SpringerLink
Log in

In-phase and anti-phase bursting dynamics and synchronisation scenario in neural network by varying coupling phase

  • Research
  • Published:
Journal of Biological Physics Aims and scope Submit manuscript

A Correction to this article was published on 14 June 2023

This article has been updated

Abstract

We have analysed the synchronisation scenario and the rich spatiotemporal patterns in the network of Hindmarsh-Rose neurons under the influence of self, mixed and cross coupling of state variables which are realised by varying coupling phase. We have introduced a coupling matrix in the model to vary coupling phase. The excitatory and inhibitory couplings in the membrane potential induce in-phase and anti-phase bursting dynamics, respectively, in the two coupled system. When the off-diagonal elements of the matrix are zero, the system shows self coupling of the three variables, which helps to attain synchrony. The off-diagonal elements give cross interactions between the variables, which reduces synchrony. The stability of the synchrony attained is analysed using Lyapunov function approach. In our study, we found that self coupling in three variables is sufficient to induce chimera states in non-local coupling. The strength of incoherence and discontinuity measure validates the existence of chimera and multichimera states. The inhibitor self coupling in local interaction induces interesting patterns like Mixed Oscillatory State and clusters. The results may help in understanding the spatiotemporal communications of the brain, within the limitations of the size of the network analysed in this study.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Availability of data and material

The data that support the findings were generated from numerical simulations by the software MATLAB.

Code availability

Code for data simulation was developed by ourselves in MATLAB and will be made available on request.

Change history

References

  1. Swanson, L.W.: Brain Architecture: Understanding the Basic Plan. Oxford University Press (2012)

    Google Scholar 

  2. Bem, T., Le Feuvre, Y., Rinzel, J., Meyrand, P.: Electrical coupling induces bistability of rhythms in networks of inhibitory spiking neurons. Eur. J. Neurosci. 22(10), 2661–2668 (2005)

    Article  Google Scholar 

  3. Bem, T., Rinzel, J.: Short duty cycle destabilizes a half-center oscillator, but gap junctions can restabilize the anti-phase pattern. J. Neurophysiol. 91(2), 693–703 (2004)

    Article  Google Scholar 

  4. Njitacke, Z.T., Doubla, I.S., Kengne, J., Cheukem, A.: Coexistence of firing patterns and its control in two neurons coupled through an asymmetric electrical synapse. Chaos (Woodbury, NY) 30(2), 023101 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  5. Martin, E.A., Lasseigne, A.M., Miller, A.C.: Understanding the molecular and cell biological mechanisms of electrical synapse formation. Front. Neuroanat. 14, 12 (2020)

    Article  Google Scholar 

  6. Lee, E., Terman, D.: Stable antiphase oscillations in a network of electrically coupled model neurons. SIAM J. Appl. Dyn. Syst. 12(1), 1–27 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bashkirtseva, I., Ryashko, L., Pisarchik, A.N.: Stochastic transitions between in-phase and anti-phase synchronization in coupled map-based neural oscillators. Commun. Nonlinear Sci. Numer. Simul. 95, 105611 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  8. Protachevicz, P.R., Hansen, M., Iarosz, K.C., Caldas, I.L., Batista, A.M., Kurths, J.: Emergence of neuronal synchronisation in coupled brain areas. Front. Comput. Neurosci. 15, 35 (2021)

    Article  Google Scholar 

  9. Jia, B., Wu, Y., He, D., Guo, B., Xue, L.: Dynamics of transitions from anti-phase to multiple in-phase synchronizations in inhibitory coupled bursting neurons. Nonlinear Dyn. 93(3), 1599–1618 (2018)

    Article  Google Scholar 

  10. Korotkov, A.G., Kazakov, A.O., Levanova, T.A., Osipov, G.V.: The dynamics of ensemble of neuron-like elements with excitatory couplings. Commun. Nonlinear Sci. Numer. Simul. 71, 38–49 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Usha, K., Subha, P.: Hindmarsh-Rose neuron model with memristors. Biosystems 178, 1–9 (2019)

    Article  MATH  Google Scholar 

  12. Masoliver, M., Malik, N., Scholl, E., Zakharova, A.: Coherence resonance in a network of FitzHugh-Nagumo systems: interplay of noise, time-delay, and topology. Chaos: An Interdisciplinary Journal of Nonlinear Science 27(10), 101102 (2017)

  13. Boaretto, B., Budzinski, R., Prado, T., Kurths, J., Lopes, S.: Neuron dynamics variability and anomalous phase synchronization of neural networks. Chaos: An Interdisciplinary Journal of Nonlinear Science 28(10), 106304 (2018)

  14. Tang, J., Ma, J., Yi, M., **a, H., Yang, X.: Delay and diversity-induced synchronization transitions in a small-world neuronal network. Phys. Rev. E 83(4), 046207 (2011)

    Article  ADS  Google Scholar 

  15. Pakdaman, K., Mestivier, D.: Noise induced synchronization in a neuronal oscillator. Physica D 192(1–2), 123–137 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Colgin, L.L.: Rhythms of the hippocampal network. Nat. Rev. Neurosci. 17(4), 239–249 (2016)

    Article  Google Scholar 

  17. Lee, E., Terman, D.: Stability of antiphase oscillations in a network of inhibitory neurons. SIAM J. Appl. Dyn. Syst. 14(1), 448–480 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Vicente, R., Gollo, L.L., Mirasso, C.R., Fischer, I., Pipa, G.: Dynamical relaying can yield zero time lag neuronal synchrony despite long conduction delays. Proc. Natl. Acad. Sci. U.S.A. 105(44), 17157–17162 (2008)

    Article  ADS  Google Scholar 

  19. Lewis, C.M., Baldassarre, A., Committeri, G., Romani, G.L., Corbetta, M.: Learning sculpts the spontaneous activity of the resting human brain. Proc. Natl. Acad. Sci. U.S.A. 106(41), 17558–17563 (2009)

    Article  ADS  Google Scholar 

  20. Shmueli, K., van Gelderen, P., de Zwart, J.A., Horovitz, S.G., Fukunaga, M., Jansma, J.M., Duyn, J.H.: Low-frequency fluctuations in the cardiac rate as a source of variance in the resting-state fMRI BOLD signal. Neuroimage 38(2), 306–320 (2007)

    Article  Google Scholar 

  21. Corbetta, M., Shulman, G.L.: Control of goal-directed and stimulus driven attention in the brain. Nat. Rev. Neurosci. 3(3), 201–215 (2002)

    Article  Google Scholar 

  22. Simpson, J.R., Snyder, A.Z., Gusnard, D.A., Raichle, M.E.: Emotion induced changes in human medial prefrontal cortex: I. during cognitive task performance. Proc. Natl. Acad. Sci. U.S.A. 98(2), 683–687 (2001)

  23. Mantini, D., Perrucci, M.G., Del Gratta, C., Romani, G.L., Corbetta, M.: Electrophysiological signatures of resting state networks in the human brain. Proc. Natl. Acad. Sci. U.S.A. 104(32), 13170–13175 (2007)

    Article  ADS  Google Scholar 

  24. Horovitz, S.G., Braun, A.R., Carr, W.S., Picchioni, D., Balkin, T.J., Fukunaga, M., Duyn, J.H.: Decoupling of the brain’s default mode network during deep sleep. Proc. Natl. Acad. Sci. U.S.A. 106(27), 11376–11381 (2009)

    Article  ADS  Google Scholar 

  25. Cymbalyuk, G.S., Nikolaev, E., Borisyuk, R.: In-phase and antiphase self-oscillations in a model of two electrically coupled pacemakers. Biol. Cybern. 71, 153–160 (1994)

    Article  MATH  Google Scholar 

  26. Merrison-Hort, R., Borisyuk, R.: The emergence of two anti-phase oscillatory neural populations in a computational model of the parkinsonian globus pallidus. Front. Comput. Neurosci. 7, 173 (2013)

    Article  Google Scholar 

  27. Li, D., Zhou, C.: Organization of anti-phase synchronization pattern in neural networks: what are the key factors? Front. Syst. Neurosci. 5, 100 (2011)

    Article  Google Scholar 

  28. Koulierakis, I., Verganelakis, D.A., Omelchenko, I., Zakharova, A., Scholl, E., Provata, A.: Structural anomalies in brain networks induce dynamical pacemaker effects. Chaos: An Interdisciplinary Journal of Nonlinear Science 30(11), 113137 (2020)

  29. Wolfrum, M., Omel’chenko, E.: Chimera states are chaotic transients. Phys. Rev. E 84(1), 015204 (2011)

    Article  ADS  Google Scholar 

  30. Majhi, S., Bera, B.K., Ghosh, D., Perc, M.: Chimera states in neuronal networks: A review. Phys. Life Rev. 28, 100–121 (2019)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  32. Simo, G.R., Louodop, P., Ghosh, D., Njougouo, T., Tchitnga, R., Cerdeira, H.A.: Traveling chimera patterns in a two-dimensional neuronal network. Phys. Lett. A 127519 (2021)

  33. Dudkowski, D., Czolczynski, K., Kapitaniak, T.: Traveling chimera states for coupled pendula. Nonlinear Dyn. 95(3), 1859–1866 (2019)

    Article  MATH  Google Scholar 

  34. Majhi, S., Ghosh, D.: Alternating chimeras in networks of ephaptically coupled bursting neurons. Chaos: An Interdisciplinary Journal of Nonlinear Science 28(8), 083113 (2018)

  35. Zhang, Y., Nicolaou, Z.G., Hart, J.D., Roy, R., Motter, A.E.: Critical switching in globally attractive chimeras. Phys. Rev. X 10(1), 011044 (2020)

    Article  ADS  Google Scholar 

  36. Usha, K., Subha, P., Nayak, C.R.: The route to synchrony via drum head mode and mixed oscillatory state in star coupled Hindmarsh-Rose neural network. Chaos, Solitons Fractals 108, 25–31 (2018)

    Article  ADS  MATH  Google Scholar 

  37. Bandyopadhyay, B., Khatun, T., Dutta, P.S., Banerjee, T.: Symmetry breaking by power-law coupling. Chaos, Solitons Fractals 139, 110289 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  38. Remi, T., Subha, P., Usha, K.: Collective dynamics of neural network with distance dependent field coupling. Commun. Nonlinear Sci. Numer. Simul. 110, 106390 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zakharova, A., Kapeller, M., Scholl, E.: Chimera death: Symmetry breaking in dynamical networks. Phys. Rev. Lett. 112(15), 154101 (2014)

    Article  ADS  Google Scholar 

  40. Wang, Z., Xu, Y., Li, Y., Kapitaniak, T., Kurths, J.: Chimera states in coupled Hindmarsh-Rose neurons with \(\alpha\)-stable noise. Chaos Solitons Fractals 148, 110976 (2021)

  41. Kundu, S., Bera, B.K., Ghosh, D., Lakshmanan, M.: Chimera patterns in three-dimensional locally coupled systems. Phys. Rev. E 99(2), 022204 (2019)

    Article  ADS  Google Scholar 

  42. Zhang, Y., Motter, A.E.: Mechanism for strong chimeras. Phys. Rev. Lett. 126(9), 094101 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  43. Asllani, M., Siebert, B.A., Arenas, A., Gleeson, J.P.: Symmetry-breaking mechanism for the formation of cluster chimera patterns. Chaos: An Interdisciplinary Journal of Nonlinear Science 32(1), 013107 (2022)

  44. Shanahan, M.: Metastable chimera states in community-structured oscillator networks. Chaos: An Interdisciplinary Journal of Nonlinear Science 20(1), 013108 (2010)

  45. Budzinski, R.C., Nguyen, T.T., Joan, J., Minac, J., Sejnowski, T.J., Muller, L.E.: Geometry unites synchrony, chimeras, and waves in nonlinear oscillator networks. Chaos: An Interdisciplinary Journal of Nonlinear Science 32(3), 031104 (2022)

  46. Omelchenko, I., Omel’chenko, E., Hovel, P., Scholl, E.: When nonlocal coupling between oscillators becomes stronger: patched synchrony or multichimera states. Phys. Rev. Lett. 110(22), 224101 (2013)

    Article  ADS  Google Scholar 

  47. Wang, Z., Liu, Z.: A brief review of chimera state in empirical brain networks. Front. Physiol. 11, 724 (2020)

    Article  Google Scholar 

  48. Pinto, R.D., Varona, P., Volkovskii, A., Szucs, A., Abarbanel, H.D., Rabinovich, M.I.: Synchronous behavior of two coupled electronic neurons. Phys. Rev. E 62(2), 2644 (2000)

    Article  ADS  Google Scholar 

  49. Erichsen, R., Jr, Mainieri, M., Brunnet, L.: Periodicity and chaos in electrically coupled Hindmarsh-Rose neurons. Phys. Rev. E 74(6), 061906 (2006)

    Article  ADS  Google Scholar 

  50. Pecora, L.M., Carroll, T.L.: Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80(10), 2109 (1998)

    Article  ADS  Google Scholar 

  51. Krasovskii, N.N.: Stability of Motion. Stanford University Press (1963)

  52. 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 

  53. Hussain, I., Jafari, S., Ghosh, D., Perc, M.: Synchronization and chimeras in a network of photosensitive FitzHugh-Nagumo neurons. Nonlinear Dynamics, 1–11 (2021)

  54. Zhou, P., Hu, X., Zhu, Z., Ma, J.: What is the most suitable Lyapunov function? Chaos, Solitons Fractals 150, 111154 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  55. Joshi, S.K.: Synchronization of coupled Hindmarsh-Rose neuronal dynamics: Analysis and experiments. Express Briefs, IEEE Transactions on Circuits and Systems II (2021)

    Google Scholar 

  56. Hindmarsh, J.L., Rose, R.: A model of neuronal bursting using three coupled first order differential equations. Proceedings of the Royal society of London. Series B. Biological sciences 221(1222), 87-102 (1984)

  57. Usha, K., Subha, P.: Star-coupled Hindmarsh-Rose neural network with chemical synapses. Int. J. Mod. Phys. C 29(03), 1850023 (2018)

    Article  ADS  Google Scholar 

  58. Usha, K., Subha, P.: Energy feedback and synchronous dynamics of Hindmarsh-Rose neuron model with memristor. Chin. Phys. B 28(2), 020502 (2019)

    Article  ADS  Google Scholar 

  59. Remi, T., Subha, P., Usha, K.: Controlling phase synchrony in the mean field coupled Hindmarsh-Rose neurons. Int. J. Mod. Phys. C 33(05), 2250058 (2022)

    Article  ADS  MathSciNet  Google Scholar 

  60. Buric, N., Todorovic, K., Vasovic, N.: Synchronization of bursting neurons with delayed chemical synapses. Phys. Rev. E 78(3), 036211 (2008)

    Article  ADS  Google Scholar 

  61. Usha, K., Subha, P.: Collective dynamics and energy aspects of star coupled Hindmarsh-Rose neuron model with electrical, chemical and field couplings. Nonlinear Dyn. 96(3), 2115–2124 (2019)

    Article  MATH  Google Scholar 

  62. Erichsen, R., Jr, Brunnet, L.: Multistability in networks of Hindmarsh-Rose neurons. Phys. Rev. E 78(6), 061917 (2008)

    Article  ADS  Google Scholar 

  63. Shi, X., Wang, Z.: Adaptive synchronization of time delay Hindmarsh-Rose neuron system via self-feedback. Nonlinear Dyn. 69(4), 2147–2153 (2012)

    Article  MathSciNet  Google Scholar 

  64. Buscarino, A., Frasca, M., Branciforte, M., Fortuna, L., Sprott, J.C.: Synchronization of two Rossler systems with switching coupling. Nonlinear Dyn. 88(1), 673–683 (2017)

    Article  Google Scholar 

  65. Yamakou, M.E.: Chaotic synchronization of memristive neurons: Lyapunov function versus Hamilton function. Nonlinear Dyn. 101(1), 487–500 (2020)

    Article  MathSciNet  Google Scholar 

  66. Xu, Y., Jia, Y., Ma, J., Hayat, T., Alsaedi, A.: Collective responses in electrical activities of neurons under field coupling. Sci. Rep. 8(1), 1–10 (2018)

    Google Scholar 

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

    Article  ADS  Google Scholar 

Download references

Acknowledgements

TR would like to thank UGC, India, for the research fellowship through MANF and PAS would like to acknowledge DST, India, for the financial assistance through FIST program.

Author information

Authors and Affiliations

  1. Department of Physics, Farook College University of Calicut, Kerala, India, 673632

    Thazhathethil Remi & Pallimanhiyil Abdulraheem Subha

Authors
  1. Thazhathethil Remi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pallimanhiyil Abdulraheem Subha
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the study conception and design. The work, data simulation and analysis were performed by T. Remi. P. A. Subha made substantial contributions to the conception or design of the work. The first draft of the manuscript was written by T. Remi and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Pallimanhiyil Abdulraheem Subha.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figure 1 of the original version of this article unfortunately contained a colour inverting problem which was introduced during the production process. The original article has been corrected.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Remi, T., Subha, P. In-phase and anti-phase bursting dynamics and synchronisation scenario in neural network by varying coupling phase. J Biol Phys 49, 345–361 (2023). https://doi.org/10.1007/s10867-023-09635-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10867-023-09635-1

Keywords

PACS

Search

Navigation