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

For citation:

Sysoeva M. V., Kornilov M. V., Takaishvili L. V., Matrosov V. V., Sysoev I. V. Reconstruction of integrated equations of periodically driven phase-locked loop system from scalar time series. Izvestiya VUZ. Applied Nonlinear Dynamics, 2022, vol. 30, iss. 4, pp. 391-410. DOI: 10.18500/0869-6632-2022-30-4-391-410, EDN: BPJAOD

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Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
530.182
DOI: 
10.18500/0869-6632-2022-30-4-391-410
EDN: 
BPJAOD

Reconstruction of integrated equations of periodically driven phase-locked loop system from scalar time series

Autors: 
Sysoeva Marina Vyacheslavovna, Saratov State University
Kornilov Maksim Vyacheslavovich, Saratov State University
Takaishvili Lev Vyacheslavovich, Saratov State University
Matrosov Valerij Vladimirovich, Lobachevsky State University of Nizhny Novgorod
Sysoev Ilya Vyacheslavovich, Saratov State University
Abstract: 

Purpose of this work is to develop a reconstruction technique for the equations of a phase-locked loop system under periodic external driving from a scalar time series of one variable. Methods. Instead of the original model, we reconstructed a time-integrated model. So, since it is not necessary to evaluate the second derivative of the observable numerically, the method sensitivity to observation noise has significantly decreased. The external periodic driving is approximated with a trigonometric polynomial of time, the antiderivative of which is also a trigonometric polynomial. The assumption about continuity of an unknown nonlinear function is used to construct the target function for optimization. Results. It is shown that the proposed approach gives a significant advantage over the previously developed approach to the reconstruction of non-integrated equations, allowing to achieve acceptable parameter estimates with measurement noise being about 10% of the RMS deviation of the signal even in the presence of external driving. Conclusion. The described approach significantly extends the possibilities of reconstruction of phase-locked loop systems, allowing systems to be reconstructed under arbitrary periodic driving and at the same time significantly increasing noise resistance.

Key words: 
reconstruction
phase-locked loop system
periodic driving
Acknowledgments: 
This study was supported in part by the President of the Russian Federation under Grant MD-3006.2021.1.2, and in part by the Russian Foundation for Basic Research, grant No. 19-02-00071
Reference: 
  1. Bezruchko BP, Smirnov DA. Extracting Knowledge From Time Series: An Introduction to Nonlinear Empirical Modeling. Berlin: Springer; 2010. 410 p. DOI: 10.1007/978-3-642-12601-7.
  2. Hodgkin AL, Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 1952;117(4):500–544. DOI: 10.1113/jphysiol.1952.sp004764.
  3. FitzHugh R. Impulses and physiological states in theoretical models of nerve membranes. Biophysical Journal. 1961;1(6):445–466. DOI: 10.1016/S0006-3495(61)86902-6.
  4. Nagumo J, Arimoto S, Yoshizawa S. An active pulse transmission line simulating nerve axon. Proc. IRE. 1962;50(10):2061–2070. DOI: 10.1109/JRPROC.1962.288235.
  5. Hindmarsh JL, Rose RM. A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. B. 1984;221(1222):87–102. DOI: 10.1098/rspb.1984.0024.
  6. Morris C, Lecar H. Voltage oscillations in the barnacle giant muscle fiber. Biophysical Journal. 1981;35(1):193–213. DOI: 10.1016/S0006-3495(81)84782-0.
  7. Gouesbet G, Letellier C. Global vector-field reconstruction by using a multivariate polynomial L2 approximation on nets. Phys. Rev. E. 1994;49(6):4955–4972. DOI: 10.1103/PhysRevE.49.4955.
  8. Packard NH, Crutchfield JP, Farmer JD, Shaw RS. Geometry from a time series. Phys. Rev. Lett. 1980;45(9):712–716. DOI: 10.1103/PhysRevLett.45.712.
  9. Baake E, Baake M, Bock HG, Briggs KM. Fitting ordinary differential equations to chaotic data. Phys. Rev. A. 1992;45(8):5524–5529. DOI: /10.1103/PhysRevA.45.5524.
  10. Ponomarenko VI, Prokhorov MD. Evaluating the order and reconstructing the model equations of a time-delay system. Tech. Phys. Lett. 2006;32(9):768–771. DOI: 10.1134/S1063785006090100.
  11. Ishizaka K, Flanagan JL. Synthesis of voiced sounds from a two-mass model of the vocal cords. Bell. Syst. Tech. J. 1972;51(6):1233–1268. DOI: 10.1002/j.1538-7305.1972.tb02651.x.
  12. Barfred M, Mosekilde E, Holstein-Rathlou NH. Bifurcation analysis of nephron pressure and flow regulation. Chaos. 1996;6(3):280–287. DOI: 10.1063/1.166175.
  13. Huys QJM, Ahrens MB, Paninski L. Efficient estimation of detailed single-neuron models. Journal of Neurophysiology. 2006;96(2):872–890. DOI: 10.1152/jn.00079.2006.
  14. Shalfeev VD. Investigation of the dynamics of a system of automatic phase control of frequency with a coupling capacitor in the control loop. Radiophysics and Quantum Electronics. 1968;11(3): 221–226. DOI: 10.1007/BF01033800.
  15. Mishchenko MA, Shalfeev VD, Matrosov VV. Neuron-like dynamics in phase-locked loop. Izvestiya VUZ. Applied Nonlinear Dynamics. 2012;20(4):122–130 (in Russian). DOI: 10.18500/0869-6632-2012-20-4-122-130.
  16. Matrosov VV, Mishchenko MA, Shalfeev VD. Neuron-like dynamics of a phase-locked loop. The European Physical Journal Special Topics. 2013;222(10):2399–2405. DOI: 10.1140/epjst/e2013- 02024-9.
  17. Mishchenko MA, Bolshakov DI, Matrosov VV. Instrumental implementation of a neuronlike generator with spiking and bursting dynamics based on a phase-locked loop. Tech. Phys. Lett. 2017;43(7):596–599. DOI: 10.1134/S1063785017070100.
  18. Sysoev IV, Prokhorov MD, Ponomarenko VI, Bezruchko BP. Reconstruction of ensembles of coupled time-delay systems from time series. Phys. Rev. E. 2014;89(6):062911. DOI: 10.1103/PhysRevE.89.062911.
  19. Sysoeva MV, Sysoev IV, Ponomarenko VI, Prokhorov MD. Reconstructing the neuron-like oscillator equations modeled by a phase-locked system with delay from scalar time series. Izvestiya VUZ. Applied Nonlinear Dynamics. 2020;28(4):397–413 (in Russian). DOI: 10.18500/0869-6632- 2020-28-4-397-413.
  20. Sysoeva MV, Sysoev IV, Prokhorov MD, Ponomarenko VI, Bezruchko BP. Reconstruction of coupling structure in network of neuron-like oscillators based on a phase-locked loop. Chaos, Solitons & Fractals. 2021;142:110513. DOI: 10.1016/j.chaos.2020.110513.
  21. Luttjohann A, Pape HC. Regional specificity of cortico-thalamic coupling strength and directionality during waxing and waning of spike and wave discharges. Scientific Reports. 2019;9(1):2100. DOI: 10.1038/s41598-018-37985-7.
  22. Smirnov DA, Sysoev IV, Seleznev EP, Bezruchko BP. Reconstructing nonautonomous system models with discrete spectrum of external action. Tech. Phys. Lett. 2003;29(10):824–827. DOI: 10.1134/1.1623857.
  23. Mishchenko MA. Neuron-like model on the basis of the phase-locked loop. Vestnik of Lobachevsky University of Nizhni Novgorod. 2011;5(3):279–282 (in Russian).
  24. Mishchenko MA, Zhukova NS, Matrosov VV. Excitability of neuron-like generator under pulse stimulation. Izvestiya VUZ. Applied Nonlinear Dynamics. 2018;26(5):6–19 (in Russian). DOI: 10.18500/0869-6632-2018-26-5-6-19.
  25. Mishchenko MA, Kovaleva NS, Polovinkin AV, Matrosov VV. Excitation of phase-controlled oscillator by pulse sequence. Izvestiya VUZ. Applied Nonlinear Dynamics. 2021;29(2):240–253 (in Russian). DOI: 10.18500/0869-6632-2021-29-2-240-253.
  26. Zhang J, Davidson RM, Wei M, Loew LM. Membrane electric properties by combined patch clamp and fluorescence ratio imaging in single neurons. Biophysical Journal. 1998;74(1):48–53. DOI: 10.1016/S0006-3495(98)77765-3.
  27. Jin L, Han Z, Platisa J, Wooltorton JRA, Cohen LB, Pieribone VA. Single action potentials and subthreshold electrical events imaged in neurons with a fluorescent protein voltage probe. Neuron. 2012;75(5):779–785. DOI: 10.1016/j.neuron.2012.06.040.
  28. Savitzky A, Golay MJE. Smoothing and differentiation of data by simplified least squares procedures. Anal. Chem. 1964;36(8):1627–1639. DOI: 10.1021/ac60214a047.
  29. Bezruchko BP, Smirnov DA. Constructing nonautonomous differential equations from experimental time series. Phys. Rev. E. 2001;63(1):016207. DOI: 10.1103/PhysRevE.63.016207.
  30. Sysoev IV, Sysoeva MV, Ponomarenko VI, Prokhorov MD. Neural-like dynamics in a phase-locked loop system with delayed feedback. Tech. Phys. Lett. 2020;46(7):710–712. DOI: 10.1134/S1063785020070287.
  31. Sysoev IV. Reconstruction of ensembles of generalized Van der Pol oscillators from vector time series. Physica D. 2018;384–385:1–11. DOI: 10.1016/j.physd.2018.07.004.
  32. Sysoev IV, Bezruchko BP. Noise robust approach to reconstruction of van der Pol-like oscillators and its application to Granger causality. Chaos. 2021;31(8):083118. DOI: 10.1063/5.0056901.
Received: 
22.10.2021
Accepted: 
14.12.2021
Published: 
01.08.2022
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