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Chimera states and synchronization behavior in multilayer memristive neural networks

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Abstract

To study the effect of electromagnetic induction on the spatiotemporal dynamic behavior of neural networks, in this paper, we have mainly studied both the synchronization behavior and the evolution of chimera states (CS) in coupled neural networks. To do this, a multilayer memristive neural network was constructed by selecting the Hindmarsh–Rose neurons as the network nodes, and the effect of electromagnetic induction is introduced by using the cubic flux-controlled memristive model as synapse. For simplicity, the following coupling scheme is adopted: only the coupling connections for the neurons between different layers are considered with memristive synapses, while those neurons in each layer are still bidirectional coupled with the classical electrical synapses. It is found that, by adjusting the coupled strength of electrical synapses and the parameters of memristive synapses, the coexistence behavior of coherent and incoherent states, i.e., the CS, could appear in each layer. It is interesting that the CS are also found in inter-layer memristive synapse network. Furthermore, we have discussed the synchronization behavior in this multilayer memristive neural network, one can find when the whole multilayer network is in a synchronization state, not only the spatiotemporal consistency of the CS in each layer neural networks is observed, but also the memductance of all memristive synapses is completely synchronized. Our results suggest that the electromagnetic induction may play an important role in regulating the dynamic behavior of neural networks, and the introduction of memristive synapse into the biological neural network will provide useful clues for revealing the memory behavior of the neural system in human brain.

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References

  1. Wang, W., Tang, M., Zhang, H.F., et al.: Epidemic spreading on complex networks with general degree and weight distributions. Phys. Rev. E 90(4), 042803 (2014)

    Article  Google Scholar 

  2. Zhang, J.Q., Wang, C.D., Wang, M.S., et al.: Firing patterns transition induced by system size in coupled Hindmarsh Rose neural system. Neurocomputing 74(17), 2961–2966 (2011)

    Article  Google Scholar 

  3. Chen, H.S., Li, G.F., Zhang, H.F., et al.: Optimal allocation of resources for suppressing epidemic spreading on networks. Phys. Rev. E 96(1), 012321 (2017)

    Article  Google Scholar 

  4. Zhang, J.Q., Huang, S.F., Pang, S.T., et al.: Optimizing calculations of coupling matrix in Hindmarsh Rose neural network. Nonlinear Dyn. 84(3), 1303–1310 (2016)

    Article  MathSciNet  Google Scholar 

  5. Mookherjee, R., Hobbs, B.F., Friesz, T.L., et al.: Dynamic oligopolistic competition on an electric power network with ramping costs and joint sales constraints. J. Ind. Manag. Optim 4(3), 425–452 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jiang, P.Q., Wang, B.H., Bu, S.L.: Hyperchaotic synchronization in deterministic small-world dynamical networks. Int. J. Mod. Phys. B 18(17n19), 2674–2679 (2004)

    Article  MATH  Google Scholar 

  7. Liu, Q.P., Wang, X.F.: Social learning in networks with time-varying topologies. Asian J. Control 16(5), 1342–1349 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Wang, W.X., Wang, B.H., Yin, C.Y., et al.: Traffic dynamics based on local routing protocol on a scale-free network. Phys. Rev. E 73(2), 026111 (2006)

    Article  Google Scholar 

  9. Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys 74(1), 47–98 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45(2), 167–256 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Shepherd, G.M.: The synaptic organization of the brain. Q. Rev. Biol. 3, 145–162 (2004)

    Google Scholar 

  12. Xu, M.M., Zhou, J., Lu, J.A., et al.: Synchronizability of two-layer networks. Eur. Phys. J. B 88(9), 240 (2015)

    Article  Google Scholar 

  13. Sun, X.J., Lei, J.Z., Perc, M., et al.: Burst synchronization transitions in a neuronal network of subnetworks. Chaos 21(1), 054101 (2011)

    Article  Google Scholar 

  14. Ning, D., Wu, X.Q., Lu, J.A., et al.: Driving-based generalized synchronization in two-layer networks via pinning control. Chaos 25(11), 113104 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang, X., Hou, Z.Q., Yu, W.S., et al.: Online scale adaptive visual tracking based on multilayer convolutional features. IEEE Trans. Cybern. 99, 1–13 (2017)

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Lv, M., Wang, C.N., Ren, G.D., et al.: Model of electrical activity in a neuron under magnetic flow effect. Nonlinear Dyn. 85, 1479–1490 (2016)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Bao, B.C., Qian, H., Wang, J., et al.: Numerical analyses and experimental validations of coexisting multiple attractors in Hopfield neural network. Nonlinear Dyn. 90(4), 2359–2369 (2017)

    Article  MathSciNet  Google Scholar 

  21. Li, Q.D., Tang, S., Zeng, H.Z., et al.: On hyperchaos in a small memristive neural network. Nonlinear Dyn. 78(2), 1087–1099 (2014)

    Article  MATH  Google Scholar 

  22. Jo, S.H., Chang, T., Ebong, I., et al.: Nanoscale memristor device as synapse in neuromorphic systems. Nano Lett. 10, 1297–1301 (2010)

    Article  Google Scholar 

  23. Wu, F.Q., Wang, C.N., Jin, W.Y., et al.: Dynamical responses in a new neuron model subjected to electromagnetic induction and phase noise. Phys. A 469, 81–88 (2017)

    Article  MathSciNet  Google Scholar 

  24. Ma, J., Wu, F.Q., Wang, C.N.: Synchronization behaviors of coupled neurons under electromagnetic radiation. Int. J. Modem Phys. B. 31, 1650251 (2017)

    Article  MathSciNet  Google Scholar 

  25. Ma, J., Lv, M., Zhou, P.: Phase synchronization between two neurons induced by coupling of electromagnetic field. Appl. Math. Comput. 307, 321–328 (2017)

    MathSciNet  Google Scholar 

  26. Xu, Y., Jia, Y., Ma, J., et al.: Synchronization between neurons coupled by memristor. Chaos Soliton Fract. 104, 435–442 (2017)

    Article  Google Scholar 

  27. Kuramoto, Y., Battogtokh, D.: Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Physics 4, 385 (2013)

    Google Scholar 

  28. Abrams, D.M., Strogatz, S.H.: Chimera states for coupled oscillators. Phys. Rev. Lett. 93(17), 174102 (2004)

    Article  Google Scholar 

  29. Rattenborg, N.C., Amlaner, C.J., Lima, S.L.: Behavioral, neurophysiological and evolutionary perspectives on unihemispheric sleep. Neurosci. Biobehav. Rev. 24(8), 817–842 (2000)

    Article  Google Scholar 

  30. Wang, H.Q., Li, X.: Synchronization and chimera states of frequency-weighted Kuramoto-oscillator networks. Phys. Rev. E 83(6), 066214 (2011)

    Article  Google Scholar 

  31. Zhu, Y., Zheng, Z.G., Yang, J.Z.: Chimera states on complex networks. Phys. Rev. E 89(2), 022914 (2014)

    Article  Google Scholar 

  32. Kundu, S., Majhi, S., Bera, B.K., et al.: Chimera states in two-dimensional networks of locally coupled oscillators. Phys. Rev. E 97(2), 022201 (2018)

    Article  MathSciNet  Google Scholar 

  33. Omelchenko, I., Provata, A., Hizanidis, J., et al.: Robustness of chimera states for coupled FitzHugh–Nagumo oscillators. Phys. Rev. E 91(2), 022917 (2015)

    Article  MathSciNet  Google Scholar 

  34. Glaze, T.A., Lewis, S., Bahar, S.: Chimera states in a Hodgkin–Huxley model of thermally sensitive neurons. Chaos 26(8), 083119 (2016)

    Article  MathSciNet  Google Scholar 

  35. Bera, B.K., Ghosh, D., Lakshmanan, M.: Chimera states in bursting neurons. Phys. Rev. E 93(1), 012205 (2016)

    Article  MathSciNet  Google Scholar 

  36. Yeldesbay, A., Pikovsky, A., Rosenblum, M.: Chimeralike states in an ensemble of globally coupled oscillators. Phys. Rev. Lett. 112(14), 144103 (2014)

    Article  Google Scholar 

  37. Ratas, I., Pyragas, K.: Symmetry breaking in two interacting populations of quadratic integrate-and-fire neurons. Phys. Rev. E 96(4), 042212 (2017)

    Article  MathSciNet  Google Scholar 

  38. Bera, B.K., Ghosh, D.: Chimera states in purely local delay-coupled oscillators. Phys. Rev. E 93(5), 052223 (2016)

    Article  MathSciNet  Google Scholar 

  39. Gopal, R., Chandrasekar, V.K., Venkatesan, A., et al.: Observation and characterization of chimera states in coupled dynamical systems with nonlocal coupling. Phys. Rev. E 89(5), 052914 (2014)

    Article  Google Scholar 

  40. Maistrenko, Y., Sudakov, O., Osiv, O., et al.: Chimera states in three dimensions. New J. Phys. 17(7), 073037 (2015)

    Article  Google Scholar 

  41. Tian, C.H., Bi, H.J., Zhang, X.Y., et al.: Asymmetric couplings enhance the transition from chimera state to synchronization. Phys. Rev. E 96(5), 052209 (2017)

    Article  Google Scholar 

  42. Feng, Y.E., Li, H.H.: The dependence of chimera states on initial conditions. Chin. Phys. Lett. 32(6), 060502 (2015)

    Article  Google Scholar 

  43. Kalle, P., Sawicki, J., Zakharova, A., et al.: Chimera states and the interplay between initial conditions and non-local coupling. Chaos 27(3), 033110 (2017)

    Article  Google Scholar 

  44. Yao, N., Zheng, Z.G.: Chimera states in spatiotemporal systems: theory and applications. Int. J. Modern Phys. B 30(7), 1630002 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  45. Gambuzza, L.V., Frasca, M.: Pinning control of chimera states. Phys. Rev. E 94(2), 022306 (2016)

    Article  MathSciNet  Google Scholar 

  46. Sieber, J., Oleh, E., Wolfrum, M.: Controlling unstable chaos: stabilizing chimera states by feedback. Phys. Rev. Lett. 112(5), 054102 (2014)

    Article  Google Scholar 

  47. Isele, T., Hizanidis, J., Provata, A., et al.: Controlling chimera states: the influence of excitable units. Phys. Rev. E 93(2), 022217 (2016)

    Article  Google Scholar 

  48. Shepelev, I.A., Bukh, A.V., Strelkova, G.I., et al.: Chimera states in ensembles of bistable elements with regular and chaotic dynamics. Nonlinear Dyn. 90(4), 2317–2330 (2017)

    Article  MathSciNet  Google Scholar 

  49. Kapitaniak, T., Kuzma, P., Wojewoda, J., et al.: Imperfect chimera states for coupled pendula. Sci. Rep. 4, 6379 (2014)

    Article  Google Scholar 

  50. Xie, J., Knobloch, E., Kao, H.C.: Multicluster and traveling chimera states in nonlocal phase-coupled oscillators. Phys. Rev. E 90(2), 022919 (2014)

    Article  Google Scholar 

  51. Bera, B.K., Ghosh, D., Banerjee, T.: Imperfect traveling chimera states induced by local synaptic gradient coupling. Phys. Rev. E 94(1), 012215 (2016)

    Article  Google Scholar 

  52. Sethia, G.C., Sen, A., Johnston, G.L.: Amplitude-mediated chimera states. Phys. Rev. E 88(4), 042917 (2013)

    Article  Google Scholar 

  53. Li, B.W., Dierckx, H.: Spiral wave chimeras in locally coupled oscillator systems. Phys. Rev. E 93(2), 020202 (2016)

    Article  Google Scholar 

  54. Maksimenko, V.A., Goremyko, M.V., Makarov, V.V., et al.: Excitation and suppression of chimeric states in the multilayer network of oscillators with nonlocal coupling. Bull. Russ. Acad. Sci. Phys. 81(1), 110–113 (2017)

    Article  Google Scholar 

  55. Majhi, S., Perc, M., Ghosh, D.: Chimera states in uncoupled neurons induced by a multilayer structure. Sci. Rep. 6, 39033 (2016)

    Article  Google Scholar 

  56. Kasatkin, D.V., Yanchuk, S., Schll, E., et al.: Self-organized emergence of multilayer structure and chimera states in dynamical networks with adaptive couplings. Phys. Rev. E 96(6), 062211 (2017)

    Article  Google Scholar 

  57. Majhi, S., Perc, M., Ghosh, D.: Chimera states in a multilayer network of coupled and uncoupled neurons. Chaos 27(7), 073109 (2017)

    Article  MathSciNet  Google Scholar 

  58. Hizanidis, J., Kouvaris, N.E., Zamora-López, G., et al.: Chimera-like states in modular neural networks. Sci. Rep. 6, 19845 (2016)

    Article  Google Scholar 

  59. Hindmarsh, J.L., Rose, R.M.: A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. 221, 87–102 (1984)

    Article  Google Scholar 

  60. Bao, B.C., Wang, N., Xu, Q.: A simple third-order memristive band pass filter chaotic circuit. IEEE Trans. Circuits Syst. II Express Briefs 99, 1–1 (2016)

    Google Scholar 

  61. Xu, Q., Lin, Y., Bao, B.C., Chen, M.: Multiple attractors in a non-ideal active voltage-controlled memristor based Chua circuit. Chaos, Solitons Fractals 83, 186–200 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  62. Tang, J., Zhang, J., Ma, J., et al.: Astrocyte calcium wave induces seizure-like behavior in neuron network. Sci. China Technol. Sci. 60(7), 1–8 (2017)

    Article  Google Scholar 

  63. Xu, X.: Complicated dynamics of a ring neural network with time delays. J. Phys. A-Math. Theor. 41(41), 035102 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  64. Xu, F., Zhang, J.Q., Fang, T., et al.: Synchronous dynamics in neural system coupled with memristive synapse. Nonlinear Dyn. 92(3), 1395–1402 (2018)

    Article  Google Scholar 

  65. Rakshit, S., Majhi, S., Bera, B.K., et al.: Time-varying multiplex network: intralayer and interlayer synchronization. Phys. Rev. E 96(6), 062308 (2017)

    Article  Google Scholar 

  66. Leyva, I., Sevilla-Escoboza, R., Sendiña-Nadal, I., et al.: Inter-layer synchronization in non-identical multi-layer networks. Sci. Rep. 7, 45475 (2017)

    Article  Google Scholar 

  67. Sun, X.J., Li, G.F.: Synchronization transitions induced by partial time delay in a excitatory inhibitory coupled neuronal network. Nonlinear Dyn. 89(4), 1–12 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The project supported by the Natural Science Foundation of Anhui Province, China (No. 1508085MA15), the Key project of cultivation of leading talents in Universities of Anhui Provence (No. gxbjZD2016014), the Innovation and practice research project of graduate students of Anhui Normal University, China (No. 2017cxsj045), and the project of Academic and technical leaders candidate of Anhui Province (2017H117).

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Authors and Affiliations

  1. College of Physics and Electronic Information, Anhui Normal University, Wuhu, 241000, China

    Fei Xu, Jiqian Zhang, Shoufang Huang & Tingting Fang

  2. School of Mathematics and Computation Science, Anqing Normal University, Anqing, 246011, China

    Meng Jin

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  1. Fei Xu
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Xu, F., Zhang, J., Jin, M. et al. Chimera states and synchronization behavior in multilayer memristive neural networks. Nonlinear Dyn 94, 775–783 (2018). https://doi.org/10.1007/s11071-018-4393-9

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