, W.J. (ed.) Sources and Development of Mathematical Software, pp. 88–111. Prentice-Hall, Upper Saddle River (1984)Google Scholar
  45. 45.
    Millman, K.J., Aivazis, M.: Python for scientists and engineers. Comput. Sci. Eng. 13, 9–12 (2011)CrossRefGoogle Scholar
  46. 46.
    Rosenstein, M.T., Collins, J.J., De Luca, C.J.: A practical method for calculating largest Lyapunov exponents from small data sets. Physica D: Nonlinear Phenom. 65, 117–134 (1993)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Baake, E., Baake, M., Bock, H., Briggs, K.: Fitting ordinary differential equations to chaotic data. Phys. Rev. A 45(8), 5524–5529 (1992)CrossRefGoogle Scholar
  48. 48.
    Sysoev, I.V., Smirnov, D.A., Bezruchko, B.P.: Identification of chaotic systems with hidden variables (modified Bock’s algorithm). Chaos, Solitons Fractal. 29, 82–90 (2006)CrossRefGoogle Scholar
  49. 49.
    Smirnov, D.A., Sysoev, I.V., Seleznev, E.P., Bezruchko, B.P.: Global reconstruction from nonstationary data. Tech. Phys. Lett. 29(10), 824–827 (2003)CrossRefGoogle Scholar
  50. 50.
    Lüttjohann, A., van Luijtelaar, G.: The dynamics of cortico-thalamo-cortical interactions at the transition from pre-ictal to ictal lfps in absence epilepsy. Neurobiol. Dis. 47, 47–60 (2012)CrossRefGoogle Scholar
  51. 51.
    Coenen, A.M.L., van Luijtelaar, E.L.J.M.: Genetic animal models for absence epilepsy: a review of the WAG/Rij strain of rats. Behav. Genet. 33, 635–655 (2003)CrossRefGoogle Scholar
  52. 52.
    Jun, J.J., Steinmetz, N.A., Siegle, J.H., Denman, D.J., Bauza, M., Barbarits, B., Lee, A.K., Anastassiou, C.A., Çağatay Aydın, A.A., Barbic, M., Blanche, T.J., Bonin, V., Couto, J., Dutta, B., Gratiy, S.L., Gutnisky, D.A., Häusser, M., Karsh, B., Ledochowitsch, P., Lopez, C.M., Mitelut, C., Musa, S., Okun, M., Pachitariu, M., Putzeys, J., Rich, P.D., Rossant, C., lung Sun, W., Svoboda, K., Carandini, M., Harris, K.D., Koch, C., O’Keefe, J., Harris, T.D.: Fully integrated silicon probes for high-density recording of neural activity. Nature 551, 232–236 (2017)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Saratov State UniversitySaratovRussia
  2. 2.Saratov Branch of the Institute of Radioengineering and Electronics of Russian Academy of SciencesSaratovRussia

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