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Reconstruction of ensembles of nonlinear neurooscillators with Sigmoid coupling function. (English) Zbl 1432.34047
Summary: 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).
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
37M10 Time series analysis of dynamical systems
HYBRJ; minpack; Python
Full Text: DOI
[1] Boccaletti, S., Latora, V., Morenod, Y., Chavez, M., Hwang, D.-U.: Complex networks: structure and dynamics. Phys. Rep. 424, 175-308 (2006) · Zbl 1371.82002
[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)
[3] Sompolinsky, H., Crisanti, A., Sommers, H.E.: Chaos in random neural networks. Phys. Rev. Lett. 61(3), 259-262 (1988)
[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)
[5] Shandilya, S.G., Timme, M.: Inferring network topology from complex dynamics. N. J. Phys. 13(1), 013004 (2011)
[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)
[7] Pikovsky, A.: Reconstruction of a neural network from a time series of firing rates. Phys. Rev. E 93, 062313 (2016)
[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) · Zbl 1243.93025
[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) · Zbl 1348.93087
[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)
[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)
[12] Porta, A., Faes, L.: Wiener-Granger causality in network physiology with applications to cardiovascular control and neuroscience. Proceedings of the IEEE, 12 (2015)
[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) · Zbl 1123.62316
[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) · Zbl 1355.37095
[15] Baccala, L., Sameshima, K.: Partial directed coherence: a new concept in neural structure determination. Biol. Cybern. 84, 463-474 (2001) · Zbl 1160.92306
[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) · Zbl 1160.92314
[17] Rosenblum, M.G., Pikovsky, A.S.: Detecting direction of coupling in interacting oscillators. Phys. Rev. E 64, 045202(R) (2001)
[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)
[19] Koutlis, C., Kugiumtzis, D.: Discrimination of coupling structures using causality networks from multivariate time series. Chaos 26, 093120 (2016)
[20] Kralemann, B., Pikovsky, A., Rosenblum, M.: Reconstructing phase dynamics of oscillator networks. Chaos 21, 025104 (2011) · Zbl 1317.34057
[21] Wu, X., Wang, W., Zheng, W.X.: Inferring topologies of complex networks with hidden variables. Phys. Rev. E 86, 046106 (2012)
[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)
[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)
[24] De Feo, O., Carmeli, C.: Estimating interdependences in networks of weakly coupled deterministic systems. Phys. Rev. E 77(2), 026711 (2008)
[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) · Zbl 1281.93100
[26] Shemyakin, V., Haario, H.: Online identification of large-scale chaotic system. Nonlinear Dyn. 93(2), 961-975 (2018)
[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)
[28] Han, X., Shen, Z., Wang, W.-X., Di, Z.: Robust reconstruction of complex networks from sparse data. Phys. Rev. Lett. 114, 28701 (2015)
[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) · Zbl 1355.94013
[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)
[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)
[32] Gouesbet, G., Meunier-Guttin-Cluzel, G., Menard, O.: Chaos and its Reconstruction. Nova Science Publishers, New York (2003)
[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) · Zbl 1210.00041
[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) · Zbl 1359.70089
[35] Timme, M., Casadiego, J.: Revealing networks from dynamics: an introduction. J. Phys. A Math. Theor. 47, 343001 (2014) · Zbl 1305.92036
[36] Smirnov, D.A.: Quantifying causal couplings via dynamical effects: a unifying perspective. Phys. Rev. E 90, 062921 (2014)
[37] Richards, F.J.: A flexible growth function for empirical use. J. Exp. Bot. 10(2), 290-300 (1959)
[38] Levenberg, K.: A method for the solution of certain non-linear problems in least squares. Q. Appl. Math. 2, 164-168 (1944) · Zbl 0063.03501
[39] Marquardt, D.: An algorithm for least-squares estimation of nonlinear parameters. SIAM J. Appl. Math. 11(2), 431-441 (1963) · Zbl 0112.10505
[40] Coleman, T.F., Li, Y.: An interior trust region approach for nonlinear minimization subject to bounds. SIAM J. Opt. 6, 418-445 (1996) · Zbl 0855.65063
[41] Kera, H., Hasegawa, Y.: Noise-tolerant algebraic method for reconstruction of nonlinear dynamical systems. Nonlinear Dyn. 85(1), 675-692 (2016) · Zbl 1349.93097
[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)
[43] Savitzky, A., Golay, M.: Smoothing and differentiation of data by simplified least squares procedures. Anal. Chem. 38(8), 1627-1639 (1964)
[44] Moré, JJ; Sorensen, DC; Hillstrom, KE; Garbow, BS; Cowell, WJ (ed.), The minpack project, 88-111 (1984), Upper Saddle River
[45] Millman, K.J., Aivazis, M.: Python for scientists and engineers. Comput. Sci. Eng. 13, 9-12 (2011)
[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) · Zbl 0779.58030
[47] Baake, E., Baake, M., Bock, H., Briggs, K.: Fitting ordinary differential equations to chaotic data. Phys. Rev. A 45(8), 5524-5529 (1992)
[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) · Zbl 1147.93326
[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)
[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)
[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)
[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)
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