Izvestiya of Saratov University.

Physics

ISSN 1817-3020 (Print)
ISSN 2542-193X (Online)


For citation:

Fateev I. S., Polezhaev A. A. Chimera states in systems of superdiffusively coupled neurons. Izvestiya of Saratov University. Physics , 2024, vol. 24, iss. 4, pp. 328-339. DOI: 10.18500/1817-3020-2024-24-4-328-339, EDN: AKRGLX

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
25.12.2024
Full text:
(downloads: 26)
Language: 
Russian
Article type: 
Review
UDC: 
530.182
EDN: 
AKRGLX

Chimera states in systems of superdiffusively coupled neurons

Autors: 
Fateev Ilya Sergeevich, P. N. Lebedev Physical Institute of the Russian Academy of Sciences
Polezhaev Andrey A., P. N. Lebedev Physical Institute of the Russian Academy of Sciences
Abstract: 

Background and Objectives: One of the most intriguing collective phenomena, which arise in systems of coupled oscillators of different nature, are chimera states. They are characterized by the emergence of coordinated spatial synchronization and desynchronization, in an initially homogeneous system. Materials and Methods: This paper discusses the results of studies of one-dimensional and two-dimensional systems of interacting neurons organized on the basis of the fractional Laplace operator and the superdiffusion kinetic mechanism. Their use significantly extends the possibilities of describing chimera-like phenomena from the position of the classical reaction-diffusion approach. Due to mathematical brevity and its ability to reproduce almost all known scenarios of point neural activity, Hindmarsh–Rose model functions were used as a nonlinear part. Results: The studies under discussion demonstrate that one-dimensional and two-dimensional systems, two and three-component reaction-superdiffusion equations organized on the basis the fractional Laplace operator are able to reproduce chimera states. Dynamic regimes in the parameter space of the fractional Laplace operator exponents associated with the shape-forming features of networks of interacting neurons have been analyzed. Parameter regions of synchronization modes, modes of incoherent behavior, and chimera states are discussed. Conclusion: The results of the presented studies can be used in computational neuroscience tasks and various interdisciplinary studies as an alternative to existing network models.

Acknowledgments: 
The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”.
Reference: 
  1. Kuramoto Y., Battogtokh D. Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. arXiv preprint cond-mat/0210694, 2002. https://doi.org/10.48550/arXiv.cond-mat/0210694
  2. Abrams D. M., Strogatz S. H. Chimera states for coupled oscillators. Physical Review Letters, 2004, vol. 93, no. 17, pp. 174102. https://doi.org/10.1103/PhysRevLett.93.174102
  3. Zakharova A. Chimera patterns in networks. Switzerland, Springer, 2020. 233 p. https://doi.org/10.1007/978-3-030-21714-3
  4. Maistrenko Y. L., Vasylenko A., Sudakov O., Levchenko R., Maistrenko V. L. Cascades of multiheaded chimera states for coupled phase oscillators. International Journal of Bifurcation and Chaos, 2014, vol. 24, no. 08, pp. 1440014. https://doi.org/10.1142/S0218127414400148
  5. Martens E. A., Thutupalli S., Fourrière A., Hallatschek O. Chimera states in mechanical oscillator networks. Proceedings of the National Academy of Sciences, 2013, vol. 110, no. 26, pp. 10563–10567. https://doi.org/10.1073/pnas.1302880110
  6. Viktorov E. A., Habruseva T., Hegarty S. P., Huyet G., Kelleher B. Coherence and incoherence in an optical comb. Physical Review Letters, 2014, vol. 112, no. 22, pp. 224101. https://doi.org/10.1103/PhysRevLett.112.224101
  7. Tinsley M. R., Nkomo S., Showalter K. Chimera and phase-cluster states in populations of coupled chemical oscillators. Nature Physics, 2012, vol. 8, no. 9, pp. 662–665. https://doi.org/10.1038/nphys2371
  8. Bera B. K., Ghosh D., Lakshmanan M. Chimera states in bursting neurons. Physical Review E, 2016, vol. 93, no. 1, pp. 012205. https://doi.org/10.1103/PhysRevE.93.012205
  9. Wang Z., Xu Y., Li Y., Kapitaniak T., Kurths J. Chimera states in coupled Hindmarsh–Rose neurons with α-stable noise. Chaos, Solitons & Fractals, 2021, vol. 148, pp. 110976. https://doi.org/10.1016/j.chaos.2021.110976
  10. Hizanidis J., Kanas V. G., Bezerianos A., Bountis T. Chimera states in networks of nonlocally coupled Hindmarsh–Rose neuron models. International Journal of Bifurcation and Chaos, 2014, vol. 24, no. 03, pp. 1450030. https://doi.org/10.1142/S0218127414500308
  11. Majhi S., Bera B. K., Ghosh D., Perc M. Chimera states in neuronal networks: A review. Physics of Life Reviews, 2019, vol. 28, pp. 100–121. https://doi.org/10.1016/j.plrev.2018.09.003
  12. Parastesh F., Jafari S., Azarnoush H., Shahriari Z., Wang Z., Boccaletti S., Perc M. Chimeras. Physics Reports, 2021, vol. 898, pp. 1–114. https://doi.org/10.1016/j.physrep.2020.10.003
  13. Huo S., Tian C., Kang L., Liu Z. Chimera states of neuron networks with adaptive coupling. Nonlinear Dynamics, 2019, vol. 96, pp. 75–86. https://doi.org/10.1007/s11071-019-04774-4
  14. Bera B. K., Ghosh D. Chimera states in purely local delay-coupled oscillators. Physical Review E, 2016, vol. 93, no. 5, pp. 052223. https://doi.org/10.1103/PhysRevE.93.052223
  15. Fateev I., Polezhaev A. Chimera states in a chain of superdiffusively coupled neurons. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2023, vol. 33, no. 10, pp. 103110. https://doi.org/10.1063/5.0168422
  16. Kundu S., Ghosh D. Higher-order interactions promote chimera states. Physical Review E, 2022, vol. 105, no. 4, pp. L042202. https://doi.org/10.1103/PhysRevE.105.L042202
  17. Qin H., Ma J., Wang C., Chu R. Autapse-induced target wave, spiral wave in regular network of neurons. Science China Physics, Mechanics & Astronomy, 2014, vol. 57, pp. 1918–1926. https://doi.org/10.1007/s11433-014-5466-5
  18. Jun M., He-Ping Y., Yong L., Shi-Rong L. Development and transition of spiral wave in the coupled Hindmarsh–Rose neurons in two-dimensional space. Chinese Physics B, 2009, vol. 18, no. 1, pp. 98–105. https://doi.org/10.1088/1674-1056/18/1/017
  19. Huang X., Xu W., Liang J., Takagaki K., Gao X., Wu J. Y. Spiral wave dynamics in neocortex. Neuron, 2010, vol. 68, no. 5, pp. 978–990. https://doi.org/10.1016/j.neuron.2010.11.007
  20. Wu J. Y., Huang X., Zhang C. Propagating waves of activity in the neocortex: What they are, what they do. The Neuroscientist, 2008, vol. 14, no. 5, pp. 487–502. https://doi.org/10.1177/1073858408317066
  21. Shepelev I. A., Bukh A. V., Muni S. S., Anishchenko V. S. Role of solitary states in forming spatiotemporal patterns in a 2D lattice of van der Pol oscillators. Chaos, Solitons & Fractals, 2020, vol. 135, pp. 109725. https://doi.org/10.1016/j.chaos.2020.109725
  22. Rybalova E., Bukh A., Strelkova G., Anishchenko V. Spiral and target wave chimeras in a 2D lattice of map-based neuron models. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2019, vol. 29, no. 10, pp. 101104. https://doi.org/10.1063/1.5126178
  23. Fateev I., Polezhaev A. Chimera states in a lattice of superdiffusively coupled neurons. Chaos, Solitons & Fractals, 2024, vol. 181, pp. 114722. https://doi.org/10.1016/j.chaos.2024.114722
  24. Kasimatis T., Hizanidis J., Provata A. Three-dimensional chimera patterns in networks of spiking neuron oscillators. Physical Review E, 2018, vol. 97, no. 5, pp. 052213. https://doi.org/10.1103/PhysRevE.97.052213
  25. Klages R., Radons G., Sokolov I. M. Anomalous transport. Weinheim, Wiley-VCH Verlag, 2008. 608 p. https://doi.org/10.1002/9783527622979
  26. Ramakrishnan B., Parastesh F., Jafari S., Rajagopal K., Stamov G., Stamova I. Synchronization in a multiplex network of nonidentical fractional-order neurons. Fractal and Fractional, 2022, vol. 6, no. 3, pp. 169. https://doi.org/10.3390/fractalfract6030169
  27. Yan B., Parastesh F., He S., Rajagopal K., Jafari S., Perc M. Interlayer and intralayer synchronization in multiplex fractional-order neuronal networks. Fractals, 2022, vol. 30, no. 10, pp. 22401946. https://doi.org/10.1142/S0218348X22401946
  28. Giresse T. A., Crepin K. T., Martin T. Generalized synchronization of the extended Hindmarsh–Rose neuronal model with fractional order derivative. Chaos, Solitons & Fractals, 2019, vol. 118, pp. 311–319. https://doi.org/10.1016/j.chaos.2018.11.028
  29. Buzsáki G., Mizuseki K. The log-dynamic brain: How skewed distributions affect network operations. Nature Reviews Neuroscience, 2014, vol. 15, no. 4, pp. 264–278. https://doi.org/10.1038/nrn3687
  30. Cossell L., Iacaruso M. F., Muir D. R., Houlton R., Sader E. N., Ko H., Hofer S. B., Mrsic-Flogel T. D. Functional organization of excitatory synaptic strength in primary visual cortex. Nature, 2015, vol. 518, no. 7539, pp. 399–403. https://doi.org/10.1038/nature14182
  31. Song S., Sjöström P. J., Reigl M., Nelson S., Chklovskii D. B. Highly nonrandom features of synaptic connectivity in local cortical circuits. PLoS Biology, 2005, vol. 3, no. 3, pp. e68. https://doi.org/10.1371/journal.pbio.0030068
  32. Hilgetag C. C., Goulas A. Is the brain really a small-world network? Brain Structure and Function, 2016, vol. 221, pp. 2361–2366. https://doi.org/10.1007/s00429-015-1035-6
  33. Beggs J. M., Plenz D. Neuronal avalanches in neocortical circuits. Journal of Neuroscience, 2003, vol. 23, no. 35, pp. 11167–11177. https://doi.org/10.1523/JNEUROSCI.23-35-11167.2003
  34. Barabási A. L., Albert R. Emergence of scaling in random networks. Science, 1999, vol. 286, no. 5439, pp. 509–512. https://doi.org/10.1126/science.286.5439.509
  35. Baronchelli A., Radicchi F. Lévy 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
  36. Wardak A., Gong P. Fractional diffusion theory of balanced heterogeneous neural networks. Physical Review Research, 2021, vol. 3, no. 1, pp. 013083. https://doi.org/10.1103/PhysRevResearch.3.013083
  37. Lee H. G. A second-order operator splitting Fourier spectral method for fractional-in-space reaction–diffusion equations. Journal of Computational and Applied Mathematics, 2018, vol. 333, pp. 395–403. https://doi.org/10.1016/j.cam.2017.09.007
  38. Liu F., Turner I., Anh V., Yang Q., Burrage K. A numerical method for the fractional Fitzhugh–Nagumo monodomain model. Anziam Journal, 2012, vol. 54, pp. C608–C629. https://doi.org/10.21914/anziamj.v54i0.6372
  39. Chen G., Gong P. A spatiotemporal mechanism of visual attention: Superdiffusive motion and theta oscillations of neural population activity patterns. Science Advances, 2022, vol. 8, no. 16, pp. Eabl4995. https://doi.org/10.1126/sciadv.abl4995
  40. Qi Y., Gong P. Fractional neural sampling as a theory of spatiotemporal probabilistic computations in neural circuits. Nature Communications, 2022, vol. 13, no. 1, pp. 4572. https://doi.org/10.1038/s41467-022-32279-z
  41. Samko S. G., Kilbas A. A, Marichev O. I. Fractional integrals and derivatives: Theory and applications. Switzerland, Gordon and Breach, 1993. 976 p.
  42. Zhuang P., Liu F., Anh V., Turner I. Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM Journal on Numerical Analysis, 2009, vol. 47, no. 3, pp. 1760–1781. https://doi.org/10.1137/080730597
  43. Liu F., Chen S., Turner I., Burrage K., Anh V. Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term. Open Phys., 2013, vol. 11, no. 10, pp. 1221–1232. https://doi.org/10.2478/s11534-013-0296-z
  44. Li B. W., Dierckx H. Spiral wave chimeras in locally coupled oscillator systems. Physical Review E, 2016, vol. 93, no. 2, pp. 020202. https://doi.org/10.1103/PhysRevE.93.020202
  45. Garcia-Ojalvo J., Elowitz M. B., Strogatz S. H. Modeling a synthetic multicellular clock: Repressilators coupled by quorum sensing. Proceedings of the National Academy of Sciences, 2004, vol. 101, no. 30, pp. 10955–10960. https://doi.org/10.1073/pnas.0307095101
  46. Gonze D., Bernard S., Waltermann C., Kramer A., Herzel H. Spontaneous synchronization of coupled circadian oscillators. Biophysical Journal, 2005, vol. 89, no. 1, pp. 120–129. https://doi.org/10.1529/biophysj.104.058388
  47. Gopal R., Chandrasekar V. K., Venkatesan A., Lakshmanan M. Observation and characterization of chimera states in coupled dynamical systems with nonlocal coupling. Physical Review E, 2014, vol. 89, no. 5, pp. 052914. https://doi.org/10.1103/PhysRevE.89.052914
  48. Kundu S., Majhi S., Muruganandam P., Ghosh D. Diffusion induced spiral wave chimeras in ecological system. The European Physical Journal Special Topics, 2018, vol. 227, pp. 983–993. https://doi.org/10.1140/epjst/e2018-800011-1
Received: 
09.04.2024
Accepted: 
30.07.2024
Published: 
25.12.2024