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Coexistence of Multiple Stable States and Bursting Oscillations in a 4D Hopfield Neural Network

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Abstract

Neurons are regarded as basic, structural and functional units of the central nervous system. They play an active role in the collection, storing and transferring of the information during signal processing in the brain. In this paper, we investigate the dynamics of a model of a 4D autonomous Hopfield neural network (HNN). Our analyses highlight complex phenomena such as chaotic oscillations, periodic windows, hysteretic dynamics, the coexistence of bifurcations and bursting oscillations. More importantly, it has been found several sets of synaptic weight for which the proposed HNN displays multiple coexisting stable states including three disconnected attractors. Besides the phenomenon of coexistence of attractors, the bursting phenomenon characterized by homoclinic/Hopf cycle–cycle bursting via homoclinic/fold hysteresis loop is observed. This contribution represents the first case where the later phenomenon (bursting oscillations) occurs in an autonomous HNN. Also, PSpice simulations are used to support the results of the previous analyses.

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References

  1. B. Bao, H. Qian, J. Wang, Q. Xu, M. Chen, H. Wu, Y. Yu, Numerical analyses and experimental validations of coexisting multiple attractors in Hopfield neural network. Nonlinear Dyn. 90(4), 2359–2369 (2017)

    MathSciNet  Google Scholar 

  2. B. Bao, H. Qian, Q. Xu, M. Chen, J. Wang, Y. Yu, Coexisting behaviors of asymmetric attractors in hyperbolic-type memristor based Hopfield neural network. Front. Comput. Neurosci. 11, 81 (2017)

    Google Scholar 

  3. B. Bao, C. Chen, H. Bao, X. Zhang, Q. Xu, M. Chen, Dynamical effects of neuron activation gradient on Hopfield neural network: numerical analyses and hardware experiments. Int. J. Bifurc. Chaos 29(04), 1930010 (2019)

    MathSciNet  MATH  Google Scholar 

  4. H. Bao, A. Hu, W. Liu, B. Bao, Hidden bursting firings and bifurcation mechanisms in memristive neuron model with threshold electromagnetic induction. IEEE Trans. Neural Netw. Learn. Syst. (2019). https://doi.org/10.1109/TNNLS.2019.2905137

    Article  Google Scholar 

  5. H. Bao, W. Liu, A. Hu, Coexisting multiple ring patterns in two adjacent neurons coupled by memristive electromagnetic induction. Nonlinear Dyn. 95(1), 43–56 (2019)

    Google Scholar 

  6. J. Cao, Global exponential stability of Hopfield neural networks. Int. J. Syst. Sci. 32(2), 233–236 (2001)

    MathSciNet  MATH  Google Scholar 

  7. J. Cao, M. Xiao, Stability and Hopf bifurcation in a simplified bam neural network with two time delays. IEEE Trans. Neural Networks 18(2), 416–430 (2007)

    MathSciNet  Google Scholar 

  8. M.F. Danca, N. Kuznetsov, Hidden chaotic sets in a Hopfield neural system. Chaos Solitons Fractals 103, 144–150 (2017)

    MathSciNet  MATH  Google Scholar 

  9. H. Gu, Biological experimental observations of an unnoticed chaos as simulated by the Hindmarsh–Rose model. PLoS ONE 8(12), e81759 (2013)

    Google Scholar 

  10. R.C. Hilborn, Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers (Oxford University Press on Demand, Oxford, 2000)

    MATH  Google Scholar 

  11. J.J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. 81(10), 3088–3092 (1984)

    MATH  Google Scholar 

  12. C. Huang, Z. Yang, T. Yi, X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities. J. Differ. Equ. 256(7), 2101–2114 (2014)

    MathSciNet  MATH  Google Scholar 

  13. C. Huang, R. Su, J. Cao, S. Xiao, Asymptotically stable high-order neutral cellular neural networks with proportional delays and D operators. Math. Comput. Simul. (2019). https://doi.org/10.1016/j.matcom.2019.06.001

    Article  Google Scholar 

  14. E.M. Izhikevich, Neural excitability, spiking and bursting. Int. J. Bifurc. Chaos 10(06), 1171–1266 (2000)

    MathSciNet  MATH  Google Scholar 

  15. E.M. Izhikevich, Which model to use for cortical spiking neurons? IEEE Trans. Neural Networks 15(5), 1063–1070 (2004)

    Google Scholar 

  16. J. Kengne, Z.N. Tabekoueng, V.T. Kamdoum, A.N. Nguomkam, Periodicity, chaos, and multiple attractors in a memristor-based Shinriki’s circuit. Chaos Interdiscip. J. Nonlinear Sci. 25(10), 103126 (2015)

    MathSciNet  MATH  Google Scholar 

  17. J. Kengne, Z.N. Tabekoueng, H. Fotsin, Coexistence of multiple attractors and crisis route to chaos in autonomous third order Duffing–Holmes type chaotic oscillators. Commun. Nonlinear Sci. Numer. Simul. 36, 29–44 (2016)

    MathSciNet  MATH  Google Scholar 

  18. Q. Li, X. Yang, Complex dynamics in a simple Hopfield-type neural network, in International symposium on neural networks (Springer, Berlin, 2005), pp. 357–362

  19. Q. Li, S. Tang, H. Zeng, T. Zhou, On hyperchaos in a small memristive neural network. Nonlinear Dyn. 78(2), 1087–1099 (2014)

    Article  MATH  Google Scholar 

  20. K. Mineeja, R.P. Ignatius, Spatiotemporal activities of a pulse-coupled biological neural network. Nonlinear Dyn. 92(4), 1881–1897 (2018)

    Google Scholar 

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

    MATH  Google Scholar 

  22. H.S. Nik, S. Effati, J. Saberi-Nadja, Ultimate bound sets of a hyperchaotic system and its application in chaos synchronization. Complexity 20(4), 30–44 (2015)

    MathSciNet  Google Scholar 

  23. S. Panahi, Z. Aram, S. Jafari, J. Ma, J. Sprott, Modeling of epilepsy based on chaotic artificial neural network. Chaos, Solitons Fractals 105, 150–156 (2017)

    MathSciNet  Google Scholar 

  24. V.T. Pham, S. Jafari, S. Vaidyanathan, C. Volos, X. Wang, A novel memristive neural network with hidden attractors and its circuitry implementation. Sci. China Technol. Sci. 59(3), 358–363 (2016)

    Google Scholar 

  25. H. Qiu, X. Chen, W. Liu, G. Zhou, Y. Wang, J. Lai, A fast 1-solver and its applications to robust face recognition. J. Ind. Manag. Optim. (JIMO) 8, 163–178 (2012)

    MathSciNet  MATH  Google Scholar 

  26. Z.N. Tabekoueng, J. Kengne, Complex dynamics of a 4D Hopfield neural networks (HNNS) with a nonlinear synaptic weight: coexistence of multiple attractors and remerging Feigenbaum trees. AEU Int. J. Electron. Commun. 93, 242–252 (2018)

    Google Scholar 

  27. Z.N. Tabekoueng, J. Kengne, Nonlinear dynamics of three-neurons-based Hopfield neural networks (HNNS): remerging Feigenbaum trees, coexisting bifurcations and multiple attractors. J. Circuits Syst. Comput. 28(07), 1950121 (2019)

    Google Scholar 

  28. Z.N. Tabekoueng, J. Kengne, L.K. Kengne, Antimonotonicity, chaos and multiple coexisting attractors in a simple hybrid diode-based jerk circuit. Chaos Solitons Fractals 105, 77–91 (2017)

    MathSciNet  Google Scholar 

  29. Z.N. Tabekoueng, J. Kengne, H. Fotsin, A plethora of behaviors in a memristor based Hopfield neural networks (hnns). Int. J. Dyn. Control 7(1), 36–52 (2019)

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  31. Y. Wang, G. Zhou, L. Caccetta, W. Liu, An alternative lagrange-dual based algorithm for sparse signal reconstruction. IEEE Trans. Signal Process. 59(4), 1895–1901 (2010)

    Google Scholar 

  32. Z. Wang, J. Cao, Z. Guo, L. Huang, Generalized stability for discontinuous complex valued Hopfield neural networks via differential inclusions. Proc. R. Soc. A 474(2220), 20180507 (2018)

    MathSciNet  MATH  Google Scholar 

  33. A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining Lyapunov exponents from a time series. Physica D 16(3), 285–317 (1985)

    MathSciNet  MATH  Google Scholar 

  34. X. Wu, J. Ma, L. Yuan, Y. Liu, Simulating electric activities of neurons by using pspice. Nonlinear Dyn. 75(1–2), 113–126 (2014)

    MathSciNet  Google Scholar 

  35. Q. Xu, Z. Song, H. Bao, M. Chen, B. Bao, Two-neuron-based non-autonomous memristive Hopfield neural network: numerical analyses and hardware experiments. AEU Int. J. Electron. Commun. 96, 66–74 (2018)

    Google Scholar 

  36. Y. Xu, Y. Jia, M. Ge, L. Lu, L. Yang, X. Zhan, Effects of ion channel blocks on electrical activity of stochastic Hodgkin–Huxley neural network under electromagnetic induction. Neurocomputing 283, 196–204 (2018)

    Google Scholar 

  37. X.S. Yang, Q. Yuan, Chaos and transient chaos in simple Hopfield neural networks. Neurocomputing 69(1–3), 232–241 (2005)

    Google Scholar 

  38. P. Zheng, W. Tang, J. Zhang, Some novel double-scroll chaotic attractors in Hopfield networks. Neurocomputing 73(10–12), 2280–2285 (2010)

    Google Scholar 

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Acknowledgements

The authors are grateful to the referee for his/her helpful comments.

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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

    Z. Tabekoueng Njitacke

  2. Unité de Recherche d’Automatique et Informatique Appliquée (URAIA), Department of Electrical Engineering, IUT-FV Bandjoun, University of Dschang, Bandjoun, Cameroon

    Z. Tabekoueng Njitacke & J. Kengne

  3. Unité de Recherche de Matière Condensée, d’Electronique et de Traitement du Signal (URMACETS), Department of Physics, University of Dschang, P.O. Box 67, Dschang, Cameroon

    H. B. Fotsin

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  1. Z. Tabekoueng Njitacke
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  2. J. Kengne
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  3. H. B. Fotsin
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Correspondence to Z. Tabekoueng Njitacke.

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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, 3424–3444 (2020). https://doi.org/10.1007/s00034-019-01324-6

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  • DOI: https://doi.org/10.1007/s00034-019-01324-6

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