Coexisting multi-stable patterns in memristor synapse-coupled Hopfield neural network with two neurons

  • Chengjie Chen
  • Jingqi Chen
  • Han Bao
  • Mo Chen
  • Bocheng BaoEmail author
Original Paper


When possessing a potential difference between two neurons, an electromagnetic induction current appears in the Hopfield neural network (HNN), which can be emulated by a flux-controlled memristor synapse. Thus, a three-order two-neuron-based autonomous memristive HNN is presented in this paper, which is the lowest order and has not been reported in the previous studies. With the mathematical model, the detailed stability analyses for the line equilibrium are executed, so that the fold and Hopf bifurcation sets and stability region distributions in the parameter plane are obtained. Furthermore, numerical results of coexisting bifurcation patterns are investigated, which are confirmed effectively by local basins of attraction and phase plane plots. The numerical results demonstrate coexisting multi-stable patterns of the spiral chaotic patterns with different dynamic amplitudes, periodic patterns with different periodicities, and stable resting patterns with different positions in the memristive HNN. Besides, the circuit synthesis and breadboard experiments are performed to well validate the numerical simulations.


Hopfield neural network (HNN) Memristor synapse Coexisting multi-stable patterns Line equilibrium Circuit synthesis 



This work was supported by the grants from the National Natural Science Foundations of China under 51777016, 61601062, 61801054, and 11602035, and the Natural Science Foundations of Jiangsu Province, China under BK20160282.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest. These authors contribute equally to this work.


  1. 1.
    Hopfield, J.J.: Neurons with graded response have collective computational properties like those of 2-state neurons. Proc. Natl. Acad. Sci. USA 81(10), 3088–3092 (1984)zbMATHGoogle Scholar
  2. 2.
    Korn, H., Faure, P.: Is there chaos in the brain II. Experimental evidence and related models. C. R. Biol. 326(9), 787–840 (2003)Google Scholar
  3. 3.
    Yang, X.S., Huang, Y.: Complex dynamics in simple Hopfield neural networks. Chaos 16(3), 033114 (2006)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Babloyantz, A., Lourenco, C.: Brain chaos and computation. Int. J. Neural Syst. 7(4), 461–471 (1996)Google Scholar
  5. 5.
    Laskowski, Ł.: A novel hybrid-maximum neural network in stereo-matching process. Neural Comput. Appl. 23(7), 2435–2450 (2013)Google Scholar
  6. 6.
    Yang, J., Wang, L.D., Wang, Y., Guo, T.T.: A novel memristive Hopfield neural network with application in associative memory. Neurocomputing 227, 142–148 (2017)Google Scholar
  7. 7.
    Brosch, T., Neumann, H.: Computing with a canonical neural circuits model with pool normalization and modulating feedback. Neural Comput. 26(12), 2735–2789 (2014)MathSciNetGoogle Scholar
  8. 8.
    Mathias, A.C., Rech, P.C.: Hopfield neural network: the hyperbolic tangent and the piecewise-linear activation functions. Neural Netw. 34(10), 42–45 (2012)Google Scholar
  9. 9.
    Bao, B.C., Li, Q.D., Wang, N., Xu, Q.: Multistability in Chua’s circuit with two stable node-foci. Chaos 26(4), 043111 (2016)MathSciNetGoogle Scholar
  10. 10.
    Chen, M., Xu, Q., Lin, Y., Bao, B.C.: Multistability induced by two symmetric stable node-foci in modified canonical Chua’s circuit. Nonlinear Dyn. 87(2), 789–802 (2017)Google Scholar
  11. 11.
    Negou, A.N., Kengne, J.: Dynamic analysis of a unique jerk system with a smoothly adjustable symmetry and nonlinearity: reversals of period doubling, offset boosting and coexisting bifurcations. AEÜ Int. J. Electron. Commun. 90, 1–19 (2018)Google Scholar
  12. 12.
    Li, C.B., Sprott, J.C.: An infinite 3-D quasiperiodic lattice of chaotic attractors. Phys. Lett. A 382(8), 581–587 (2018)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Pham, V.T., Ouannas, A., Volos, C.K., Kapitaniak, T.: A simple fractional order chaotic system without equilibrium and its synchronization. AEÜ Int. J. Electron. Commun. 86, 69–76 (2018)Google Scholar
  14. 14.
    Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N.V., Leonov, G.A., Prasad, A.: Hidden attractors in dynamical systems. Phys. Rep. 637, 1–50 (2016)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ma, J., Wu, F.G., Ren, G.D., Tang, J.: A class of initials dependent dynamical systems. Appl. Math. Comput. 298, 65–76 (2017)MathSciNetGoogle Scholar
  16. 16.
    Xu, Q., Lin, Y., Bao, B.C., Chen, M.: Multiple attractors in a non-ideal active voltage-controlled memristor based Chua’s circuit. Chaos Solitons Fractals 83, 186–200 (2016)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kengne, J., Negou, A.N., Tchiotsop, D.: Antimonotonicity, chaos and multiple attractors in a novel autonomous memristor-based jerk circuit. Nonlinear Dyn. 88, 2589–2608 (2017)Google Scholar
  18. 18.
    Bao, B.C., Xu, L., Wang, N., Bao, H., Xu, Q., Chen, M.: Third-order RLCM-four-elements-based chaotic circuit and its coexisting bubbles. AEÜ Int. J. Electron. Commun. 94, 26–35 (2018)Google Scholar
  19. 19.
    Zheng, P.S., Tang, W.S., Zhang, J.X.: Some novel double-scroll chaotic attractors in Hopfield networks. Neurocomputing 73, 2280–2285 (2010)Google Scholar
  20. 20.
    Li, Q.D., Yang, X.S., Yang, F.Y.: Hyperchaos in Hopfield-type neural networks. Neurocomputing 67, 275–280 (2005)Google Scholar
  21. 21.
    Yuan, Q., Li, Q.D., Yang, X.S.: Horseshoe chaos in a class of simple Hopfield neural networks. Chaos Solitons Fractals 39, 1522–1529 (2009)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Huang, W.Z., Huang, Y.: Chaos, bifurcations and robustness of a class of Hopfield neural networks. Int. J. Bifurc. Chaos 21(3), 885–895 (2011)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Rech, P.C.: Period-adding and spiral organization of the periodicity in a Hopfield neural network. Int. J. Mach. Learn. Cybern. 6(1), 1–6 (2015)Google Scholar
  24. 24.
    Bao, B.C., Qian, H., Wang, J., Xu, Q., Chen, M., Wu, H.G., Yu, Y.J.: Numerical analyses and experimental validations of coexisting multiple attractors in Hopfield neural network. Nonlinear Dyn. 90(4), 2359–2369 (2017)MathSciNetGoogle Scholar
  25. 25.
    Njitacke, Z.T., Kengne, J.: Complex dynamics of a 4D Hopfield neural networks (HNNs) with a nonlinear synaptic weight: coexistence of multiple attractors and remerging Feigenbaum trees. AEÜ Int. J. Electron. Commun. 93, 242–252 (2018)Google Scholar
  26. 26.
    Danca, M.F., Kuznetsov, N.V.: Hidden chaotic sets in a Hopfield neural system. Chaos Solitons Fractals 103, 144–150 (2017)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Li, Q.D., Tang, S., Zeng, H.Z., Zhou, T.T.: On hyperchaos in a small memristive neural network. Nonlinear Dyn. 78(2), 1087–1099 (2014)zbMATHGoogle Scholar
  28. 28.
    Xu, Q., Song, Z., Bao, H., Chen, M., Bao, B.C.: Two-neuron-based non-autonomous memristive Hopfield neural network: numerical analyses and hardware experiments. AEÜ Int. J. Electron. Commun. 96, 66–74 (2018)Google Scholar
  29. 29.
    Pham, V.T., Jafari, S., Vaidyanathan, S., Volos, C.K., Wang, X.: A novel memristive neural network with hidden attractors and its circuitry implementation. Sci. China Technol. Sci. 59, 358–363 (2016)Google Scholar
  30. 30.
    Bao, B.C., Qian, H., Xu, Q., Chen, M., Wang, J., Yu, Y.J.: Coexisting behaviors of asymmetric attractors in hyperbolic-type memristor based Hopfield neural network. Front. Comput. Neurosci. 11, 1–14 (2017). Article 81Google Scholar
  31. 31.
    Hu, X.Y., Liu, C.X., Liu, L., Ni, J.K., Yao, Y.P.: Chaotic dynamics in a neural network under electromagnetic radiation. Nonlinear Dyn. 91(3), 1541–1554 (2018)Google Scholar
  32. 32.
    Eshraghian, K., Kavehei, O., Cho, K.R., Chappell, J.M., Iqbal, A., Al-Sarawi, S.F., Abbott, D.: Memristive device fundamentals and modeling: applications to circuits and systems simulation. Proc. IEEE 100(6), 1991–2007 (2012)Google Scholar
  33. 33.
    Wang, Z., Joshi, S., Savel’Ev, S.E., Jiang, H., Rivu, M., Lin, P., Hu, M., Ge, N., Strachan, J.P., Li, Z., Wu, Q., Barnell, M., Li, G.L., Xin, H.L., Williams, R.S., Xia, Q., Yang, J.J.: Memristors with diffusive dynamics as synaptic emulators for neuromorphic computing. Nat. Mater. 16(1), 101–108 (2017)Google Scholar
  34. 34.
    Kumar, S., Strachan, J.P., Williams, R.S.: Chaotic dynamics in nanoscale \(\text{ NbO }_{2}\) Mott memristor for analogue computing. Nature 548(7667), 318–321 (2017)Google Scholar
  35. 35.
    Serb, A., Bill, J., Khiat, A., Berdan, R., Legenstein, R., Prodromakis, T.: Unsupervised learning in probabilistic neural networks with multi-state metal-oxide memristive synapses. Nat. Commun. 7, 12611 (2016)Google Scholar
  36. 36.
    Wu, J., Xu, Y., Ma, J.: Lévy noise improves the electrical activity in a neuron under electromagnetic radiation. PLoS ONE 12, e0174330 (2017)Google Scholar
  37. 37.
    Ma, J., Lv, M., Zhou, P., Xu, Y., Hayat, T.: Phase synchronization between two neurons induced by coupling of electromagnetic field. Appl. Math. Comput. 307, 321–328 (2017)MathSciNetGoogle Scholar
  38. 38.
    Ge, M.Y., Jia, Y., Xu, Y., Yang, L.J.: Mode transition in electrical activities of neuron driven by high and low frequency stimulus in the presence of electromagnetic induction and radiation. Nonlinear Dyn. 91(1), 515–523 (2018)Google Scholar
  39. 39.
    Lu, L.L., Jia, Y., Liu, W.H., Yang, L.J.: Mixed stimulus-induced mode selection in neural activity driven by high and low frequency current under electromagnetic radiation. Complexity 2017, 7628537 (2017)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Xu, F., Zhang, J., Fang, T., Huang, S., Wang, M.: Synchronous dynamics in neural system coupled with memristive synapse. Nonlinear Dyn. 92(3), 1395–1402 (2018)Google Scholar
  41. 41.
    Xu, Y., Jia, Y., Ma, J., Alsaedi, A., Ahmad, B.: Synchronization between neurons coupled by memristor. Chaos Solitons Fractals 104, 435–442 (2017)Google Scholar
  42. 42.
    Bao, H., Liu, W., Hu, A.H.: Coexisting multiple firing patterns in two adjacent neurons coupled by memristive electromagnetic induction. Nonlinear Dyn. 95(1), 43–56 (2019)Google Scholar
  43. 43.
    Bao, B.C., Hu, A.H., Bao, H., Xu, Q., Chen, M., Wu, H.G.: Three-dimensional memristive Hindmarsh–Rose neuron model with hidden coexisting asymmetric behaviors. Complexity 2018, 3872573 (2018)Google Scholar
  44. 44.
    Xu, Q., Zhang, Q.L., Bao, B.C., Hu, Y.H.: Non-autonomous second-order memristive chaotic circuit. IEEE Access 5(1), 21039–21045 (2017)Google Scholar
  45. 45.
    Bao, B.C., Jiang, T., Xu, Q., Chen, M., Wu, H.G., Hu, Y.H.: Coexisting infinitely many attractors in active band-pass filter-based memristive circuit. Nonlinear Dyn. 86(3), 1711–1723 (2016)Google Scholar
  46. 46.
    Ma, J., Zhang, G., Hayat, T., Ren, G.D.: Model electrical activity of neuron under electric field. Nonlinear Dyn. (2018)
  47. 47.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica D 16(3), 285–317 (1985)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Strelioff, C.C., Hübler, A.W.: Medium-term prediction of chaos. Phys. Rev. Lett. 96(4), 044101 (2006)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Information Science and EngineeringChangzhou UniversityChangzhouChina
  2. 2.College of Automation EngineeringNanjing University of Aeronautics and AstronauticsNanjingChina

Personalised recommendations