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Nonlinear Dynamics of Complex Neurophysiologic Systems Within a Quantum-Chaos Geometric Approach

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Advances in Methods and Applications of Quantum Systems in Chemistry, Physics, and Biology

Part of the book series: Progress in Theoretical Chemistry and Physics ((PTCP,volume 33))

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Abstract

It is presented an advanced, uniform, quantum-chaos-geometric approach to analysis, computing simulation and prediction of a nonlinear dynamics of the complex neurophysiological systems (such as the ensembles fluctuations of spontaneous Parkinsonian tremor and fluctuations of the local potential etc.) with elements of a deterministic chaos. The approach is based on the combined application of the complex quantum chaos and dynamical systems theory methods, such as a multi-fractal formalism, mutual information approach, correlation integral analysis, false nearest neighbour algorithm, the Lyapunov exponent’s and Kolmogorov’s entropy analysis, surrogate data method, prediction models, including the neural networks algorithms, algorithms of optimized trajectories etc. It has been numerically studied a chaotic dynamics of the complex neurophysiological systems (the ensembles fluctuations of spontaneous Parkinsonian tremor and fluctuations of the local potential). New advanced numerical data on topological and dynamical invariants of the system studied, in particular, the correlation, embedding, Kaplan-York dimensions, the Lyapunov exponent’s and Kolmogorov’s entropy etc. are listed and analyzed.

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References

  1. Brändas E, Elander N (eds) (1989) Resonances. In: Springer lecture notes in physics, vol 325. Springer, Berlin

    Google Scholar 

  2. Brändas E (1997) In: Prigogine I, Rice S (eds) Advances in chemical physics: resonances, instability, and irreversibility, vol 99. Wiley, pp 211–244

    Google Scholar 

  3. Glushkov AV (2012) Methods of a chaos theory. Astroprint, Odessa

    Google Scholar 

  4. Zaslavsky GM (2005) Hamiltonian chaos and fractional dynamics. Oxford Univ. Press, Oxford

    Google Scholar 

  5. Hastings AM, Hom C, Ellner S, Turchin P, Godfray Y (1993) Chaos in ecology: is mother nature a strange attractor? Annu Rev Ecol Syst 24:1–33

    Article  Google Scholar 

  6. Grassberger P, Procaccia (1983) Measuring the strangeness of strange attractors. Phys D 9:189–208

    Article  Google Scholar 

  7. Gallager RG (1986) Information theory and reliable communication. Wiley, New York

    Google Scholar 

  8. Glushkov AV, Ternovsky VB, Buyadzhi VV, Prepelitsa GP (2014) Geometry of a relativistic quantum chaos: new approach to dynamics of quantum systems in electromagnetic field and uniformity and charm of a chaos. Proc Int Geom Center 7(4):60–71

    Google Scholar 

  9. Prepelitsa GP, Glushkov AV, Lepikh YaI, Buyadzhi VV, Ternovsky VB, Zaichko PA (2014) Chaotic dynamics of non-linear processes in atomic and molecular systems in electromagnetic field and semiconductor and fiber laser devices: new approaches, uniformity and charm of chaos. Sensor Electron Microsyst Technol 11(4)43–57

    Google Scholar 

  10. Glushkov AV, Khetselius OYu, Kruglyak YuA, Ternovsky VB (2014) Calculational methods in quantum geometry and chaos theory. P.3. OSENU (TEC), Odessa

    Google Scholar 

  11. Glushkov AV (2008) Relativistic quantum theory. Relativistic quantum mechanics of atomic systems. Astroprint, Odessa

    Google Scholar 

  12. Khetselius OYu (2008) Hyperfine structure of atomic spectra. Astroprint, Odessa

    Google Scholar 

  13. Khetselius OYu (2011) Quantum structure of electroweak interaction in heavy finite Fermi-systems. Astroprint, Odessa

    Google Scholar 

  14. Svinarenko AA, Glushkov AV, Khetselius OYu, Ternovsky VB, Dubrovskaya YuV, Kuznetsova AA, Buyadzhi VV (2017) Theoretical spectroscopy of rare-earth elements: spectra and autoionization resonance. In: Jose EA (ed) Rare earth element. InTech, Orjuela, pp 83–104. https://doi.org/10.5772/intechopen.69314

  15. Glushkov AV, Ivanov LN (1992) DC strong field stark effect: consistent quantum mechanical approach. J Phys B: At Mol Opt Phys 26:L379–386

    Article  Google Scholar 

  16. Glushkov AV, Ivanov LN (1992) Radiation decay of atomic states: atomic residue polarization and gauge noninvariant contributions. Phys Lett A 170:33–36

    Article  CAS  Google Scholar 

  17. Ivanova EP, Ivanov LN, Glushkov AV, Kramida A (1985) High order corrections in the relativistic perturbation theory with the model zeroth approximation, Mg-like and Ne-like ions. Phys Scripta 32:513–522

    Article  CAS  Google Scholar 

  18. Khetselius OYu (2009) Relativistic perturbation theory calculation of the hyperfine structure parameters for some heavy-element isotopes. Int J Quantum Chem 109:3330–3335

    Article  CAS  Google Scholar 

  19. Khetselius OYu, Lopatkin YuM, Dubrovskaya YuV, Svinarenko AA (2010) Sensing hyperfine-structure, electroweak interaction and parity non-conservation effect in heavy atoms and nuclei: new nuclear-QED approach. Sensor Electr Microsyst Technol 7(2):11–19

    Article  Google Scholar 

  20. Glushkov AV, Dan’kov SV, Prepelitsa G, Polischuk VN, Efimov AV (1997) Qed theory of nonlinear interaction of the complex atomic systems with laser field multi-photon resonances. J Tech Phys 38(2):219–222

    CAS  Google Scholar 

  21. Buyadzhi VV, Zaichko PA, Gurskaya MY, Kuznetsova AA, Ponomarenko EL, Ternovsky VB (2017) Relativistic theory of excitation and ionization of Rydberg atomic systems in a black-body radiation field. J Phys: Conf Ser 810:012047

    Google Scholar 

  22. Svinarenko AA, Khetselius OYu, Buyadzhi VV, Florko TA, Zaichko PA, Ponomarenko EL (2014) Spectroscopy of Rydberg atoms in a black-body radiation field: relativistic theory of excitation and ionization. J Phys: Conf Ser 548:012048

    Google Scholar 

  23. Buyadzhi VV, Zaichko PA, Antoshkina OA, Kulakli TA, Prepelitsa GP, Ternovsky VB, Mansarliysky VF (2017) Computing of radiation parameters for atoms and multicharged ions within relativistic energy approach: advanced code. J Phys: Conf Ser 905:012003

    Google Scholar 

  24. Aru J, Aru J, Priesemann V, Wibral M, Lana L, Pipa G, Singer W, Vicente R (2015) Untangling cross-frequency coupling in neuroscience. Curr Opin Neurobiol 31:51–61

    Article  CAS  PubMed  Google Scholar 

  25. Berman JI, McDaniel J, Liu S, Cornew L, Gaetz W, Roberts TP, Edgar JC (2012) Variable bandwidth filtering for improved sensitivity of cross-frequency coupling metrics. Brain Connect 2(3):155–163

    Article  PubMed  PubMed Central  Google Scholar 

  26. Besserve M, Schölkopf B, Logothetis NK, Panzeri S (2010) Causal relationships between frequency bands of extracellular signals in visual cortex revealed by an information theoretic analysis. J Comput Neurosci 29(3):547–566

    Article  PubMed  PubMed Central  Google Scholar 

  27. Bigdely-Shamlo N, Mullen T, Kothe C, Su K-M, Robbins KA (2015) The PREP pipeline: standardized preprocessing for large-scale EEG analysis. Front Neuroinform 9:16

    Article  PubMed  PubMed Central  Google Scholar 

  28. La Tour TD (2018) Non-linear models for neurophysiological time series. Université Paris-Saclay Espace Technologique/Immeuble Discovery Route de l’Orme aux Merisiers RD 128/91190 Saint-Aubin, France

    Google Scholar 

  29. Glushkov AV, Khetselius OYu, Brusentseva SV, Zaichko PA, Ternovsky VB (2014) Studying interaction dynamics of chaotic systems within a non-linear prediction method: application to neurophysiology. In: Balicki J (ed) Advances in neural networks, fuzzy systems and artificial intelligence. Recent advances in computer engineering, vol 21. WSEAS Press, Gdansk, pp 69–75

    Google Scholar 

  30. Glushkov AV, Khetselius OYu, Bunuakova YuYa, Buyadzhi VV, Brusentseva SV, Zaichko PA (2014) Sensing interaction dynamics of chaotic systems within a chaos theory and microsystem technology geomath with application to neurophysiological systems. Sensor Electron Microsyst Technol 11(3):62–69. Prepriont OSENU N AMP-3

    Google Scholar 

  31. Glushkov AV, Buyadzhi VV, Ternovsky VB, Ignatenko AV, Kuznetsova AA, Mashkantsev A (2018) A chaos-dynamical approach to analysis, processing and forecasting measurements data of the chaotic quantum and laser systems and sensors. Sens Electron Microsyst Technol 15(4):41–49

    Google Scholar 

  32. Glushkov AV, Buyadzhi VV, Kvasikova AS, Ignatenko AV, Kuznetsova AA, Prepelitsa GP and Ternovsky VB (2017) Non-linear chaotic dynamics of quantum systems: molecules in an electromagnetic field and laser systems. In: Tadjer A, Pavlov R, Maruani J, Brändas E, Delgado-Barrio G (eds) Quantum systems in physics, chemistry, and biology. Advances in concepts and applications. Progress in theoretical chemistry and physics, vol 30, Chap 10. Springer, pp 169–180

    Google Scholar 

  33. Ignatenko AV, Buyadzhi AA, Buyadzhi VV, Kuznetsova AA, Mashkantsev AA, Ternovsky EV (2019) Nonlinear chaotic dynamics of quantum systems. In: Molecules in an electromagnetic field. Advances in quantum chemistry, vol 78, Chap 7. Elsevier, pp 149–170. https://doi.org/10.1016/bs.aiq.2018.06.006

  34. Glushkov AV, Prepelitsa GP, Svinarenko AA, Zaichko PA (2013) Studying interaction dynamics of the non-linear vibrational systems within non-linear prediction method (application to quantum autogenerators). In: Awrejcewicz J, Kazmierczak M, Olejnik P, Mrozowski J (eds) Dynamical systems theory, vol T1. Wyd. Politech. Łódz., Łódz, pp 467–477

    Google Scholar 

  35. Glushkov AV, Svinarenko AA, Buyadzhi VV, Zaichko PA, Ternovsky VB (2014) Chaos-geometric attractor and quantum neural networks approach to simulation chaotic evolutionary dynamics during perception process. In: Balicki J (ed) Advances in neural networks, fuzzy systems and artificial intelligence. Recent advances in computer engineering, vol 21. WSEAS Pub., Gdansk, pp 143–150

    Google Scholar 

  36. Khetselius OYu, Glushkov AV, Stepanenko SN, Svinarenko AA, Bunyakova YuYa, Buyadzhi VV (2019) Sensing and analysis of radioactive radon 222Rn concentration chaotic variability in an atmosphere environment. Sensor Electron Microsyst Technol 16(4):27–36

    Google Scholar 

  37. Glushkov AV, Khokhlov VN, Tsenenko IA (2004) Atmospheric teleconnection patterns: wavelet analysis. Nonlinear Proc Geophys 11:285–296

    Google Scholar 

  38. Khetselius OYu (2013) Forecasting evolutionary dynamics of chaotic systems using advanced non-linear prediction method. In: Awrejcewicz J, Kazmierczak M, Olejnik P, Mrozowski J (eds) Dynamical systems applications, vol T2. Lodz, Polland, pp 145–152

    Google Scholar 

  39. Glushkov AV, Khetselius OYu, Svinarenko AA and Prepelitsa GP (2011) Energy approach to atoms in a laser field and quantum dynamics with laser pulses of different shape. In: Duarte FJ (ed) Coherence and ultrashort pulsed emission. Intech, Vienna, pp 101–130

    Google Scholar 

  40. Glushkov AV, Malinovskaya SV, Shpinareva IM, Kozlovskaya VP, Gura VI (2005) Quantum stochastic modelling energy transfer and effect of rotational and v-t relaxation on multi-photon excitation and dissociation for CF3Br molecules. Int J Quantum Chem 104(4):512–516

    Article  CAS  Google Scholar 

  41. Glushkov AV, Malinovskaya SV, Svinarenko AA, Vitavetskaya LA (2005) Detection of spectral hierarchy, quantum chaos, chaotic diffusion effects and dynamical stabilization in multiphoton atomic dynamics with intense laser radiation field. Sensor Electron Microsyst Technol 2(2):29–37

    Article  Google Scholar 

  42. Glushkov AV, Khetselius OYu, Brusentseva S, Duborez A (2014) Modeling chaotic dynamics of complex systems with using chaos theory, geometric attractors, and quantum neural networks. Proc Int Geom Center 7(3):87–94

    Google Scholar 

  43. Mañé R (1981) On the dimensions of the compact invariant sets of certain non-linear maps. In: Dynamical systems and turbulence. Lecture notes in mathematics, vol 898. Springer, Berlin, pp 230–242

    Google Scholar 

  44. Sano M, Sawada Y (1995) Measurement of the Lyapunov spectrum from a chaotic time series. Phys Rev Lett 55:1082–1086

    Article  Google Scholar 

  45. Zaslavsky GM (2002) Chaos, fractional kinetics, and anomalous transport. Phys Rep 371:461–580

    Article  Google Scholar 

  46. Friedrich H, Wintgen D (1989) The hydrogen atom in a uniform magnetic field—an example of chaos. Phys Rep 183:37–84

    Article  CAS  Google Scholar 

  47. Berman GP, Bulgakov EN, Holm DD (1995) Nonlinear resonance and dynamical chaos in a diatomic molecule driven by a resonant IR field. Phys Rev A 52:3074–3080

    Article  CAS  PubMed  Google Scholar 

  48. Bezruchko BP, Ponomarenko VI, Prokhorov MD, Smirnov DA, Tass PA (2008) Modeling and diagnostics of nonlinear oscillatory systems using chaotic time series analysis (applications in neurophysiology). Phys Uspekhi 178(3):323–328

    Google Scholar 

  49. Gottwald GA, Melbourne (2004) A new test for chaos in deterministic systems. Proc R Soc Lond Ser A Math Phys Sci 460:603–611

    Article  Google Scholar 

  50. Packard N, Crutchfield J, Farmer J, Shaw R (1988) Geometry from a time series. Phys Rev Lett 45:712–716

    Article  Google Scholar 

  51. Abarbanel H (1996) Analysis of observed chaotic data. Springer, New York

    Book  Google Scholar 

  52. Kennel M, Brown R, Abarbanel H (1992) Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys Rev A 45:3403–3411

    Article  CAS  PubMed  Google Scholar 

  53. Mandelbrot BB (1983) The fractal geometry of nature. W. H. Freeman and Co, San Francisco

    Book  Google Scholar 

  54. Prigogine I (1980) From being to becoming. Freeman, New York

    Google Scholar 

  55. Arnold VI (1978) Mathematical methods of classical mechanics. Acad. Press, New York

    Book  Google Scholar 

  56. Kenneth F (2003) Fractal geometry: mathematical foundations and applications. Wiley, Chichester

    Google Scholar 

  57. May RM (1995) Necessity and chance: deterministic chaos in ecology and evolution. Bull Am Math Soc 32:291–308

    Article  Google Scholar 

  58. Fraser AM, Swinney (1986) Independent coordinates for strange attractors from mutual information. Phys Rev A 33(2):1134–1140

    Article  CAS  Google Scholar 

  59. Schreiber T (1999) Interdisciplinary application of nonlinear time series methods. Phys Rep 308(1):1–64

    Article  Google Scholar 

  60. Takens F (1981) Detecting strange attractors in turbulence. In: Dynamical systems and turbulence. Lecture notes in mathematics, vol 898. Springer, Berlin, pp 366–381

    Google Scholar 

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Authors and Affiliations

  1. Odessa State Environmental University, L’vovskaya str., bld. 15, Odessa, 65016, Ukraine

    Alexander V. Glushkov & Olga Yu. Khetselius

Authors
  1. Alexander V. Glushkov
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  2. Olga Yu. Khetselius
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Corresponding author

Correspondence to Alexander V. Glushkov .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Odessa State Environmental University, Odessa, Ukraine

    Alexander V. Glushkov

  2. Department of Mathematics, Odessa State Environmental University, Odessa, Ukraine

    Olga Yu. Khetselius

  3. LCP-MR, CNRS and Sorbonne-Universités, Paris, France

    Jean Maruani

  4. Department of Chemistry, Uppsala University, Uppsala, Sweden

    Erkki Brändas

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Glushkov, A.V., Khetselius, O.Y. (2021). Nonlinear Dynamics of Complex Neurophysiologic Systems Within a Quantum-Chaos Geometric Approach. In: Glushkov, A.V., Khetselius, O.Y., Maruani, J., Brändas, E. (eds) Advances in Methods and Applications of Quantum Systems in Chemistry, Physics, and Biology. Progress in Theoretical Chemistry and Physics, vol 33. Springer, Cham. https://doi.org/10.1007/978-3-030-68314-6_14

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