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Memristive electromagnetic induction effects on Hopfield neural network

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Due to the existence of membrane potential differences, the electromagnetic induction flows can be induced in the interconnected neurons of Hopfield neural network (HNN). To express the induction flows, this paper presents a unified memristive HNN model using hyperbolic-type memristors to link neurons. By employing theoretical analysis along with multiple numerical methods, we explore the electromagnetic induction effects on the memristive HNN with three neurons. Three cases are classified and discussed. When using one memristor to link two neurons bidirectionally, the coexisting bifurcation behaviors and extreme events are disclosed with respect to the memristor coupling strength. When using two memristors to link three neurons, the antimonotonicity phenomena of periodic and chaotic bubbles are yielded, and the initial-related extreme events are emerged. When using three memristors to link three neurons end to end, the extreme events owning prominent riddled basins of attraction are demonstrated. In addition, we develop the printed circuit board (PCB)-based hardware experiments by synthesizing the memristive HNN, and the experimental results well confirm the memristive electromagnetic induction effects. Certainly, the PCB-based implementation will benefit the integrated circuit design for large-scale Hopfield neural network in the future.

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  1. Chua, L.O.: Memristor—the missing circuit element. IEEE Trans. Circuit Theory 18(5), 507–519 (1971)

    Article  Google Scholar 

  2. Kumar, S., Strachan, J., Williams, R.: Chaotic dynamics in nanoscale NbO2 Mott memristors for analogue computing. Nature 548, 318–321 (2017)

    Article  Google Scholar 

  3. Li, C., Min, F.H., Li, C.B.: Multiple coexisting attractors of the serial-parallel memristor-based chaotic system and its adaptive generalized synchronization. Nonlinear Dyn. 94, 2785–2806 (2018)

    Article  Google Scholar 

  4. Xie, W.L., Wang, C.H., Lin, H.R.: A fractional-order multistable locally active memristor and its chaotic system with transient transition, state jump. Nonlinear Dyn. 104, 4523–4541 (2021)

    Article  Google Scholar 

  5. Miranda, E., Milano, G., Ricciardi, C.: Modeling of short-term synaptic plasticity effects in ZnO nanowire-based memristors using a potentiation-depression rate balance equation. IEEE Trans. Nanotechnol. 19, 609–612 (2020)

    Article  Google Scholar 

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

    Article  Google Scholar 

  7. Bao, H., Chen, C.J., Hu, Y.H., Chen, M., Bao, B.C.: Two-dimensional piecewise-linear neuron model. IEEE Trans. Circuits Syst. II Exp. Briefs 68(4), 1453−1457 (2021)

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

    Article  Google Scholar 

  9. Sun, J.W., Xiao, X., Yang, Q.F., Liu, P., Wang, Y.F.: Memristor-based Hopfield network circuit for recognition and sequencing application. AEÜ-Int. J. Electron. Commun. 134, 153698 (2021)

  10. Hong, Q.H., Yan, R., Wang, C.H., Sun, J.R.: Memristive circuit implementation of biological nonassociative learning mechanism and its applications. IEEE Trans. Biomed. Circ. Syst. 14(5), 1036–1050 (2020)

    Article  Google Scholar 

  11. Ma, J., Tang, J.: A review for dynamics in neuron and neuronal network. Nonlinear Dyn. 89(3), 1569–1578 (2017)

    Article  MathSciNet  Google Scholar 

  12. 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 (2017)

    Article  Google Scholar 

  13. Parastesh, F., Azarnoush, H., Jafari, S., Hatef, B., Perc, M., Repnik, R.: Synchronizability of two neurons with switching in the coupling. Appl. Math. Comput. 350, 217–223 (2019)

    MathSciNet  MATH  Google Scholar 

  14. Fang, T.T., Zhang, J.Q., Huang, S.F., Xu, F., Wang, M.S., Yang, H.: Synchronous behavior among different regions of the neural system induced by electromagnetic radiation. Nonlinear Dyn. 98(17), 1267–1274 (2019)

    Article  Google Scholar 

  15. Parastesh, F., Rajagopal, K., Alsaadi, F.E., Hayat, T., Pham, V.T., Hussain, I.: Birth and death of spiral waves in a network of Hindmarsh-Rose neurons with exponential magnetic flux and excitable media. Appl. Math. Comput. 354, 377–384 (2019)

    MathSciNet  MATH  Google Scholar 

  16. Takembo, C.N., Mvogo, A., Fouda, H.P.E., Kofané, T.C.: Effect of electromagnetic radiation on the dynamics of spatiotemporal patterns in memristor-based neuronal network. Nonlinear Dyn. 95, 1067–1078 (2018)

    Article  Google Scholar 

  17. Bao, H., Hu, A.H., Liu, W.B., Bao, B.C.: Hidden bursting firings and bifurcation mechanisms in memristive neuron model with threshold electromagnetic induction. IEEE Trans. Neural Netw. Learn. Syst. 31(2), 502–511 (2020)

    Article  Google Scholar 

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

    Article  Google Scholar 

  19. Li, R.H., Wang, Z.H., Dong, E.Z.: A new locally active memristive synapse-coupled neuron model. Nonlinear Dyn. 104, 4459–4475 (2021)

    Article  Google Scholar 

  20. 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 

  21. Zhou, Q., Wei, D.Q.: Collective dynamics of neuronal network under synapse and field coupling. Nonlinear Dyn. (2021).

    Article  Google Scholar 

  22. Yang, K., Duan, Q.X., Wang, Y.H., Zhang, T., Yang, Y.C., Huang, R.: Transiently chaotic simulated annealing based on intrinsic nonlinearity of memristors for efficient solution of optimization problems. Sci. Adv. 6(33), eaba9901 (2020)

  23. Pu, Y., Yi, Z., Zhou, J.: Fractional Hopfield neural networks: Fractional dynamic associative recurrent neural networks. IEEE Trans. Neural Netw. Learn. Syst. 28(10), 2319–2333 (2017)

    Article  MathSciNet  Google Scholar 

  24. Cai, F.X., Kumar, Suhas., Vaerenbergh, T.V., et al.: Power-efficient combinatorial optimization using intrinsic noise in memristor Hopfield neural networks. Nat. Electron. 3(7), 409−418 (2020)

  25. Wang, Z., Parastesh, F., Rajagopal, K., Hamarash, I.I., Hussain, I.: Delay-induced synchronization in two coupled chaotic memristive Hopfield neural networks. Chaos Solit. Fractals 134, 109702 (2020)

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

  27. Lin, H.R., Wang. C.H., Hong, Q.H., Sun, Y.C.: A multi-stable memristor and its application in a neural network. IEEE Trans. Circuits Syst. II Exp. Briefs 67(12), 3472−3476 (2020)

  28. 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, 1541–1554 (2018)

    Article  Google Scholar 

  29. Lin, H.R., Wang, C.H., Tan, Y.M.: Hidden extreme multistability with hyperchaos and transient chaos in a Hopfield neural network affected by electromagnetic radiation. Nonlinear Dyn. 99, 2369–2386 (2020)

    Article  Google Scholar 

  30. Chen, C.J., Chen, J.Q., Bao, H., Chen, M., Bao, B.C.: Coexisting multi-stable patterns in memristor synapse-coupled Hopfield neural network with two neurons. Nonlinear Dyn. 95(4), 3385–3399 (2019)

    Article  Google Scholar 

  31. Chen, C.J., Bao, H., Chen, M., Xu, Q., Bao, B.C.: Non-ideal memristor synapse-coupled bi-neuron Hopfield neural network: Numerical simulations and breadboard experiments. AEÜ-Int. J. Electron. Commun. 111, 152894 (2019)

  32. Chen, M., Chen, C.J., Bao, B.C., Xu, Q.: Initial sensitive dynamics in memristor synapse-coupled Hopfield neural network. J. Electron. Inf. Technol. 42(4), 870–877 (2020)

    Google Scholar 

  33. Ma, C., Mou, J., Yang, F., Yan, H.Z.: A fractional-order hopfield neural network chaotic system and its circuit realization. Eur. Phys. J. Plus 135, 100 (2020)

    Article  Google Scholar 

  34. Rajagopal, K., Munoz-Pacheco, J.M., Pham, V.T., Hoang, D.V., Alsaadi, F.E., Alsaadi, F.E.: A Hopfield neural network with multiple attractors and its FPGA design. Eur. Phys. J. Spec. Top. 227, 811–820 (2018)

    Article  Google Scholar 

  35. Yu, F., Zhang, Z.N., Shen, H., Huang, Y.Y., Cai, S., Jin, J., Du, S.C.: Design and FPGA implementation of a pseudo-random number generator based on a Hopfield neural network under electromagnetic radiation, Front. Phys. 9, 690651 (2021)

  36. Zheng, P.S., Tang, W.S., Zhang, J.X.: Dynamic analysis of unstable Hopfield networks. Nonlinear Dyn. 61(3), 399–406 (2010)

    Article  MathSciNet  Google Scholar 

  37. Khalil, H.K.: Nonlinear systems, 3rd edn. Prentice-Hall, Upper Saddle River, NJ, USA (2002)

    MATH  Google Scholar 

  38. Chen, T.P., Amari, S.I.: Stability of asymmetric Hopfield networks. IEEE Trans. Neural Netw. Learn. Syst. 12(1), 159–163 (2001)

    Article  Google Scholar 

  39. Silva, C.P.: Shil’nikov’s theorem-a tutorial. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 40(10), 675−682 (1993)

  40. Ansmann, G., Karnatak, R., Lehnertz, K., Feudel, U.: Extreme events in excitable systems and mechanisms of their generation. Phys. Rev. E 88, 052911 (2013)

  41. Karnatak, R., Ansmann, G., Feudel, U., Lehnertz, K.: Route to extreme events in excitable systems. Phys. Rev. E 90, 022917 (2014)

  42. Saha, A., Feudel, U.: Riddled basins of attraction in systems exhibiting extreme events. Chaos 28(3), 033610 (2018)

  43. Mishra, A., Kingston, S.L., Hens, C., Kapitaniak, T., Feudel, U., Dana, S.K.: Routes to extreme events in dynamical systems: dynamical and statistical characteristics. Chaos 30(6), 063114 (2020)

  44. Ray, A., Rakshit, S., Basak, G.K., Dana, S.K., Ghosh, D.: Understanding the origin of extreme events in El Niño southern oscillation. Phys. Rev. E 101, 062210 (2020)

  45. Bao, H., Zhu, D., Liu, W.B., Xu, Q., Chen, M., Bao, B.C.: Memristor synapse-based Morris−Lecar model: Bifurcation analyses and FPGA-based validations for periodic and chaotic bursting/spiking firings. Int. J. Bifurcation Chaos 30(3), 2050045 (2020)

    Article  MathSciNet  Google Scholar 

  46. Ribar, L., Sepulchre, R.: Neuromodulation of neuromorphic circuits. IEEE Trans. Circuits Syst. I, Reg. Papers 66(8), 3028−3040 (2019)

  47. Haghiri, S., Naderi, A., Ghanbari, B., Ahmadi, A.: High speed and low digital resources implementation of Hodgkin-Huxley neuronal model using base-2 functions. IEEE Trans. Circuits Syst. I, Reg. Papers 68(1), 275−287 (2021)

  48. Jokar, E., Abolfathi, H., Ahmadi, A., Ahmadi, M.: An efficient uniform-segmented neuron model for large-scale neuromorphic circuit design: Simulation and FPGA synthesis results. IEEE Trans. Circuits Syst. I, Reg. Papers 66(6), 2336−2349 (2019)

  49. Li, K.X., Bao, H., Li, H.Z., Ma, J., Hua, Z.Y., Bao, B.C.: Memristive Rulkov neuron model with magnetic induction effects. IEEE Trans. Ind. Informat. (2021).

    Article  Google Scholar 

  50. Lin, H.R., Wang, C.H., Chen, C.J., Sun, Y.C., Zhou, C., Xu, C., Hong, Q.H.: Neural bursting and synchronization emulated by neural networks and circuits. IEEE Trans. Circuits Syst. I, Reg. Papers (2021).

  51. Bao, B.C., Jiang, T., Wang, G.Y., Jin, P.P., Bao, H., Chen, M.: Two-memristor-based Chua’s hyperchaotic circuit with plane equilibrium and its extreme multistability. Nonlinear Dyn. 89(2), 1157–1171 (2017)

    Article  Google Scholar 

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This work was supported by the National Natural Science Foundation of China under Grant Nos. 61971228 and 51777016, the Natural Science Foundation of Henan Province under Grant No. 202300410351, and the Key Scientific Research of Colleges and Universities in Henan Province under Grant No. 21A120007.

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Correspondence to Fuhong Min.

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Chen, C., Min, F., Zhang, Y. et al. Memristive electromagnetic induction effects on Hopfield neural network. Nonlinear Dyn 106, 2559–2576 (2021).

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