The European Physical Journal Special Topics

, Volume 228, Issue 11, pp 2371–2379 | Cite as

Spiral wave in a two-layer neuronal network

  • Yu Feng
  • Abdul Jalil M. Khalaf
  • Fawaz E. Alsaadi
  • Tasawar Hayat
  • Viet-Thanh PhamEmail author
Regular Article
Part of the following topical collections:
  1. Diffusion Dynamics and Information Spreading in Multilayer Networks


Multi-layer networks are quite fundamental to study structural and functional properties of various biological systems. In this study, a two-layer neuronal network is considered and the interactions between the spiral pattern in one layer and a homogeneous state in the other layer, under the effect of inter-layer bidirectional connection are investigated. Spiral wave has been confirmed to play a significant role in many complex systems. In this regard, in laminar structured systems in particular, it is crucial to study the dynamics of the spiral wave affected by the inter-layer interactions. Here, for each layer (sub-network), a regular network with eight-neighbor connection is designed. For the local dynamics of each neuron, the magnetic Fitzhugh–Nagumo (FN) neuronal model is introduced. The results show that depending on the level of interactions between the two layers, four different types of collective electrical activity can occur. When the inter-layer connection is weak, the layer with spiral pattern does not change while the homogeneous state of the other layer is broken by a blurred spiral pattern. As the inter-layer connection is strengthened, the dynamics of the spiral wave changes significantly, leading to unstable spiral wave and spiral turbulence. However, by a further increase in the inter-layer coupling strength, the spiral wave does not exist at all.


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  1. 1.
    J. Ma, J. Tang, Sci. China Technol. Sci. 58, 2038 (2015)CrossRefGoogle Scholar
  2. 2.
    J. Ma, J. Tang, Nonlinear Dyn. 89, 1569 (2017)CrossRefGoogle Scholar
  3. 3.
    B.K. Bera, D. Ghosh, M. Lakshmanan, Phys. Rev. E 93, 012205 (2016)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    R. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    X. Sun, M. Perc, J. Kurths, Chaos: Interdiscip. J. Nonlinear Sci. 27, 053113 (2017)CrossRefGoogle Scholar
  6. 6.
    S. Rakshit, B.K. Bera, D. Ghosh, Phys. Rev. E 98, 032305 (2018)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    S. Rakshit, B.K. Bera, D. Ghosh, S. Sinha, Phys. Rev. E 97, 052304 (2018)ADSCrossRefGoogle Scholar
  8. 8.
    A.-L. Barabási, R. Albert, Science 286, 509 (1999)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    M.-B. Hu, R. Jiang, Y.-H. Wu, W.-X. Wang, Q.-S. Wu, Physica A 387, 4967 (2008)ADSCrossRefGoogle Scholar
  10. 10.
    Q. Liu, X. Wang, Asian J. Control 16, 1342 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    V. Berec, Chaos, Solitons Fractals 86, 75 (2016)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    K. Rajagopal, F. Parastesh, H. Azarnoush, B. Hatef, S. Jafari, V. Berec, Chaos: Interdiscip. J. Nonlinear Sci. 29, 043109 (2019)CrossRefGoogle Scholar
  13. 13.
    Z. Shahriari, F. Parastesh, M. Jalili, V. Berec, J. Ma, S. Jafari, Europhys. Lett. 125, 60001 (2019)ADSCrossRefGoogle Scholar
  14. 14.
    S. Kundu, S. Majhi, B.K. Bera, D. Ghosh, M. Lakshmanan, Phys. Rev. E 97, 022201 (2018)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    M. Kurant, P. Thiran, Phys. Rev. Lett. 96, 138701 (2006)ADSCrossRefGoogle Scholar
  16. 16.
    X.-Y. Luo, J. Chen, M. Li, M.-B. Hu, Int. J. Theor. Appl. Finance (IJTAF) 29, 1 (2018)Google Scholar
  17. 17.
    S. Majhi, M. Perc, D. Ghosh, Sci. Rep. 6, 39033 (2016)ADSCrossRefGoogle Scholar
  18. 18.
    S. Rakshit, S. Majhi, B.K. Bera, S. Sinha, D. Ghosh, Phys. Rev. E 96, 062308 (2017)ADSCrossRefGoogle Scholar
  19. 19.
    S. Majhi, T. Kapitaniak, D. Ghosh, Chaos: Interdiscip. J. Nonlinear Sci. 29, 013108 (2019)CrossRefGoogle Scholar
  20. 20.
    S. Zhou, S. Xu, L. Wang, Z. Liu, G. Chen, X. Wang, Phys. Rev. E 98, 012303 (2018)ADSCrossRefGoogle Scholar
  21. 21.
    M. Kurant, P. Thiran, P. Hagmann, Phys. Rev. E 76, 026103 (2007)ADSCrossRefGoogle Scholar
  22. 22.
    K. Ding, Z. Rostami, S. Jafari, B. Hatef, Complexity 2018, 6427870 (2018)Google Scholar
  23. 23.
    E.R. Kandel, J.H. Schwartz, T.M. Jessell, S.A. Siegelbaum, A.J. Hudspeth, S. Mack, Principles of neural science (McGraw-hill, New York, 2000)Google Scholar
  24. 24.
    R. Chen, C. Wen, R. Fu, J. Li, J. Wu, PLoS One 13, e0208029 (2018)CrossRefGoogle Scholar
  25. 25.
    M. De Domenico, A. Solé-Ribalta, E. Cozzo, M. Kivelä, Y. Moreno, M.A. Porter, S. Gòmez, A. Arenas, Phys. Rev. X 3, 041022 (2013)Google Scholar
  26. 26.
    C. Wang, J. Ma, Int. J. Mod. Phys. B 32, 1830003 (2018)ADSCrossRefGoogle Scholar
  27. 27.
    S. Majhi, M. Perc, D. Ghosh, Chaos: Interdiscip. J. Nonlinear Sci. 27, 073109 (2017)CrossRefGoogle Scholar
  28. 28.
    S. Majhi, D. Ghosh, Chaos: Interdiscip. J. Nonlinear Sci. 28, 083113 (2018)CrossRefGoogle Scholar
  29. 29.
    S. Majhi, B.K. Bera, D. Ghosh, M. Perc, Phys. Life Rev. 28, 100 (2019)ADSCrossRefGoogle Scholar
  30. 30.
    A. Bewersdorff, P. Borckmans, S.C. Müller, Chemical pattern formation, in Fluid sciences and materials science in space (Springer, 1987), pp. 257–89Google Scholar
  31. 31.
    A.R. Nayak, A. Panfilov, R. Pandit, Phys. Rev. E 95, 022405 (2017)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Z. Rostami, S. Jafari, Cognit. Neurodyn. 12, 235 (2018)CrossRefGoogle Scholar
  33. 33.
    H. Dierckx, H. Verschelde, A.V. Panfilov, Chaos: Interdiscip. J. Nonlinear Sci. 27, 093912 (2017)CrossRefGoogle Scholar
  34. 34.
    Z. Rostami, K. Rajagopal, A.J.M. Khalaf, S. Jafari, M. Perc, M. Slavinec, Physica A 509, 1162 (2018)ADSCrossRefGoogle Scholar
  35. 35.
    Y. Deng, B.Y. Liu, T. Wu, Y.Y. Shangguan, J. Ma, J. Tang, Europhys. Lett. 119, 58002 (2017)ADSCrossRefGoogle Scholar
  36. 36.
    Z. Rostami, S. Jafari, M. Perc, M. Slavinec, Nonlinear Dyn. 94, 679 (2018)CrossRefGoogle Scholar
  37. 37.
    D. Olmos-Liceaga, H. Ocejo-Monge, Chaos, Solitons Fractals 99, 162 (2017)ADSCrossRefGoogle Scholar
  38. 38.
    S. Kundu, S. Majhi, P. Muruganandam, D. Ghosh, Eur. Phys. J. Special Topics 227, 983 (2018)ADSCrossRefGoogle Scholar
  39. 39.
    F. Wu, C. Wang, Y. Xu, J. Ma, Sci. Rep. 6, 28 (2016)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Yu Feng
    • 1
  • Abdul Jalil M. Khalaf
    • 2
  • Fawaz E. Alsaadi
    • 3
  • Tasawar Hayat
    • 4
    • 5
  • Viet-Thanh Pham
    • 6
    Email author
  1. 1.Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal UniversityYulin, GuangxiP.R. China
  2. 2.Ministry of Higher Education and Scientific ResearchBaghdadIraq
  3. 3.Department of Information TechnologyFaculty of Computing and IT, King Abdulaziz UniversityJeddahSaudi Arabia
  4. 4.Department of MathematicsQuaid-I-Azam University 45320IslamabadPakistan
  5. 5.NAAM Research Group, King Abdulaziz UniversityJeddahSaudi Arabia
  6. 6.Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang UniversityHo Chi Minh CityVietnam

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