Nonlinear Dynamics

, Volume 87, Issue 3, pp 1879–1899 | Cite as

Nonlinear feedback coupling in Hindmarsh–Rose neurons

Original Paper


We analyse the model of neurons described by Hindmarsh–Rose (H–R) system with various coupling schemes. We have examined the scope of synchronization, anti-phase synchronization and amplitude death for linear indirect synaptic coupling of the H–R neurons. The work is extended to coupling of the form nonlinear cubic feedback. The coupling between two neurons using memristor is also examined. A memristor is now identified as the fourth fundamental circuit element which can be considered as an electrical synapse. Mutual coupling of H–R systems using cubic flux-controlled memristor exhibits the properties of bursting and amplitude death. With unidirectional coupling one neuron exhibits tonic spiking or bursting while the other neuron shows amplitude death. Mutual coupling with quadratic flux-controlled memristor model shows the possibilities of synchronization, oscillation death and other interesting dynamics like near-death rare spikes. Exponential flux-controlled memristor coupling in H–R neuron presents synchronization and oscillation death. We have examined the stability of different coupled systems and the Lyapunov exponent plots. It is shown that among different memristor couplings in H–R neurons, cubic flux-controlled memristor has got highest Lyapunov exponent. The present work summarizes the simulation results of different coupling schemes in H–R neurons which show many interesting dynamical characteristics of coupled neuron cells.


Chaos Amplitude death Oscillation death Indirect synaptic coupling Nonlinear feedback Memristor coupling Stability analysis Lyapunov exponent 



We thank the referees for their constructive comments and suggestions. The authors would like to acknowledge the valuable suggestions from Mineeja K.K., Physics Department, St. Teresa’s College, Ernakulam.


  1. 1.
    Thornburg, K.S., Moller Jr., M., Roy, R., Carr, T.W., Li, R.D., Erneux, T.: Chaos and coherence in coupled lasers. Phys. Rev. E 55, 38–3865 (1997)CrossRefGoogle Scholar
  2. 2.
    Kiss, I.Z., Gaspar, V., Hudson, J.L.: Experiments on synchronization and control of chaos on coupled electrochemical oscillators. J. Phys. Chem. B 104(31), 7554–7560 (2000)CrossRefGoogle Scholar
  3. 3.
    Harrison, M.A., Lai, Y.C., Holt, R.D.: Dynamical mechanism for coexistence of dispersing species without trade-offs in spatially extended ecological systems. Phys. Rev. E 63(051905), 519051–519055 (2001)Google Scholar
  4. 4.
    Glass, L.: Synchronization and rhythmic processes in physiology. Nature 410(6825), 277–284 (2001)CrossRefGoogle Scholar
  5. 5.
    Prasad, A., Dhamala, M., Adhikari, B.M., Ramaswamy, R.: Amplitude death in nonlinear oscillators with nonlinear coupling. Phys. Rev. E 81, 027201 (2010)CrossRefGoogle Scholar
  6. 6.
    Koseska, A., Volkov, E., Kurths, J.: Oscillation quenching mechanisms: amplitude vs. oscillation death. Phys. Rep. 531, 173–199 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ahn, S., Rubchinsky, L.L.: Short desynchronization episodes prevail in synchronous dynamics of human brain rhythms. Chaos 23, 1–7 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Arumugam, E.M.E., Spano, M.L.: A chimeric path to neuronal synchronization. Chaos Interdiscip. J. nonlinear Sci. 25, 013107 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Resmi, V., Ambika, G., Amritkar, R.E.: General mechanism for amplitude death in coupled systems. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 84, 046212 (2011)Google Scholar
  10. 10.
    Sharma, A., Sharma, P.R., Shrimali, M.D.: Amplitude death in nonlinear oscillators with indirect coupling. Phys. Lett. Sect. A Gen. Atom. Solid State Phys. 376, 1562–1566 (2012)MATHGoogle Scholar
  11. 11.
    Saha, D.C.: On the synchronization of synaptically coupled nonlinear oscillators: theory and experiment. Annu. Rev. Chaos Theory Bifurc. Dyn. Syst. 6, 1–29 (2016)Google Scholar
  12. 12.
    Koch, A.J., et al.: Biological pattern formation: from basic mechanisms to complex structures. Rev. Mod. Phys. 66, 1481–1507 (1994)CrossRefGoogle Scholar
  13. 13.
    Arbib, M.A.: Handbook of Brain Theory and Neural Network. MIT Press, Cambridge (2002)Google Scholar
  14. 14.
    Zou, W., Senthilkumar, D.V., Nagao, R., Kiss, I.Z., Tang, Y., Koseska, A., Duan, J., Kurths, J.: Restoration of rhythmicity in diffusively coupled dynamical networks. Nat. Commun. 6, 7709 (2015)CrossRefGoogle Scholar
  15. 15.
    Yu, H., Peng, J.: Chaotic synchronization and control in nonlinear-coupled Hindmarsh-Rose neural systems. Chaos Solitons Fractals 29, 342–348 (2006)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Volos, C.K., Kyprianidis, I.M., Stouboulos, I.N., Tlelo-Cuautle, E., Vaidyanathan, S.: Memristor: a new concept in synchronization of coupled neuromorphic circuits. J. Eng. Sci. Technol. Rev. 8, 157–173 (2015)Google Scholar
  17. 17.
    Matthews, P.C., Mirollo, R.E., Strogatz, S.H.: Complex time-delay systems: theory and applications. Phys. D 52 (1991)Google Scholar
  18. 18.
    Prasad, A.: Time-varying interaction leads to amplitude death in coupled nonlinear oscillators. Pramana J. Phys. 81, 407–415 (2013)CrossRefGoogle Scholar
  19. 19.
    Suresh, K., Sabarathinam, S., Thamilmaran, K., Kurths, J., Dana, S.K.: A common lag scenario in quenching of oscillation in coupled oscillators. Chaos Interdiscip. J. Nonlinear Sci. 26, 083104 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hens, C.R., Olusola, O.I., Pal, P., Dana, S.K.: Oscillation death in diffusively coupled oscillators by local repulsive link. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 88, 034902 (2013)Google Scholar
  21. 21.
    Wang, C., He, Y., Ma, J., Huang, L.: Parameters estimation, mixed synchronization, and antisynchronization in chaotic systems. Complexity 20, 64–73 (2014)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ma, J., Song, X., Jin, W., Wang, C.: Autapse-induced synchronization in a coupled neuronal network. Chaos Solitons Fractals 80, 31–38 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Song, X., Wang, C., Ma, J., Tang, J.: Transition of electric activity of neurons induced by chemical and electric autapses. Sci. China Technol. Sci. 58, 1007–1014 (2015)CrossRefGoogle Scholar
  24. 24.
    Ma, J., Tang, J.: A review for dynamics of collective behaviours of network of neurons. Sci. China Technol. Sci. 58, 2038–2045 (2015)CrossRefGoogle Scholar
  25. 25.
    Ma, J., Qin, H., Song, X., Chu, R.: Pattern selection in neuronal network driven by electric autapses with diversity in time delays. Int. J. Mod. Phys. B 29, 1450239 (2015)CrossRefGoogle Scholar
  26. 26.
    Song, Y., Wen, S.: Synchronization control of stochastic memristor-based neural networks with mixed delays. Neurocomputing 156, 121–128 (2015)CrossRefGoogle Scholar
  27. 27.
    Berdan, R., Vasilaki, E., Khiat, A., Indiveri, G., Serb, A., Prodromakis, T.: Emulating short-term synaptic dynamics with memristive devices. Sci. Rep. 6, 18639 (2016)CrossRefGoogle Scholar
  28. 28.
    Merrikh-Bayat, F., Bagheri-Shouraki, S.: Efficient neuro-fuzzy system and its memristor crossbar-based hardware implementation. 34 (2011). https://arxiv.orgabs/1103.1156
  29. 29.
    Wei, W., Min, Z.: Chaotic dynamics and its analysis of Hindmarsh-Rose neurons by Shil’nikov approach. Chin. Phys. B 24(8), 1–6 (2015)CrossRefGoogle Scholar
  30. 30.
    Hodgkin, A., Huxley, A.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Phisiol. (Lond.) 117, 500–544 (1952)CrossRefGoogle Scholar
  31. 31.
    Fitzhugh, R.: Mathematical models for excitation and propagation in nerve. Biological Engineering, vol. 5, pp. 1–85. Mc Graw (1969)Google Scholar
  32. 32.
    Morris, C., Lecar, H.: Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35(1), 193–213 (1981)CrossRefGoogle Scholar
  33. 33.
    Hindmarsh, J.L., Rose, R.M.: A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. Biol. 221, 87–102 (1984)CrossRefGoogle Scholar
  34. 34.
    Alias, L.A., Pai, M.S., George, M.M.V.: Near death experiences (NDE) of cardiac arrest survivors-a phenomenological study. Mater. Methods 3, 216–220 (2015)Google Scholar
  35. 35.
    Zeren, T., Özbek, M., Kutlu, N., Akilli, M.: Significance of using a nonlinear analysis technique, the Lyapunov exponent, on the understanding of the dynamics of the cardiorespiratory system in rats. Turk. J. Med. Sci. 46, 159–165 (2016)CrossRefGoogle Scholar
  36. 36.
    Terman, D., Kopell, N., Bose, A.: Dynamics of two mutually coupled slow inhibitory neurons. Phys. D Nonlinear Phenom. 117, 241–275 (1998)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Fang, X.: Chaotic synchronization of Hindmarsh–Rose neurons coupled by cubic nonlinear feedback. In: Advances in Cognitive Neurodynamics ICCN 2007, pp. 315–320 (2007)Google Scholar
  38. 38.
    Chua, L.: Memristive devices and systems. Proc. IEEE 64(2), 209–223 (1976)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Mazumder, P.: Memristors: devices, models and application. Proc. IEEE 100(6), 1911–1919 (2012)CrossRefGoogle Scholar
  40. 40.
    Stockwell, R.: English words: history and structure. Cambridge Univ Press, Cambridge (2001)CrossRefGoogle Scholar
  41. 41.
    Corinto, F.: Memristor synaptic dynamics’ influence on synchronous behaviour of two Hindmarsh–Rose neurons. In: 2011 International Joint Conference on Neural Networks (IJCNN), pp. 2403–2408. IEEE (2011)Google Scholar
  42. 42.
    Johnson, R.C.: Missing link’memristor created. EE Times. 04 (2008)Google Scholar
  43. 43.
    Abdurahman, A., Jiang, H., Teng, Z.: Finite-time synchronization for memristor-based neural networks with time-varying delays. Neural Netw. 69, 20–28 (2015)CrossRefGoogle Scholar
  44. 44.
    Wang, H.T., Chen, Y.: Firing dynamics of an autaptic neuron. Chin. Phys. B 24(12) (2015). doi: 10.1088/1674-1056/24/12/128709
  45. 45.
    Guo, Q.: Properties of quadratic flux-controlled and charge-controlled memristor. Adv. Eng. Res. 2352–5401 (2015). doi: 10.2991/ameii.15.2015.269
  46. 46.
    Lee, U., Borjigin, J.: Surge of neurophysiological coherence and connectivity in the dying brain. PNAS 110(35), 144432–144437 (2013)CrossRefGoogle Scholar
  47. 47.
    Klotz, I.: Brain waves surge moments before death. Chin. Phys. B 24(8), 118401 (2009)Google Scholar
  48. 48.
    Yong, E.: In: Dying brains, signs of heightened consciousness. PNAS (2013)Google Scholar
  49. 49.
    Wei, L.: Exponential flux-controlled memristor model and its floating emulator. Chin. Phys. B 24(11) (2015)Google Scholar
  50. 50.
    Friston, K.J.: Book review: brain function, nonlinear coupling, and neuronal transients. Neuroscientist 7, 406–418 (2001)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of PhysicsSt. Teresa’s collegeErnakulamIndia

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