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Complex dynamics and autapse-modulated information patterns in memristive Wilson neurons

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Abstract

This paper introduces and investigates the dynamics of an improved Wilson neuron model with a memristive autapse. The study of the stability of the equilibrium points of the proposed model revealed that it can move from a rest state to a firing state through a Hopf bifurcation. Therefore, the improved model experiences self-excited dynamics. The study of the dynamics of the proposed model revealed neuronal behaviors such as quiescent, bursting, spiking, and hysteretic dynamics characterized by the coexistence of two and three firing patterns for the same set of parameters. In addition, the dynamics of modulated information patterns via the membrane potential related to potential information coding in a network of 500 neurons are presented and discussed. Planar impulse wave solutions of the generic model as initial conditions result in a localized unstable wave pattern under autaptic coupling and magnetic flux time delay. High memristive autaptic coupling and magnetic flux time delay have antagonistic effects on the spatiotemporal patterns. Our results suggest that high autaptic coupling could be a possible route to chaotic and chimera-like behaviors in the network, while high magnetic flux time delay could be an efficient bifurcation parameter in controlling the chaotic and chimera-like behaviors. Finally, a microcontroller implementation of the introduced memristive Wilson neuron has been addressed based on the 32-bit STM32F407ZE development board. Results obtained from the experimental investigations of the model agreed with the numerical ones based on the physically captured phase trajectories and time-domain waveforms.

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Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

References

  1. Bao, H., Liu, W., Ma, J., Wu, H.: Memristor initial-offset boosting in memristive hr neuron model with hidden firing patterns. Int. J. Bifurc. Chaos 30(10), 2030029 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chay, T.R.: Chaos in a three-variable model of an excitable cell. Phys. D Nonlinear Phenom. 16(2), 233–242 (1985)

    Article  MATH  Google Scholar 

  3. Chua, L.: Everything you wish to know about memristors but are afraid to ask. In: Handbook of Memristor Networks, pp. 89–157. Springer (2019)

  4. Doubla Isaac, S., Njitacke, Z.T., Kengne, J.: Effects of low and high neuron activation gradients on the dynamics of a simple 3d Hopfield neural network. Int. J. Bifurc. Chaos 30(11), 2050159 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hindmarsh, J., Rose, R.: A model of the nerve impulse using two first-order differential equations. Nature 296(5853), 162–164 (1982)

    Article  Google Scholar 

  6. Hindmarsh, J.L., Rose, R.: A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. Ser. B Biol. Sci. 221(1222), 87–102 (1984)

    Google Scholar 

  7. Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117(4), 500 (1952)

    Article  Google Scholar 

  8. Imani, M.A., Ahmadi, A., RadMalekshahi, M., Haghiri, S.: Digital multiplierless realization of coupled Wilson neuron model. IEEE Trans. Biomed. Circuits Syst. 12(6), 1431–1439 (2018)

    Article  Google Scholar 

  9. Izhikevich, E.: Fitzhugh–Nagumo model. Scholarpedia 1(9), 1349 (2006)

    Article  Google Scholar 

  10. Izhikevich, E.M.: Simple model of spiking neurons. IEEE Trans. Neural Netw. 14(6), 1569–1572 (2003)

    Article  MathSciNet  Google Scholar 

  11. Ju, Z., Lin, Y., Chen, B., Wu, H., Chen, M., Xu, Q.: Electromagnetic radiation induced non-chaotic behaviors in a Wilson neuron model. Chin. J. Phys. 6, 66 (2022)

    MathSciNet  Google Scholar 

  12. Liu, Y., Ma, J., Xu, Y., Jia, Y.: Electrical mode transition of hybrid neuronal model induced by external stimulus and electromagnetic induction. Int. J. Bifurc. Chaos 29(11), 1950156 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lv, M., Wang, C., Ren, G., Ma, J., Song, X.: Model of electrical activity in a neuron under magnetic flow effect. Nonlinear Dyn. 85(3), 1479–1490 (2016)

    Article  Google Scholar 

  14. Ma, J., Yang, Z.Q., Yang, L.J., Tang, J.: A physical view of computational neurodynamics. J. Zhejiang Univ. Sci. A 20(9), 639–659 (2019)

    Article  Google Scholar 

  15. Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. Wiley, New York (2008)

    MATH  Google Scholar 

  16. Njitacke, Z.T., Isaac, S.D., Nestor, T., Kengne, J.: Window of multistability and its control in a simple 3d Hopfield neural network: application to biomedical image encryption. Neural Comput. Appl. 33(12), 6733–6752 (2021)

    Article  Google Scholar 

  17. Njitacke, Z.T., Koumetio, B.N., Ramakrishnan, B., Leutcho, G.D., Fozin, T.F., Tsafack, N., Rajagopal, K., Kengne, J.: Hamiltonian energy and coexistence of hidden firing patterns from bidirectional coupling between two different neurons. Cogn. Neurodyn. 66, 1–18 (2021)

    Google Scholar 

  18. Nouri, M., Hayati, M., Serrano-Gotarredona, T., Abbott, D.: A digital neuromorphic realization of the 2-d Wilson neuron model. IEEE Trans. Circuits Syst. II Express Briefs 66(1), 136–140 (2018)

    Google Scholar 

  19. Pisarchik, A.N., Jaimes-Reátegui, R., García-Vellisca, M.: Asymmetry in electrical coupling between neurons alters multistable firing behavior. Chaos Interdiscip. J. Nonlinear Sci. 28(3), 033605 (2018)

  20. Qi, Y., Watts, A., Kim, J., Robinson, P.A.: Firing patterns in a conductance-based neuron model: bifurcation, phase diagram, and chaos. Biol. Cybern. 107(1), 15–24 (2013)

  21. Tabekoueng Njitacke, Z., Kengne, J., Fotsin, H.B.: Coexistence of multiple stable states and bursting oscillations in a 4d Hopfield neural network. Circuits Syst. Signal Process. 39(7), 3424–3444 (2020)

    Article  MATH  Google Scholar 

  22. Tabekoueng Njitacke, Z., Laura Matze, C., Fouodji Tsotsop, M., Kengne, J.: Remerging Feigenbaum trees, coexisting behaviors and bursting oscillations in a novel 3d generalized Hopfield neural network. Neural Process. Lett. 52(1), 267–289 (2020)

    Article  Google Scholar 

  23. Takembo, C.N., Mvogo, A., Ekobena Fouda, H.P., Kofané, T.C.: Effect of electromagnetic radiation on the dynamics of spatiotemporal patterns in memristor-based neuronal network. Nonlinear Dyn. 95(2), 1067–1078 (2019)

    Article  MATH  Google Scholar 

  24. Takembo, C.N., Nyifeh, P., Fouda, H.E., Kofane, T.: Modulated wave pattern stability in chain neural networks under high-low frequency magnetic radiation. Phys. A Stat. Mech. Appl. 66, 126891 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  25. Tsumoto, K., Kitajima, H., Yoshinaga, T., Aihara, K., Kawakami, H.: Bifurcations in Morris–Lecar neuron model. Neurocomputing 69(4–6), 293–316 (2006)

    Article  Google Scholar 

  26. Wang, C., Guo, S., Xu, Y., Ma, J., Tang, J., Alzahrani, F., Hobiny, A.: Formation of autapse connected to neuron and its biological function. Complexity 6, 66 (2017)

    MathSciNet  MATH  Google Scholar 

  27. Wilson, H.R.: Simplified dynamics of human and mammalian neocortical neurons. J. Theor. Biol. 200(4), 375–388 (1999)

    Article  Google Scholar 

  28. Wise, S., Shadmehr, R., Ramachandran, V.: Encyclopedia of the human brain. Retrieved 32 May 2003 (2002)

  29. Xu, Q., Ju, Z., Ding, S., Feng, C., Chen, M., Bao, B.: Electromagnetic induction effects on electrical activity within a memristive Wilson neuron model. Cogn. Neurodyn. 66, 1–11 (2022)

    Google Scholar 

  30. Xu, Q., Ju, Z., Feng, C., Wu, H., Chen, M.: Analogy circuit synthesis and dynamics confirmation of a bipolar pulse current-forced 2d Wilson neuron model. Eur. Phys. J. Spec. Top. 230(7), 1989–1997 (2021)

    Article  Google Scholar 

  31. Xu, Q., Liu, T., Feng, C.T., Bao, H., Wu, H.G., Bao, B.C.: Continuous non-autonomous memristive Rulkov model with extreme multistability. Chin. Phys. B 30(12), 128702 (2021)

    Article  Google Scholar 

  32. Xu, Q., Tan, X., Zhu, D., Bao, H., Hu, Y., Bao, B.: Bifurcations to bursting and spiking in the Chay neuron and their validation in a digital circuit. Chaos Solitons Fract. 141, 110353 (2020)

    Article  MathSciNet  Google Scholar 

  33. Zhang, G., Wang, C., Alzahrani, F., Wu, F., An, X.: Investigation of dynamical behaviors of neurons driven by memristive synapse. Chaos Solitons Fract. 108, 15–24 (2018)

    Article  MathSciNet  Google Scholar 

  34. Zhao, X., Kim, J., Robinson, P.A., Rennie, C.J.: Low dimensional model of bursting neurons. J. Comput. Neurosci. 36(1), 81–95 (2014)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work is partially funded by the Polish National Science Center under the Grant OPUS 14 No. 2017/27/B/ST8/01330.

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Authors and Affiliations

  1. Department of Electrical and Electronic Engineering, College of Technology (COT), University of Buea, P.O. Box 63, Buea, Cameroon

    Zeric Tabekoueng Njitacke & Clovis Ntahkie Takembo

  2. Department of Automation, Biomechanics, and Mechatronics, Faculty of Mechanical engineering, Lodz University of Technology, Łódź, Poland

    Zeric Tabekoueng Njitacke & Jan Awrejcewicz

  3. Research Unit of Condensed Matter, Department of Physics, Faculty of Sciences, Electronics and Signal Processing (URMACETS), University of Dschang, P.O. Box 67, Dschang, Cameroon

    Bernard Nzoko Koumetio

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  1. Zeric Tabekoueng Njitacke
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Correspondence to Zeric Tabekoueng Njitacke.

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Njitacke, Z.T., Takembo, C.N., Koumetio, B.N. et al. Complex dynamics and autapse-modulated information patterns in memristive Wilson neurons. Nonlinear Dyn 110, 2793–2804 (2022). https://doi.org/10.1007/s11071-022-07738-3

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  • DOI: https://doi.org/10.1007/s11071-022-07738-3

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