Skip to main content
SpringerLink
Log in

Dynamics of a Chain of Interacting Neurons with Nonlocal Coupling, Given by Laplace Operator of Fractional and Variable Orders with Nonlinear Hindmarsh–Rose Model Functions

  • Published:
Bulletin of the Lebedev Physics Institute Aims and scope Submit manuscript

Abstract

A model describing the dynamics of a one-dimensional chain of interacting nonlocally coupled neurons, based on properties of the fractional Laplacian with constant and variable order is proposed. As a result of numerical simulation of the action potential propagation for various types of nonlocal interactions (with a constant set of parameters corresponding to nonlinear functions of the Hindmarsh—Rose model, as well as diffusion coefficients), the appearance of various spatiotemporal modes is established. At given parameters, in the case of classical diffusion, the general trend to synchronous behavior is observed. In the superdiffusion case, the modes of cluster excitation of the action potential are formed. For systems with variable-order fractional Laplacian, spatial anisotropy of arising structures is characteristic. It is shown that the introduction of long-range couplings realized in the system by introducing the fractional Laplacian can provide additional possibilities of describing dynamic properties of interacting neurons.

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

Access this article

Log in via an institution

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.

REFERENCES

  1. Sharma, S.K., Mondal, A., Mondal, A., Upadhyay, R.K., and Ma, J., Synchronization and pattern formation in a memristive diffusive neuron model, Int. J. Bifurc. Chaos, 2021, vol. 31, pp. 1–16. https://doi.org/10.1142/S0218127421300305

    Article  MathSciNet  MATH  Google Scholar 

  2. Mondal, A., Upadhyay, R.K., Mondal, A., and Sharma, S.K., Emergence of Turing patterns and dynamic visualization in excitable neuron model, Appl. Math. Comput., 2022, vol. 423, pp. 1–13. https://doi.org/10.1016/j.amc.2022.127010

    Article  MathSciNet  MATH  Google Scholar 

  3. Wang, K., Teng, Z., and Jiang, H., Adaptive synchronization in an array of linearly coupled neural networks with reaction—diffusion terms and time delays, Commun. Nonlinear Sci. Numer. Simul., 2012, vol. 17, pp. 3866–875. https://doi.org/10.1016/j.cnsns.2012.02.020

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Jun, M., He-Ping, Y., Yong, L., and Shi-Rong, L., Development and transition of spiral wave in the coupled Hindmarsh—Rose neurons in two-dimensional space, Chin. Phys. B, 2009, vol. 18, pp. 98–105. https://doi.org/10.1088/1674-1056/18/1/017

    Article  ADS  Google Scholar 

  5. Keane, A. and Gong, P., Propagating waves can explain irregular neural dynamics, J. Neurosci., 2015, vol. 35, pp. 1591–1605. https://doi.org/10.1523/JNEUROSCI.1669-14.2015

    Article  Google Scholar 

  6. Townsend, R.G. and Gong, P., Detection and analysis of spatiotemporal patterns in brain activity, PLoS Comput. Biol., 2015, vol. 14, pp. 1–29. https://doi.org/10.1371/journal.pcbi.1006643

    Article  Google Scholar 

  7. Durand, D.M., Park, E., and Jensen, A.L., Potassium diffusive coupling in neural networks, Phil. Trans. R. Soc. B, 2010, vol. 365, pp. 2347–2362. https://doi.org/10.1098/rstb.2010.0050

    Article  Google Scholar 

  8. Amari, S.I., Dynamics of pattern formation in lateral-inhibition type neural fields, Biological Cybernetics, 1977, vol. 27, pp. 77–87. https://doi.org/10.1007/BF00337259

    Article  MathSciNet  MATH  Google Scholar 

  9. Amari, S.I., Topographic organization of nerve fields, Bull. Math. Biol., 1980, vol. 42, pp. 339–364. https://doi.org/10.1016/S0092-8240(80)80055-3

    Article  MathSciNet  MATH  Google Scholar 

  10. Ermentrout, B., Neural networks as spatio-temporal pattern-forming systems, Rep. Progr. Phys., 1998, vol. 61, p. 353. https://doi.org/10.1088/0034-4885/61/4/002

    Article  ADS  Google Scholar 

  11. Richardson, K.A., Schiff, S.J., and Gluckman, B.J., Control of traveling waves in the mammalian cortex, Phys. Rev. Lett., 2005, vol. 94, p. 028103. https://doi.org/10.1103/PhysRevLett.94.028103

  12. Huang, X., Troy, W.C., Yang, Q., Ma, H., Laing, C.R., Schiff, S.J., and Wu, J.Y., Spiral waves in disinhibited mammalian neocortex, J. Neurosci., 2004, vol. 24, pp. 9897–9902. https://doi.org/10.1523/JNEUROSCI.2705-04.2004

    Article  Google Scholar 

  13. Njitacke, Z.T., Muni, S.S., Seth, S., Awrejcewicz, J., and Kengne, J., Complex dynamics of a heterogeneous network of Hindmarsh—Rose neurons, Phys. Scr., 2023, vol. 98, pp. 045210. https://doi.org/10.1088/1402-4896/acbdd1

  14. Muni, S.S., Njitacke, Z., Feudjio, C., Fozin, T., and Awrejcewicz J., Route to chaos and chimera states in a network of Memristive Hindmarsh—Rose neurons model with external excitation, Chaos Theory Appl., 2022, vol. 4, pp. 119–127. https://doi.org/10.51537/chaos.1144123

  15. Linaro, D., Righero, M., Biey, M., and Storace, M., Synchronization properties in networks of Hindmarsh—Rose neurons and their PWL approximations with linear symmetric coupling, 2009 IEEE International Symposium on Circuits and Systems, Taipei, 2009, pp. 1685–1688. https://doi.org/10.1109/ISCAS.2009.5118098

  16. Zheng, Q. and Shen, J., Turing instability induced by random network in FitzHugh—Nagumo model, Appl. Math. Comp., 2020, vol. 381, p. 125304. https://doi.org/10.1016/j.amc.2020.125304

  17. Hizanidis, J., Kanas, V.G., Bezerianos, A., and Bountis, T., Chimera states in networks of nonlocally coupled Hindmarsh—Rose neuron models, Int. J. Bifurc. Chaos, 2014, vol. 24, pp. 1–9. https://doi.org/10.1142/S0218127414500308

    Article  MathSciNet  MATH  Google Scholar 

  18. Bera, B.K., Ghosh, D., and Lakshmanan, M., Chimera states in bursting neurons, Phys. Rev. E, 2016, vol. 93, p. 012205. https://doi.org/10.1103/PhysRevE.93.012205

  19. Wei, Z., Parastesh, F., Azarnoush, H., Jafari, S., Ghosh, D., Perc, M., and Slavinec, M., Nonstationary chimeras in a neuronal network, Europhys. Lett., 2018, vol. 123, p. 48003. https://doi.org/10.1209/0295-5075/123/48003

    Article  Google Scholar 

  20. Wang, Z., Xu, Y., Li, Y., Kapitaniak, T., and Kurths, J., Chimera states in coupled Hindmarsh–Rose neurons with α-stable noise, Chaos, Solitons Fractals, 2021, vol. 148, p. 110976. https://doi.org/10.1016/j.chaos.2021.110976

  21. Majhi, S., Bera, B.K., and Ghosh, D., Chimera states in neuronal networks: A review, Phys. Life Rev., 2019, vol. 28, pp. 100–121. https://doi.org/10.1016/j.plrev.2018.09.003

    Article  ADS  Google Scholar 

  22. Mondal, A., Sharma, S.K., Upadhyay, R.K., Aziz-Alaoui, M.A., Kundu, P., and Hens, C., Diffusion dynamics of a conductance-based neuronal populations, Phys. Rev. E., 2019, vol. 99, pp. 1–14. https://doi.org/10.1103/PhysRevE.99.042307

    Article  Google Scholar 

  23. Mondal, A., Mondal, A., Sharma, S.K., and Upadhyay, R.K., Analysis of spatially extended excitable Izhikevich neuron model near instability, Nonlin. Dyn., 2021, vol. 105, pp. 3515–3527. https://doi.org/10.1007/s11071-021-06787-4

    Article  Google Scholar 

  24. Chen, G. and Gong, P., A spatiotemporal mechanism of visual attention: Superdiffusive motion and theta oscillations of neural population activity patterns, Sci. Adv., 2022, vol. 8, pp. 1–19. https://doi.org/10.1126/sciadv.abl4995

    Article  Google Scholar 

  25. Qi, Y. and Gong, P., Fractional neural sampling as a theory of spatiotemporal probabilistic computations in neural circuits, Nat. Commun., 2022, vol. 13, pp. 1–19. https://doi.org/10.1038/s41467-022-32279-z

    Article  Google Scholar 

  26. Beggs, J.M. and Plenz, D., Neuronal avalanches in neocortical circuits, J. Neurosci., 2003, vol. 23, pp. 11167–11177. https://doi.org/10.1523/JNEUROSCI.23-35-11167.2003

    Article  Google Scholar 

  27. Barabasi, A.L. and Albert, R., Emergence of scaling in random networks, Science, 1999, vol. 286, pp. 509–512. https://doi.org/10.1126/science.286.5439.509

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Baronchelli, A. and Radicchi, F., Levy flights in human behavior and cognition, Chaos, Solitons Fractals, 2013, vol. 56, pp. 101–105. https://doi.org/10.1016/j.chaos.2013.07.013

    Article  ADS  Google Scholar 

  29. Storace, M., Linaro, D., and de Lange, E., The Hindmarsh—Rose neuron model: bifurcation analysis and piecewise-linear approximations, Chaos: Interdiscip. J. Nonlin. Sci., 2008, vol. 18, p. 033128. https://doi.org/10.1063/1.2975967

  30. Parastesh, F., Jafari, S., Azarnoush, H., Shahriari, Z., Wang, Z., Boccaletti, S., and Chimeras, M.P., Phys. Rep., 2021, vol. 898, pp. 1–114. https://doi.org/10.1016/j.physrep.2020.10.003

  31. Klages, R., Radons, G., and Sokolov, I.M., Anomalous Transport. Foundations and Applications, Willey-VCH Verlag, 2008. https://doi.org/10.1002/9783527622979

    Book  Google Scholar 

  32. Uchaikin, V.V., Fractional Derivatives for Physicists and Engineers, Berlin: Springer, 2013.

    Book  MATH  Google Scholar 

  33. Samko, S.G., Kilbas, A.A., and Marichev, O.I., Fractional Integrals and Derivatives. Theory and Applications, Yverdon: Gordon and Breach, 1993.

    MATH  Google Scholar 

  34. Wu, Y., Xu, J., He, D., and Earn, D.J., Generalized synchronization induced by noise and parameter mismatching in Hindmarsh—Rose neurons, Chaos, Solitons Fractals, 2005, vol. 23, pp. 1605–1611. https://doi.org/10.1016/j.chaos.2004.06.077

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Li, X., Han, C., and Wang, Y., Novel patterns in fractional-in-space nonlinear coupled FitzHugh—Nagumo models with Riesz fractional derivative, Fractal Fract., 2022, vol. 6, pp. 136. https://doi.org/10.3390/fractalfract6030136

    Article  Google Scholar 

  36. Liu, F., Turner, I., Anh, V., Yang, Q., and Burrage, K., A numerical method for the fractional Fitzhugh–Nagumo monodomain model, Anziam J., 2012, vol. 54, pp. 608–629. https://doi.org/10.21914/anziamj.v54i0.6372

    Article  MathSciNet  MATH  Google Scholar 

  37. Liu, F., Zhuang, P., Turner, I., Anh, V., and Burrage, K., A semi-alternating direction method for a 2-D fractional FitzHugh—Nagumo monodomain model on an approximate irregular domain, J. Comput. Phys., 2015, vol. 293, pp. 252–263. https://doi.org/10.1016/j.jcp.2014.06.001

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Zheng, Q. and Shen, J., Turing instability induced by random network in FitzHugh—Nagumo model, Appl. Math. Comput., 2020, vol. 381, pp. 1–13. https://doi.org/10.1016/j.amc.2020.125304

    Article  MathSciNet  MATH  Google Scholar 

  39. Patnaik, S., Hollkamp, J.P., and Semperlotti, F., Applications of variable-order fractional operators: a review. Proc. R. Soc. A., 2020, vol. 476, pp. 1–32. https://doi.org/10.1098/rspa.2019.0498

    Article  MathSciNet  MATH  Google Scholar 

  40. Sun, H.G., Chen, W., Wei, H., and Chen, Y.Q., A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems, Eur. Phys. J. Spec. Top., 2011, vol. 193, pp. 185–192. https://doi.org/10.1140/epjst/e2011-01390-6

    Article  Google Scholar 

  41. Ruiz-Medina, M.D., Anh, V., and Angulo, J.M., Fractional generalized random fields of variable order, Stoch. Anal. Appl., 2004, vol. 22, pp. 775–799. https://doi.org/10.1081/SAP-120030456

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhuang, P., Liu, F., Anh, V., and Turner, I., Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, Siam J. Numer. Anal., 2009, vol. 3, pp. 1760–1781. https://doi.org/10.1137/080730597

    Article  MathSciNet  MATH  Google Scholar 

  43. Mongillo, G., Rumpel, S., and Loewenstein, Y., Intrinsic volatility of synaptic connections—a challenge to the synaptic trace theory of memory, Curr. Opin. Neurobiol., 2017, vol. 46, pp. 7–13. https://doi.org/10.1016/j.conb.2017.06.006

    Article  Google Scholar 

  44. Lee, H.G., A second-order operator splitting Fourier spectral method for fractional-in-space reaction—diffusion equations, J. Comput. Appl. Math., 2018, vol. 333, pp. 395–403. https://doi.org/10.1016/j.cam.2017.09.007

    Article  MathSciNet  MATH  Google Scholar 

  45. Bueno-Orovio, A. and Burrage, K., Complex-order fractional diffusion in reaction-diffusion systems, Commun. Nonlinear. Sci. Numer. Simul., 2023, vol. 119, p. 107120. https://doi.org/10.1016/j.cnsns.2023.107120

  46. Owolabi, K.M., Karaagac, B., and Baleanu, D., Pattern formation in superdiffusion predator-prey-like problems with integerand nonintegerorder derivatives, Math. Methods Appl. Sci., 2021, vol. 44, no. 5, pp. 4018–4036. https://doi.org/10.1002/mma.7007

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Funding

This study was supported by the Russian Science Foundation, project no. 22-21-00546.

Author information

Authors and Affiliations

  1. Lebedev Physical Institute, Russian Academy of Sciences, 119991, Moscow, Russia

    I. S. Fateev & A. A. Polezhaev

Authors
  1. I. S. Fateev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. A. Polezhaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to I. S. Fateev or A. A. Polezhaev.

Ethics declarations

The authors declare that they have no conflicts of interest.

Additional information

Translated by A. Kazantsev

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fateev, I.S., Polezhaev, A.A. Dynamics of a Chain of Interacting Neurons with Nonlocal Coupling, Given by Laplace Operator of Fractional and Variable Orders with Nonlinear Hindmarsh–Rose Model Functions. Bull. Lebedev Phys. Inst. 50, 243–252 (2023). https://doi.org/10.3103/S1068335623060039

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1068335623060039

Keywords:

Search

Navigation