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

The article published as Early Access!

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
PDF icon and_2024-5_shabunin_ranniy_dostup.pdf
(downloads: 0)
Language: 
Russian
Heading: 
Nonlinear Dynamics and Neuroscience
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-003111
EDN: 
EYKFOY

Serching the structure of couplings in a chaotic maps ensemble by means of neural networks

Autors: 
Shabunin Aleksej Vladimirovich, Saratov State University
Abstract: 

The purpose of this work is development and research of an algorithm for determining the structure of couplings of an ensemble of chaotic self-oscillating systems.

The method is based on the determination of causality by Granger and the use of direct propagation artificial neural networks trained with regularization.

Results. We have considered a method for recognition structure of couplings of a network of chaotic maps based on the Granger causality principle and artificial neural networks approach. The algorithm demonstrates its efficiency on the example of small ensembles of maps with diffusion couplings. In addition to determining the network topology, it can be used to estimate the magnitue of the couplings. Accuracy of the method essencially depends on the observed oscillatory regime. It effectively works only in the case of homogeneous space-time chaos.

Discussion. Although the method has shown its effectiveness for simple mathematical models, its applicability for real systems depends on a number of factors, such as sensitivity to noise, to possible distortion of the waveforms, the presence of crosstalks and external noise etc. These questions require additional research.

Key words: 
dynamical chaos
artificial neural networks
ensembles of maps
couplings structure identification
Reference: 
  1. Cremers J, Hubler A. Construction of differential equations from experimental data. Zeitschriftfur Naturforschung A. 1987;42(8):797–802. DOI: 10.1515/zna-1987-0805.
  2. Crutchfield JP, McNamara BS. Equations of motion from a data series. Complex Systems. 1987;1(3):417–452.
  3. Pavlov AN, Yanson NB. Application of reconstruction method to a cardiogram. Izvestiya VUZ. Applied Nonlinear Dynamics. 1997;5(1):93–108 (in Russian).
  4. Anishchenko VS, Pavlov AN. Global reconstruction in application to multichannel communication. Physical Review E. 1998;57(2):2455–2457. DOI: 10.1103/PhysRevE.57.2455.
  5. Mukhin DN, Feigin AM, Loskutov EM, Molkov YI. Modified Bayesian approach for the reconstruction of dynamical systems from time series. Physical Review E. 2006;73(3):036211. DOI: 10.1103/PhysRevE.73.036211.
  6. Bezruchko BP, Smirnov DA, Zborovsky AV, Sidak EV, Ivanov RN, Bespyatov AB. Reconstruction from time-series and a task of diagnostics. Technologies of living systems. 2007;4(3):49–56 (in Russian).
  7. Takens F. Detecting strange attractors in turbulence. Dynamical Systems and Turbulence. Lect. Notes in Math. 1980;898:366–381. DOI: 10.1007/BFB0091924.
  8. Granger CWJ. Investigating causal relations by econometric models and cross-spectral methods. Econometrica. 1969;37(3):424–438. DOI: 10.2307/1912791.
  9. Granger CWJ. Testing for causality. A personal viewpoint. Journal of Economic Dynamics and Control. 1980;2:329–352. DOI: 10.1016/0165-1889(80)90069-X.
  10. Sysoev IV. Diagnostics of connectivity by chaotic signals of nonlinear systems: solving reverse problems. Saratov: Kubik; 2019. 46 p. (in Russian).
  11. Hesse R, Molle E, Arnold M, Schack B. The use of time-variant EEG Granger causality for inspecting directed interdependencies of neural assemblies. Journal of Neuroscience Methods. 2003;124(1):27–44. DOI: 10.1016/S0165-0270(02)00366-7.
  12. Bezruchko BP, Ponomarenko VI, Prohorov MD, Smirnov DA, Tass PA. Modeling nonlinear oscillatory systems and diagnostics of coupling between them using chaotic time series analysis: applications in neurophysiology. Physics-Uspekhi. 2008;51:304–310. DOI: 10.1070/PU2008 v051n03ABEH006494.
  13. Mokhov II, Smirnov DA. Diagnostics of a cause-effect relation between solar activity and the Earth’s global surface temperature. Izvestiya, Atmospheric and Oceanic Physics. 2008;44(3): 263–272. DOI: 10.1134/S0001433808030018.
  14. Mokhov II, Smirnov DA. Empirical estimates of the influence of natural and anthropogenic factors on the global surface temperature. Doklady Earth Sciences. 2009;427(1):798–803. DOI: 10.1134/ S1028334X09050201.
  15. Sysoev IV, Karavaev AS, Nakonechny PI. Role of model nonlinearity for Granger causality based coupling estimation for pathological tremor. Izvestiya VUZ. Applied Nonlinear Dynamics. 2010;18(4):81–90. DOI: 10.18500/0869-6632-2010-18-4-81-90.
  16. Sysoeva MV, Sysoev IV. Mathematical modeling of encephalogram dynamics during epileptic seizure. Technical Physics Letters. 2012;38(2):151–154. DOI: 10.1134/S1063785012020137.
  17. Sysoev IV, Sysoeva MV. Detecting changes in coupling with Granger causality method from time series with fast transient processes. Physica D: Nonlinear Phenomena. 2015;309:9–19. DOI: 10.1016/j.physd.2015.07.005.
  18. Chen Y, Rangarajan G, Feng J, Ding M. Analyzing Multiple Nonlinear Time Series with Extended Granger Causality. Physics Letters A. 2004;324(1):26–35. DOI: 10.1016/j.physleta.2004.02.032.
  19. Marinazzo D, Pellicoro M, Stramaglia S. Nonlinear parametric model for Granger causality of time series. Physical Review E. 2006;73(6):066216. DOI: 10.1103/PhysRevE.73.066216.
  20. Kornilov МV, Sysoev IV. Recovering the Architecture of Links in a Chain of Three Unidirectionally Coupled Systems Using the Granger-Causality Test. Technical Physics Letters. 2018;44(5): 445–449. DOI: 10.21883/PJTF.2018.10.46103.17201.
  21. Haykin S. Neural Networks. New Jersey: Prentice Hall; 1999. 938 p.
  22. Galushkin AI. Neural Networks. The theory basics. Telekom; 2012. 496 p. (in Russian).
  23. de Oliveira KA, Vannucci A, Da Silva EC. Using artificial neural networks to forecast chaotic time series. Physica A. 2000;284(1–4):393–404. DOI: 10.1016/S0378-4371(00)00215-6.
  24. Antipov OI, Neganov VA. Neural network prediction and fractal analysis of the chaotic processes in discrete nonlinear systems. Doklady Physics. 2011;56(1):7–9. DOI: 10.1134/S1028335811010034.
  25. Shabunin AV. Neural network as a predictor of discrete map dynamics. Izvestiya VUZ. Applied Nonlinear Dynamics. 2014;22(5):58–72. DOI: 10.18500/0869-6632-2014-22-5-58-72.
  26. Tank A, Covert I, Foti N, Shojaie A, Fox E. Neural granger causality for nonlinear time Series. arXiv preprint arXiv:1802.05842. 2018.
  27. Tihonov AN. On incorrect linear algebra problems and a stable solution method. Reports of the USSR Academy of Sciences. 1965;163(3):591–594.
  28. Fujisaka H, Yamada T. Stability theory of synchronized motion in coupled-oscillator systems. Progress of Theoretical Physics. 1983;69(1):32–47. DOI: 10.1143/PTP.69.32.
  29. Fujisaka H, Yamada T. Stability theory of synchronized motion in coupled-oscillator systems. The mapping approach. Progress of Theoretical Physics. 1983;70(5):1240–1248. DOI: 10.1143/ PTP.70.1240.
  30. Shabunin A. Selective properties of diffusive couplings and their influence on spatiotemporal chaos. Chaos. 2021;31(7):073132. DOI: doi.org/10.1063/5.0054510.
Received: 
15.01.2024
Accepted: 
03.03.2024
Available online: 
20.06.2024
  • 91 reads