Skip to main content
SpringerLink
Log in

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

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  3. Yang, X.S., Huang, Y.: Complex dynamics in simple Hopfield neural networks. Chaos 16(3), 033114 (2006)

    Article  MATH  Google Scholar 

  4. Babloyantz, A., Lourenco, C.: Brain chaos and computation. Int. J. Neural Syst. 7(4), 461–471 (1996)

    Article  Google Scholar 

  5. Laskowski, Ł.: A novel hybrid-maximum neural network in stereo-matching process. Neural Comput. Appl. 23(7), 2435–2450 (2013)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  12. Li, C.B., Sprott, J.C.: An infinite 3-D quasiperiodic lattice of chaotic attractors. Phys. Lett. A 382(8), 581–587 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  19. Zheng, P.S., Tang, W.S., Zhang, J.X.: Some novel double-scroll chaotic attractors in Hopfield networks. Neurocomputing 73, 2280–2285 (2010)

    Article  Google Scholar 

  20. Li, Q.D., Yang, X.S., Yang, F.Y.: Hyperchaos in Hopfield-type neural networks. Neurocomputing 67, 275–280 (2005)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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 81

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  36. Wu, J., Xu, Y., Ma, J.: Lévy noise improves the electrical activity in a neuron under electromagnetic radiation. PLoS ONE 12, e0174330 (2017)

    Article  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  41. Xu, Y., Jia, Y., Ma, J., Alsaedi, A., Ahmad, B.: Synchronization between neurons coupled by memristor. Chaos Solitons Fractals 104, 435–442 (2017)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  46. Ma, J., Zhang, G., Hayat, T., Ren, G.D.: Model electrical activity of neuron under electric field. Nonlinear Dyn. https://doi.org/10.1007/s11071-018-4646-7 (2018)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  48. Strelioff, C.C., Hübler, A.W.: Medium-term prediction of chaos. Phys. Rev. Lett. 96(4), 044101 (2006)

    Article  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. School of Information Science and Engineering, Changzhou University, Changzhou, 213164, China

    Chengjie Chen, Jingqi Chen, Mo Chen & Bocheng Bao

  2. College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Han Bao

Authors
  1. Chengjie Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jingqi Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Han Bao
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mo Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Bocheng Bao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bocheng Bao.

Ethics declarations

Conflict of interest

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

Appendix

Appendix

To be helpful for more readers, the MATLAB program code for calculating the local basin of attraction in Fig. 5a is appended in Fig. 12.

Fig. 12
figure 12figure 12

MATLAB program code a sub-program code (mHNN_2N.m) b main-program code (main_program.m)

Full size image

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, C., Chen, J., Bao, H. et al. Coexisting multi-stable patterns in memristor synapse-coupled Hopfield neural network with two neurons. Nonlinear Dyn 95, 3385–3399 (2019). https://doi.org/10.1007/s11071-019-04762-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-019-04762-8

Keywords

Search

Navigation