THE DYNAMICS OF NONLINEAR OCILLATOR NEURON BY THE ACTION OF EXTERNAL NON-STATIONARY SIGNAL

R. M. Peleshchak, V. V. Lytvyn, I. R. Peleshchak

Abstract


Context. Deals with the problem of time-frequency and time dependence of the morphology of the signal at the output of nonlinear oscillatory neuron with regard to its threshold effect. The object of study is non-linear modified model of Van-der-Pol, which describes the dynamics of nonlinear oscillatory neurons under the action of various shape, frequency and amplitude of an external non-stationary signals.

Objective. Construction of nonlinear mathematical models of oscillatory dynamics of a neuron given its threshold effect under the action of a neuron to external non-stationary signals.

Method. In the approximation of Krylov-Bogoliubov-Mitropolskii the proposed method of successive approximations method for solving nonlinear differential equations of second order with quadratic nonlinearity of the unknown function with the first derivative. The proposed solution method allowed us to obtain the frequency-temporal and temporal dependence of the morphology of the signal at the output of nonlinear oscillatory neuron given its threshold effect when the effect on the neuron structure of the various input signals. Proposed coding information oscillatory nonlinear neuron on the basis of frequency modulation and decoding using the inverse operator which acts on the vector of the output signal.

Results. A non-linear model, the oscillatory neuron is implemented in the environment of computer algebra “Mathematica 7.0”. Investigation of frequency-temporal and temporal dependence of the morphology of the signal at the output of nonlinear oscillatory neuron given its threshold effect at various values of the parameters of the input non-stationary signal and different weight of synaptic connections.

Conclusions. The existence of the resonance effect in nonlinear neuron with equal external frequency non-stationary signal and frequencies of the dynamics of the neuron. It is shown that nonlinear oscillatory neuron plays the role of the frequency modulator and significantly alters the structure of the input information signal is non-stationary (shape, frequency and amplitude). Proposed coding information oscillatory nonlinear neuron on the basis of frequency modulation and decoding by using the inverse operator.

Keywords


Nonlinear oscillator neuron; frequency modulation; the morphology of the information signal; resonance effect; coding and decoding information.

References


Sauer T. Reconstruction of dynamical systems from interspike intervals / T. Sauer // Phys. Rev. Lett, 1994, Vol. 72, pp. 3811 – 3814. DOI: https://doi.org/10.1103/PhysRevLett.72.3811

Racicot D. M., Lonytin A. Interspike interval attractors from chaotically driven neuron models, Physic D, 1997, Vol. 104, pp. 184–204.

Castro R., Sauer T. Correlation dimension of attractors through interspike intervals, Phys. Rev. E., 1997, Vol. 55, pp. 287–290. DOI: https://doi.org/10.1103/PhysRevE.55.287

Sauer T. Nonlinear Dinamics and Time Series, American Mathemtical Society, 1997, Vol. 11, pp. 63–75.

Hegger R., Kantz H. Embedding of sequences of time intervals, Europhys. Lett, 1997, Vol. 38, pp. 267–272. DOI: https://doi.org/10.1209/epl/i1997-00236-0

Castro R., Sauer T. Chaotic Stochastic Resonance: Noise-Enhanced Reconstruction of Attractors, Phys. Rev. Let t., 1997, Vol. 79, pp. 1030–1033. DOI: https://doi.org/10.1103/PhysRevLett.79.1030

Pavlov A. N., Hramov A. E., Koronovskij A. A. et al Vejvlet-analiz v nejrodinamike, Uspehi fizicheskih nauk, 2012, Vol. 182, No. 9, pp. 905–939. DOI: 10.3367/UFNr.0182.201209a.0905

Tuckwell H. C. Introduction to Theoretical Neurobiology. Cambridge, Cambridge University, 1988, 304 p.

Janson N. B., Pavlov A. N., Neiman A. B. et al. Reconstruction of dynamical and geometrical properties of chaotic attractors from threshold-crossing interspike intervals, Phys. Rev. E., 1998, Vol. 58, pp. R4–R7. DOI: https://doi.org/10.1103/PhysRevE.58.R4

Pavlov A. N., Sosnovtseva O. V., Mosekilde E. et al. Extracting dynamics from threshold-crossing interspike intervals: Possibilities and limitations, Phys. Rev. E., 2000, Vol. 61, pp. 5033–5044. DOI: https://doi.org/10.1103/PhysRevE.61.5033

Pavlov A. N., Sosnovtseva O. V., Mosekilde E. et al. Chaotic dynamics from interspike intervals, Phys. Rev. E., 2001, Vol. 63, pp. 036205 (5). DOI: https://doi.org/10.1103/PhysRevE.63.036205

Bay J. S., Hemami H. Modeling of a Neural Generator with Coupled Nonlinear Oscillators, IEE Transactions biomedical engineering, 1987, Vol. BME – 34, No. 4, April, pp. 297–306. DOI: 10.1109/TBME.1987.326091

Pavlov A. N., Pavlova O. N. Primenenie vejvlet-analiza v issledovanijah struktury tochechnyh processov, Pis’ma v ZhTF, 2006, Vol. 32, No. 21, pp. 11–17.

Sugakov V. J. Osnovi sinergetiki. Kiїv:Oberegi, 2001, 288 p.

Bogoljubov M. M. Asimptotichnі metodi v teorії nelіnіjnih rіvnjan’. Kyiv, Nauka, 1992, 312 p.

Bozhokin S. V. Nepreryvnoe vejvlet-preobrazovanie i tochno reshaemaja model’ nestacionarnyh signalov, ZhTF, 2012, Vol. 82, No. 7, pp. 8–13.


GOST Style Citations


1. Sauer T. Reconstruction of dynamical systems from interspike intervals / T. Sauer // Phys. Rev. Lett. – 1994. – Vol. 72. – P. 3811–3814. DOI: https://doi.org/10.1103/PhysRevLett.72.3811

2. Racicot D. M. Interspike interval attractors from chaotically driven neuron models / D. M. Racicot, A. Lonytin // Physic D. – 1997. – Vol. 104. – P. 184–204.

3. Castro R. Correlation dimension of attractors through interspike intervals / R. Castro, T. Sauer // Phys. Rev. E. – 1997. – Vol. 55. – P. 287–290. DOI: https://doi.org/10.1103/PhysRevE.55.287

4. Sauer T. Nonlinear Dinamics and Time Series / T. Sauer // American Mathemtical Society. – 1997. – Vol. 11. – P. 63 – 75.

5. Hegger R. Embedding of sequences of time intervals / R. Hegger, H. Kantz // Europhys. Lett. – 1997. – Vol. 38. – P. 267–272. DOI: https://doi.org/10.1209/epl/i1997-00236-0

6. Castro R. Chaotic Stochastic Resonance: Noise-Enhanced Reconstruction of Attractors / R. Castro, T. Sauer // Phys. Rev. Lett. – 1997. – Vol. 79. – P. 1030–1033. DOI: https://doi.org/10.1103/PhysRevLett.79.1030

7. Вейвлет-анализ в нейродинамике / [А. Н. Павлов, А. Е. Храмов, А. А. Короновский и др.] // Успехи физических наук. – 2012. – Т. 182, № 9. – С. 905–939. DOI: 10.3367/UFNr.0182.201209a.0905

8. Tuckwell H. C. Introduction to Theoretical Neurobiology / H. C. Tuckwell. – Cambridge : Cambridge University, 1988. – 304 p.

9. Reconstruction of dynamical and geometrical properties of chaotic attractors from threshold-crossing interspike intervals / [N. B. Janson, A. N. Pavlov, A. B. Neiman et al.] // Phys. Rev. E. – 1998. – Vol. 58. – P. R4–R7. DOI: https://doi.org/10.1103/PhysRevE.58.R4

10. Extracting dynamics from threshold-crossing interspike intervals: Possibilities and limitations / [A. N. Pavlov, O. V. Sosnovtseva, E. Mosekilde et al.] // Phys. Rev. E. – 2000. – Vol. 61. – P. 5033–5044. DOI: https://doi.org/10.1103/PhysRevE.61.5033

11. Chaotic dynamics from interspike intervals / [A. N. Pavlov, O. V. Sosnovtseva, E. Mosekilde et al.] // Phys. Rev. E. – 2001. – Vol. 63. – P. 036205 (5). DOI: https://doi.org/10.1103/PhysRevE.63.036205

12. Bay J. S., Hemami H. Modeling of a Neural Generator with Coupled Nonlinear Oscillators / J. S. Bay, H. Hemami // IEE Transactions biomedical engineering. – 1987. – Vol. BME – 34, № 4, April. – P. 297 – 306. DOI: 10.1109/TBME.1987.326091

13. Павлов А. Н. Применение вейвлет-анализа в исследованиях структуры точечных процессов/ А. Н. Павлов, О. Н. Павлова // Письма в ЖТФ. – 2006. – Т. 32, вып. 21. – С. 11–17.

14. Сугаков В. Й. Основи синергетики / В. Й. Сугаков. – Київ : Обереги, 2001. – 288 с.

15. Боголюбов М. М. Асимптотичні методи  в  теорії нелінійних рівнянь / М. М. Боголюбов. – Київ : Наука, 1992. – 312 с.

16.  Божокин С. В. Непрерывное вейвлет-преобразование и точно решаемая модель нестационарных сигналов / С. В. Божокин // ЖТФ. – 2012. – Т.82, Вып. 7. – С. 8–13.




DOI: https://doi.org/10.15588/1607-3274-2017-4-11



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