Initial-induced coexisting and synchronous firing activities in memristor synapse-coupled Morris–Lecar bi-neuron network

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A memristor synapse with threshold memductance is employed to couple two neurons for representation of the electromagnetic induction effect triggered by their membrane potential difference. This paper presents a memristor synapse-coupled bi-neuron network by bidirectionally coupling two three-dimensional heterogeneous or homogeneous Morris–Lecar neurons with such a memristor synapse. The memristive bi-neuron network possesses a line equilibrium set with its stability related to the induction coefficient and memristor initial value. Coexisting firing activities in the heterogeneous memristive bi-neuron network are explored using bifurcation plots, phase plots, and time sequences, upon which the initial-induced infinitely many firing patterns including hyperchaotic, chaotic, and periodic bursting and tonic-spiking patterns are disclosed, indicating the emergence of the initial-induced extreme multistability. Furthermore, synchronous firing activities in homogeneous memristive bi-neuron network are investigated using the time sequences, synchronization transition states, and mean synchronized errors. The results demonstrate that the synchronous firing activities are associated with the induction coefficient and specially associated with the initial values of memristor synapse and coupling neurons. Finally, an FPGA-based electronic bi-neuron network is designed to experimentally confirm the memristor initial-induced coexisting firing activities.

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  1. 1.

    Ma, J., Tang, J.: A review for dynamics of collective behaviors of network of neurons. Sci. China Technol. Sci 58, 2038–2045 (2015)

  2. 2.

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

  3. 3.

    Parastesh, F., Rajagopal, K., Karthikeyan, A., Alsaedi, A., Hayat, T., Pham, V.-T.: Complex dynamics of a neuron model with discontinuous magnetic induction and exposed to external radiation. Cognit. Neurodyn. 12, 607–614 (2018)

  4. 4.

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

  5. 5.

    Wang, Y., Ma, J., Xu, Y., Wu, F., Zhou, P.: The electrical activity of neurons subject to electromagnetic induction and Gaussian white noise. Int. J. Bifurc. Chaos 27, 1750030 (2017)

  6. 6.

    Wu, F., Wang, C., Jin, W., Ma, J.: Dynamical responses in a new neuron model subjected to electromagnetic induction and phase noise. Physica A 469, 81–88 (2017)

  7. 7.

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

  8. 8.

    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)

  9. 9.

    Ren, G., Xu, Y., Wang, C.: Synchronization behavior of coupled neuron circuits composed of memristors. Nonlinear Dyn. 88, 893–901 (2017)

  10. 10.

    Xu, F., Zhang, J., Fang, T., Huang, S., Wang, M.: Synchronous dynamics in neural system coupled with memristive synapse. Nonlinear Dyn. 92, 1395–1402 (2018)

  11. 11.

    Ge, M., Jia, Y., Xu, Y., Yang, L.: 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, 515–523 (2018)

  12. 12.

    Lu, L., Jia, Y., Liu, W., Yang, L.: Mixed stimulus-induced mode selection in neural activity driven by high and low frequency current under electromagnetic radiation. Complexity 2017, 7628537 (2017)

  13. 13.

    Lv, M., Ma, J.: Multiple modes of electrical activities in a new neuron model under electromagnetic radiation. Neurocomputing 205, 375–381 (2016)

  14. 14.

    Wu, J., Ma, S.: Coherence resonance of the spiking regularity in a neuron under electromagnetic radiation. Nonlinear Dyn. 96, 1895–1908 (2019)

  15. 15.

    Bao, B., Hu, A., Bao, H., Xu, Q., Chen, M., Wu, H.: Three-dimensional memristive Hindmarsh–Rose neuron model with hidden coexisting asymmetric behaviors. Complexity 2018, 3872573 (2018)

  16. 16.

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

  17. 17.

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

  18. 18.

    Bennett, D.J., Li, Y., Harvey, P.J., Gorassini, M.: Evidence for plateau potentials in tail motoneurons of awake chronic spinal rats with spasticity. J. Neurophysiol. 86, 1972–1982 (2001)

  19. 19.

    Kim, H., Jones, K.E.: Asymmetric electrotonic coupling between the soma and dendrites alters the bistable firing behaviour of reduced models. J. Comput. Neurosci. 30, 659–674 (2011)

  20. 20.

    Bao, B., Hu, A., Xu, Q., Bao, H., Wu, H., Chen, M.: AC induced coexisting asymmetric bursters in the improved Hindmarsh–Rose model. Nonlinear Dyn. 92, 1695–1706 (2018)

  21. 21.

    Bao, B., Yang, Q., Zhu, L., Bao, H., Xu, Q., Yu, Y., Chen, M.: Chaotic bursting dynamics and coexisting multi-stable firing patterns in 3D autonomous M–L model and microcontroller-based validations. Int. J. Bifurc. Chaos 29, 1950134 (2019)

  22. 22.

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

  23. 23.

    Bao, H., Liu, W., Chen, M.: Hidden extreme multistability and dimensionality reduction analysis for an improved non-autonomous memristive FitzHugh–Nagumo circuit. Nonlinear Dyn. 96, 1879–1894 (2019)

  24. 24.

    Fozin, F.T., Kengne, J., Pelap, F.B.: Dynamical analysis and multistability in autonomous hyperchaotic oscillator with experimental verification. Nonlinear Dyn. 93, 653–669 (2018)

  25. 25.

    Pisarchik, A.N., Feudel, U.: Control of multistability. Phys. Rep. 540, 167–218 (2014)

  26. 26.

    Chen, M., Sun, M., Bao, H., Hu, Y., Bao, B.: Flux-charge analysis of two-memristor-based Chua’s circuit: dimensionality decreasing model for detecting extreme multistability. IEEE Trans. Ind. Electron 67, 2197–2206 (2020)

  27. 27.

    Sun, H., Scott, S.K., Showalter, K.: Uncertain destination dynamics. Phys. Rev. E 60, 3876–3880 (1999)

  28. 28.

    Patel, M.S., Patel, U., Sen, A., Sethia, G.C., Hens, C., Dana, S.K., Feudel, U., Showalter, K., Ngonghala, C.N., Amritkar, R.E.: Experimental observation of extreme multistability in an electronic system of two coupled Rössler oscillators. Phys. Rev. E 89, 022918 (2014)

  29. 29.

    Zhang, Y., Liu, Z., Wu, H., Chen, S., Bao, B.: Dimensionality reduction analysis for detecting initial effects on synchronization of memristor-coupled system. IEEE Access 7, 109689–109698 (2019)

  30. 30.

    Usha, K., Subha, P.: Energy feedback and synchronous dynamics of Hindmarsh–Rose neuron model with memristor. Chin. Phys. B 28, 02050 (2019)

  31. 31.

    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)

  32. 32.

    Soriano, D.C., Santos, O.V.D., Suyama, R., Fazanaro, F.I., Attux, R.: Conditional Lyapunov exponents and transfer entropy in coupled bursting neurons under excitation and coupling mismatch. Commun. Nonlinear Sci. Numer. Simul. 56, 419–433 (2018)

  33. 33.

    Wu, K., Wang, T., Wang, C., Du, T., Lu, H.: Study on electrical synapse coupling synchronization of Hindmarsh–Rose neurons under Gaussian white noise. Neural Comput. Appl. 30, 551–561 (2018)

  34. 34.

    Mostaghimi, S., Nazarimehr, F., Jafari, S., Ma, J.: Chemical and electrical synapse-modulated dynamical properties of coupled neurons under magnetic flow. Appl. Math. Comput. 348, 42–56 (2019)

  35. 35.

    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)

  36. 36.

    Ge, M., Jia, Y., Kirunda, J., Xu, Y., Shen, J., Lu, L., Liu, Y., Pei, Q., Zhan, X., Yang, L.: Propagation of firing rate by synchronization in a feed-forward multilayer Hindmarsh-Rose neural network. Neurocomputing 320, 60–68 (2018)

  37. 37.

    Eckhorn, R.: Neural mechanisms of scene segmentation: recording from the visual cortex suggest basic circuits or linking field models. IEEE Trans. Neural Netw. 10, 464–479 (1999)

  38. 38.

    Bartsch, R., Kantelhardt, J.W., Penzel, T., Havlin, S.: Experimental evidence for phase synchronization transitions in the human cardiorespiratory system. Phys. Rev. Lett. 98, 54102 (2007)

  39. 39.

    Uhlhaas, P.J., Singer, W.: Neural synchrony in brain disorders: relevance for cognitive dysfunctions and pathophysiology. Neuron 52, 155–168 (2006)

  40. 40.

    Xu, Y., Jia, Y., Wang, H., Liu, Y., Wang, P., Zhao, Y.: Spiking activities in chain neural network driven by channel noise with field coupling. Nonlinear Dyn. 95, 3237–3247 (2019)

  41. 41.

    Pinto, R.D., Varona, P., Volkovskii, A.R., Szücs, A., Abarbanel, H.D., Rabinovich, M.I.: Synchronous behavior of two coupled electronic neurons. Phys. Rev. E 62, 2644–2656 (2000)

  42. 42.

    Linaro, D., Poggi, T., Storace, M.: Experimental bifurcation diagram of a circuit-implemented neuron model. Phys. Lett. A 374, 4589–4593 (2011)

  43. 43.

    Dahasert, N., Öztürk, I., Kiliç, R.: Experimental realizations of the HR neuron model with programmable hardware and synchronization applications. Nonlinear Dyn. 70, 2343–2358 (2012)

  44. 44.

    Bilotta, E., Pantano, P., Vena, S.: Speeding up cellular neural network processing ability by embodying memristors. IEEE Trans. Neural Netw. Learn. Syst. 28, 1228–1232 (2017)

  45. 45.

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

  46. 46.

    Rajamani, V., Kim, H., Chua, L.O.: Morris–Lecar model of third-order barnacle muscle fiber is made of volatile memristors. Sci. China Inf. Sci. 61, 060426 (2018)

  47. 47.

    Shi, M., Wang, Z.: Abundant bursting patterns of a fractional-order Morris–Lecar neuron model. Commu. Nonlinear Sci. Numer. Simulat. 19, 1956–1969 (2014)

  48. 48.

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

  49. 49.

    Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice-Hall, Upper Saddle River (2002)

  50. 50.

    Liu, Y., Ren, G., Zhou, P., Hayat, T., Ma, J.: Synchronization in networks of initially independent dynamical systems. Phys. A 520, 370–380 (2019)

  51. 51.

    Buscarino, A., Frasca, M., Branciforte, M., Fortuna, L., Sprott, J.C.: Synchronization of two Rössler systems with switching coupling. Nonlinear Dyn. 88, 673–683 (2017)

  52. 52.

    Hayati, M., Nouri, M., Haghiri, S., Abbott, D.: Digital multiplierless realization of two coupled biological Morris–Lecar neuron model. IEEE Trans. Circuits Syst. I Reg. Pap. 62, 1805–1814 (2015)

  53. 53.

    Hua, Z., Zhou, B., Zhou, Y.: Sine chaotification model for enhancing chaos and its hardware implementation. IEEE Trans. Ind. Electron. 66, 1273–1284 (2018)

  54. 54.

    Rakshit, S., Bera, B.K., Perc, M., Ghosh, D.: Basin stability for chimera states. Sci. Rep. 7, 2412 (2017)

  55. 55.

    Lu, L., Jia, Y., Kirunda, J., Xu, Y., Ge, M., Pei, Q., Yang, L.: Effects of noise and synaptic weight on propagation of subthreshold excitatory postsynaptic current signal in a feed-forward neural network. Nonlinear Dyn. 95, 1673–1686 (2019)

  56. 56.

    Ge, M., Jia, Y., Xu, Y., Lu, L., Wang, H., Zhao, Y.: Wave propagation and synchronization induced by chemical autapse in chain Hindmarsh–Rose neural network. Appl. Math. Comput. 352, 136–145 (2019)

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This work was supported by the Grants from the National Natural Science Foundations of China under 51777016, 61801054, and 61601062, and the Natural Science Foundation of Jiangsu Province, China, under Grant No. BK20191451.

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Correspondence to Mo Chen.

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Bao, B., Yang, Q., Zhu, D. et al. Initial-induced coexisting and synchronous firing activities in memristor synapse-coupled Morris–Lecar bi-neuron network. Nonlinear Dyn (2019) doi:10.1007/s11071-019-05395-7

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  • Memristor synapse
  • Bi-neuron network
  • Initial value
  • Coexisting activity
  • Synchronous activity