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Ryabchenko A. Д., Rybalova E. V., Strelkova G. I. Influence of additive noise on chimera and solitary states in neural networks. Izvestiya VUZ. Applied Nonlinear Dynamics, 2024, vol. 32, iss. 1, pp. 121-140. DOI: 10.18500/0869-6632-003083, EDN: YYDPVE

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Influence of additive noise on chimera and solitary states in neural networks

Ryabchenko Andrey Дмитриевич, Saratov State University
Rybalova E. V., Saratov State University
Strelkova Galina Ivanovna, Saratov State University

The purpose of this work is to study numerically the influence of additive white Gaussian noise on the dynamics of a network of nonlocally coupled neuron models which are represented by FitzHugh–Nagumo oscillators. Depending on coupling parameters between the individual elements this network can demonstrate various spatio-temporal structures, such as chimera states, solitary states and regimes of their coexistence (combined structures). These patterns exhibit different responses against additive noise influences.

Methods. The network dynamics is explored by calculating and plotting snapshots (instantaneous spatial distributions of the coordinate values at a fixed time), space-time diagrams, projections of multidimensional attractors, mean phase velocity profiles, and spatial distributions (profiles) of cross-correlation coefficient values. We also evaluate the cross-correlation coefficient averaged over the network, the mean number of solitary nodes and the probability of settling spatio-temporal structures in the neuronal network in the presence of additive noise.

Results. It has been shown that additive noise can decrease the probability of settling regimes of solitary states and combined structures, while the probability of observing chimera states arises up to 100%. In the noisy network of FitzHugh–Nagumo oscillators exhibiting the regime of solitary states, increasing the noise intensity leads, in general case, to a decrease of the mean number of solitary nodes and the interval of coupling parameter values within which the solitary states are observed. However, there is a finite region in the coupling parameter plane, inside which the number of solitary nodes can grow in the presence of additive noise.

Conclusion. We have studied the impact of additive noise on the probability of observing chimera states, solitary states and combined structures, which coexist in the multistability region, in the network of nonlocally coupled FitzHugh–Nagumo neuron models. It has been established that chimera states represent more stable and dominating structures among the other patterns coexisting in the studied network. At the same time, the probability of settling regimes of solitary states only, the region of their existence in the coupling parameter plane and the number of solitary nodes generally decrease when the noise intensity increases.

The research was supported by the Russian Science Foundation (project No. 20-12-00119,
  1. Benzi R, Sutera A, Vulpiani A. The mechanism of stochastic resonance. Journal of Physics A: Mathematical and General. 1981;14(11):L453–L457. DOI: 10.1088/0305-4470/14/11/006.
  2. Horsthemke W, Lefever R. Noise-induced transitions in physics, chemistry, and biology. In: Noise-Induced Transitions. Vol. 15 of Springer Series in Synergetics. Berlin, Heidelberg: Springer; 1984. P. 164–200. DOI: 10.1007/3-540-36852-3_7.
  3. Neiman A. Synchronizationlike phenomena in coupled stochastic bistable systems. Physical Review E. 1994;49(4):3484–3487. DOI: 10.1103/PhysRevE.49.3484.
  4. Arnold L. Random dynamical systems. In: Johnson R, editor. Dynamical Systems. Vol. 1609 of Lecture Notes in Mathematics. Berlin, Heidelberg: Springer; 1995. P. 1–43. DOI: 10.1007/ BFb0095238.
  5. Pikovsky AS, Kurths J. Coherence resonance in a noise-driven excitable system. Physical Review Letters. 1997;78(5):775–778. DOI: 10.1103/PhysRevLett.78.775.
  6. Anishchenko VS, Neiman AB, Moss F, Shimansky-Geier L. Stochastic resonance: noise-enhanced order. Phys. Usp. 1999;42(1):7–36. DOI: 10.1070/PU1999v042n01ABEH000444.
  7. Goldobin DS, Pikovsky A. Synchronization and desynchronization of self-sustained oscillators by common noise. Physical Review E. 2005;71(4):045201. DOI: 10.1103/PhysRevE.71.045201.
  8. McDonnell MD, Ward LM. The benefits of noise in neural systems: bridging theory and experiment. Nature Reviews Neuroscience. 2011;12(7):415–425. DOI: 10.1038/nrn3061.
  9. Schimansky-Geier L, Herzel H. Positive Lyapunov exponents in the Kramers oscillator. Journal of Statistical Physics. 1993;70(1–2):141–147. DOI: 10.1007/BF01053959.
  10. Shulgin B, Neiman A, Anishchenko V. Mean switching frequency locking in stochastic bistable systems driven by a periodic force. Physical Review Letters. 1995;75(23):4157–4160. DOI: 10.1103/PhysRevLett.75.4157.
  11. Arnold L, Namachchivaya NS, Schenk-Hoppe KR. Toward an understanding of stochastic Hopf bifurcation: A case study. International Journal of Bifurcation and Chaos. 1996;6(11):1947–1975. DOI: 10.1142/S0218127496001272.
  12. Han SK, Yim TG, Postnov DE, Sosnovtseva OV. Interacting coherence resonance oscillators. Physical Review Letters. 1999;83(9):1771–1774. DOI: 10.1103/PhysRevLett.83.1771.
  13. Bashkirtseva I, Ryashko L, Schurz H. Analysis of noise-induced transitions for Hopf system with additive and multiplicative random disturbances. Chaos, Solitons & Fractals. 2009;39(1):72–82. DOI: 10.1016/j.chaos.2007.01.128.
  14. Gammaitoni L, Marchesoni F, Menichella-Saetta E, Santucci S. Stochastic resonance in bistable systems. Physical Review Letters. 1989;62(4):349–352. DOI: 10.1103/PhysRevLett.62.349.
  15. Lindner B, Schimansky-Geier L. Analytical approach to the stochastic FitzHugh-Nagumo system and coherence resonance. Physical Review E. 1999;60(6):7270–7276. DOI: 10.1103/PhysRevE. 60.7270.
  16. Kuramoto Y, Battogtokh D. Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenomena in Complex Systems. 2002;5(4):380–385.
  17. Abrams DM, Strogatz SH. Chimera states for coupled oscillators. Physical Review Letters. 2004;93(17):174102. DOI: 10.1103/PhysRevLett.93.174102.
  18. Omelchenko I, Maistrenko Y, Hovel P, Scholl E. Loss of coherence in dynamical networks: Spatial chaos and chimera states. Physical Review Letters. 2011;106(23):234102. DOI: 10.1103/ PhysRevLett.106.234102.
  19. Panaggio MJ, Abrams DM. Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity. 2015;28(3):R67. DOI: 10.1088/0951-7715/28/3/R67.
  20. Zakharova A. Chimera Patterns in Networks: Interplay between Dynamics, Structure, Noise, and Delay. Cham: Springer; 2020. 233 p. DOI: 10.1007/978-3-030-21714-3.
  21. Maistrenko Y, Penkovsky B, Rosenblum M. Solitary state at the edge of synchrony in ensembles with attractive and repulsive interactions. Physical Review E. 2014;89(6):060901. DOI: 10.1103/ PhysRevE.89.060901.
  22. Jaros P, Maistrenko Y, Kapitaniak T. Chimera states on the route from coherence to rotating waves. Physical Review E. 2015;91(2):022907. DOI: 10.1103/PhysRevE.91.022907.
  23. Bogomolov SA, Slepnev AV, Strelkova GI, Scholl E, Anishchenko VS. Mechanisms of appearance of amplitude and phase chimera states in ensembles of nonlocally coupled chaotic systems. Communications in Nonlinear Science and Numerical Simulation. 2017;43:25–36. DOI: 10.1016/ j.cnsns.2016.06.024.
  24. Panaggio MJ, Abrams DM. Chimera states on a flat torus. Physical Review Letters. 2013;110(9): 094102. DOI: 10.1103/PhysRevLett.110.094102.
  25. Sawicki J, Omelchenko I, Zakharova A, Scholl E. Chimera states in complex networks: interplay of fractal topology and delay. The European Physical Journal Special Topics. 2017;226(9):1883–1892. DOI: 10.1140/epjst/e2017-70036-8.
  26. Scholl E. Synchronization patterns and chimera states in complex networks: Interplay of topology and dynamics. The European Physical Journal Special Topics. 2016;225(6–7): 891–919. DOI: 10.1140/epjst/e2016-02646-3.
  27. Scholl E. Chimeras in physics and biology: Synchronization and desynchronization of rhythms. In: Hacker J, Lengauer T, editors. Zeit in Natur und Kultur: Vortrage anlasslich der Jahresversammlung am 20. und 21. September 2019 in Halle (Saale). Stuttgart: Wissenschaftliche Verlagsgesellschaft; 2021. P. 67–95. DOI: 10.26164/leopoldina_10_00275.
  28. Semenova N, Zakharova A, Scholl E, Anishchenko V. Does hyperbolicity impede emergence of chimera states in networks of nonlocally coupled chaotic oscillators? Europhysics Letters. 2015;112(4):40002. DOI: 10.1209/0295-5075/112/40002.
  29. Shima S, Kuramoto Y. Rotating spiral waves with phase-randomized core in nonlocally coupled oscillators. Physical Review E. 2004;69(3):036213. DOI: 10.1103/PhysRevE.69.036213.
  30. Ulonska S, Omelchenko I, Zakharova A, Scholl E. Chimera states in networks of Van der Pol oscillators with hierarchical connectivities. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2016;26(9):094825. DOI: 10.1063/1.4962913.
  31. Zakharova A, Kapeller M, Scholl E. Chimera death: Symmetry breaking in dynamical networks. Physical Review Letters. 2014;112(15):154101. DOI: 10.1103/PhysRevLett.112.154101.
  32. Menck PJ, Heitzig J, Kurths J, Schellnhuber HJ. How dead ends undermine power grid stability. Nature Communications. 2014;5(1):3969. DOI: 10.1038/ncomms4969.
  33. Motter AE, Myers SA, Anghel M, Nishikawa T. Spontaneous synchrony in power-grid networks. Nature Physics. 2013;9(3):191–197. DOI: 10.1038/nphys2535.
  34. Wang B, Suzuki H, Aihara K. Enhancing synchronization stability in a multi-area power grid. Scientific Reports. 2016;6(1):26596. DOI: 10.1038/srep26596.
  35. Hong S, Chun Y. Efficiency and stability in a model of wireless communication networks. Social Choice and Welfare. 2010;34(3):441–454. DOI: 10.1007/s00355-009-0409-1.
  36. Gonzalez-Avella JC, Cosenza MG, San Miguel M. Localized coherence in two interacting populations of social agents. Physica A: Statistical Mechanics and its Applications. 2014;399: 24–30. DOI: 10.1016/j.physa.2013.12.035.
  37. Bansal K, Garcia JO, Tompson SH, Verstynen T, Vettel JM, Muldoon SF. Cognitive chimera states in human brain networks. Science Advances. 2019;5(4):eaau8535. DOI: 10.1126/sciadv.aau8535.
  38. Majhi S, Bera BK, Ghosh D, Perc M. Chimera states in neuronal networks: A review. Physics of Life Reviews. 2019;28:100–121. DOI: 10.1016/j.plrev.2018.09.003.
  39. Scholl E. Partial synchronization patterns in brain networks. Europhysics Letters. 2022;136(1): 18001. DOI: 10.1209/0295-5075/ac3b97.
  40. Levy R, Hutchison WD, Lozano AM, Dostrovsky JO. High-frequency synchronization of neuronal activity in the subthalamic nucleus of parkinsonian patients with limb tremor. J. Neurosci. 2000;20(20):7766–7775. DOI: 10.1523/JNEUROSCI.20-20-07766.2000.
  41. Rattenborg NC, Amlaner CJ, Lima SL. Behavioral, neurophysiological and evolutionary perspectives on unihemispheric sleep. Neurosci. Biobehav. Rev. 2000;24(8):817–842. DOI: 10.1016/s0149- 7634(00)00039-7.
  42. Funahashi S, Bruce CJ, Goldman-Rakic PS. Neuronal activity related to saccadic eye movements in the monkey’s dorsolateral prefrontal cortex. J. Neurophysiol. 1991;65(6):1464–1483. DOI: 10.1152/ jn.1991.65.6.1464.
  43. Swindale NV. A model for the formation of ocular dominance stripes. Proc. R. Soc. Lond. B. 1980;208(1171):243–264. DOI: 10.1098/rspb.1980.0051.
  44. Andrzejak RG, Rummel C, Mormann F, Schindler K. All together now: Analogies between chimera state collapses and epileptic seizures. Scientific Reports. 2016;6(1):23000. DOI: 10.1038/srep23000.
  45. Malchow A-K, Omelchenko I, Scholl E, Hovel P. Robustness of chimera states in nonlocally coupled networks of nonidentical logistic maps. Physical Review E. 2018;98(1):012217. DOI: 10.1103/PhysRevE.98.012217.
  46. Bukh AV, Slepnev AV, Anishchenko VS, Vadivasova TE. Stability and noise-induced transitions in an ensemble of nonlocally coupled chaotic maps. Regular and Chaotic Dynamics. 2018;23(3):325– 338. DOI: 10.1134/S1560354718030073.
  47. Rybalova EV, Klyushina DY, Anishchenko VS, Strelkova GI. Impact of noise on the amplitude chimera lifetime in an ensemble of nonlocally coupled chaotic maps. Regular and Chaotic Dynamics. 2019;24(4):432–445. DOI: 10.1134/S1560354719040051.
  48. Rybalova E, Scholl E, Strelkova G. Controlling chimera and solitary states by additive noise in networks of chaotic maps. Journal of Difference Equations and Applications. 2022:1–22. DOI: 10.1080/10236198.2022.2118580.
  49. Rybalova E, Muni S, Strelkova G. Transition from chimera/solitary states to traveling waves. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2023;33(3):033104. DOI: 10.1063/ 5.0138207.
  50. Nechaev VA, Rybalova EV, Strelkova GI. Influence of parameters inhomogeneity on the existence of chimera states in a ring of nonlocally coupled maps. Izvestiya VUZ. Applied Nonlinear Dynamics. 2021;29(6):943–952 (in Russian). DOI: 10.18500/0869-6632-2021-29-6-943-952.
  51. Nikishina NN, Rybalova EV, Strelkova GI, Vadivasova TE. Destruction of cluster structures in an ensemble of chaotic maps with noise-modulated nonlocal coupling. Regular and Chaotic Dynamics. 2022;27(2):242–251. DOI: 10.1134/S1560354722020083.
  52. Omelchenko I, Provata A, Hizanidis J, Scholl E, Hovel P. Robustness of chimera states for coupled FitzHugh-Nagumo oscillators. Physical Review E. 2015;91(2):022917. DOI: 10.1103/PhysRevE. 91.022917.
  53. Semenova N, Zakharova A, Anishchenko V, Scholl E. Coherence-resonance chimeras in a network of excitable elements. Physical Review Letters. 2016;117(1):014102. DOI: 10.1103/PhysRevLett. 117.014102.
  54. Zakharova A, Loos S, Siebert J, Gjurchinovski A, Scholl E. Chimera patterns: influence of time delay and noise. IFAC-PapersOnLine. 2015;48(18):7–12. DOI: 10.1016/j.ifacol.2015.11.002.
  55. Loos SAM, Claussen JC, Scholl E, Zakharova A. Chimera patterns under the impact of noise. Physical Review E. 2016;93(1):012209. DOI: 10.1103/PhysRevE.93.012209.
  56. Wu H, Dhamala M. Dynamics of Kuramoto oscillators with time-delayed positive and negative couplings. Physical Review E. 2018;98(3):032221. DOI: 10.1103/PhysRevE.98.032221.
  57. Jaros P, Brezetsky S, Levchenko R, Dudkowski D, Kapitaniak T, Maistrenko Y. Solitary states for coupled oscillators with inertia. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2018;28(1):011103. DOI: 10.1063/1.5019792.
  58. Berner R, Polanska A, Scholl E, Yanchuk S. Solitary states in adaptive nonlocal oscillator networks. The European Physical Journal Special Topics. 2020;229(12–13):2183–2203. DOI: 10.1140/epjst/ e2020-900253-0.
  59. Semenova NI, Rybalova EV, Strelkova GI, Anishchenko VS. “Coherence–incoherence” transition in ensembles of nonlocally coupled chaotic oscillators with nonhyperbolic and hyperbolic attractors. Regular and Chaotic Dynamics. 2017;22(2):148–162. DOI: 10.1134/S1560354717020046.
  60. Semenova N, Vadivasova T, Anishchenko V. Mechanism of solitary state appearance in an ensemble of nonlocally coupled Lozi maps. The European Physical Journal Special Topics. 2018;227(10–11):1173–1183. DOI: 10.1140/epjst/e2018-800035-y.
  61. Schulen L, Ghosh S, Kachhvah AD, Zakharova A, Jalan S. Delay engineered solitary states in complex networks. Chaos, Solitons & Fractals. 2019;128:290–296. DOI: 10.1016/j.chaos. 2019.07.046.
  62. Mikhaylenko M, Ramlow L, Jalan S, Zakharova A. Weak multiplexing in neural networks: Switching between chimera and solitary states. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2019;29(2):023122. DOI: 10.1063/1.5057418.
  63. Rybalova E, Anishchenko VS, Strelkova GI, Zakharova A. Solitary states and solitary state chimera in neural networks. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2019;29(7):071106. DOI: 10.1063/1.5113789.
  64. Schulen L, Janzen DA, Medeiros ES, Zakharova A. Solitary states in multiplex neural networks: Onset and vulnerability. Chaos, Solitons & Fractals. 2021;145:110670. DOI: 10.1016/j.chaos. 2021.110670.
  65. Taher H, Olmi S, Scholl E. Enhancing power grid synchronization and stability through time-delayed feedback control. Physical Review E. 2019;100(6):062306. DOI: 10.1103/PhysRevE. 100.062306.
  66. Hellmann F, Schultz P, Jaros P, Levchenko R, Kapitaniak T, Kurths J, Maistrenko Y. Network induced multistability through lossy coupling and exotic solitary states. Nature Communications. 2020;11(1):592. DOI: 10.1038/s41467-020-14417-7.
  67. Berner R, Yanchuk S, Scholl E. What adaptive neuronal networks teach us about power grids. Physical Review E. 2021;103(4):042315. DOI: 10.1103/PhysRevE.103.042315.
  68. Kapitaniak T, Kuzma P, Wojewoda J, Czolczynski K, Maistrenko Y. Imperfect chimera states for coupled pendula. Scientific Reports. 2014;4(1):6379. DOI: 10.1038/srep06379.
  69. Fried I, MacDonald KA, Wilson CL. Single neuron activity in human hippocampus and amygdala during recognition of faces and objects. Neuron. 1997;18(5):753–765. DOI: 10.1016/s0896- 6273(00)80315-3.
  70. Kreiman G, Koch C, Fried I. Category-specific visual responses of single neurons in the human medial temporal lobe. Nature Neuroscience. 2000;3(9):946–953. DOI: 10.1038/78868.
  71. Rose D. Some reflections on (or by?) grandmother cells. Perception. 1996;25(8):881–886. DOI: 10.1068/p250881.
  72. Quiroga RQ, Reddy L, Kreiman G, Koch C, Fried I. Invariant visual representation by single neurons in the human brain. Nature. 2005;435(7045):1102–1107. DOI: 10.1038/nature03687.
  73. Xin Y, Zhong L, Zhang Y, Zhou T, Pan J, Xu N-L. Sensory-to-category transformation via dynamic reorganization of ensemble structures in mouse auditory cortex. Neuron. 2019;103(5):909–921. DOI: 10.1016/j.neuron.2019.06.004.
  74. Franovic I, Eydam S, Semenova N, Zakharova A. Unbalanced clustering and solitary states in coupled excitable systems. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2022;32(1): 011104. DOI: 10.1063/5.0077022.
  75. Rybalova E, Strelkova G. Response of solitary states to noise-modulated parameters in nonlocally coupled networks of Lozi maps. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2022;32(2):021101. DOI: 10.1063/5.0082431.
  76. Omelchenko I, Omel’chenko OE, Hovel P, Scholl E. When nonlocal coupling between oscillators becomes stronger: Patched synchrony or multichimera states. Physical Review Letters. 2013; 110(22):224101. DOI: 10.1103/PhysRevLett.110.224101.
  77. FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal. 1961;1(6):445–466. DOI: 10.1016/s0006-3495(61)86902-6.
  78. Nagumo J, Arimoto S, Yoshizawa S. An active pulse transmission line simulating nerve axon. Proceedings of the IRE. 1962;50(10):2061–2070. DOI: 10.1109/JRPROC.1962.288235.
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