Analysis of noise-induced bursting in two-dimensional Hindmarsh–Rose model

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List of references:

  1. И. А. Башкирцева, Т. В. Перевалова. Анализ стохастических аттракторов при бифуркации точка покоя – цикл // Автоматика и телемеханика. — 2007. — № 10. — С. 53–69.
  2. И. А. Башкирцева, Л. Б. Ряшко. Метод квазипотенциала в исследовании локальной устойчивости предельных циклов к случайнымвозм ущениям // Изв. вузов. Прикл. нелинейная динамика. — 2001. — Т. 9, № 6. — С. 104–113.
  3. И. А. Башкирцева, Л. Б. Ряшко, Е. С. Слепухина. Бифуркация расщепления стохастических циклов в модели Фицхью-Нагумо // Нелинейная динамика. — 2013. — Т. 9, № 2. — С. 295–307.
  4. А. Д. Вентцель, М. И. Фрейдлин. Флуктуации в динамических системах под действием малых случайных возмущений. — М: Наука, 1979. — 424 с.
  5. К. В. Гардинер. Стохастические методы в естественных науках. — М: Мир, 1986. — 538 с.
  6. И. И. Гихман, А. В. Скороход. Стохастические дифференциальные уравнения и их приложения. — Киев: Наукова думка, 1982. — 612 с.
  7. Г. Н. Мильштейн, Л. Б. Ряшко. Первое приближение квазипотенциала в задачах об устойчивости систем со случайными невырожденными возмущениями // Прикл. математика и механика. — 1995. — Т. 59, № 1. — С. 53–63.
  8. J. Baltanas, J. Casado. Noise-induced resonances in the Hindmarsh–Rose neuronal model // Phys. Rev. E. — 2002. — V. 65. — 6 p. — 041915. — DOI: 10.1103/PhysRevE.65.041915. — ads: 2002PhRvE..65d1915B.
  9. R. Barrio, A. Shilnikov. Parameter-sweeping techniques for temporal dynamics of neuronal systems: case study of Hindmarsh–Rose model // Journal of mathematical neuroscience. — 2011. — V. 1, no. 6. — 22 p. — MathSciNet: MR2827413.
  10. I. A. Bashkirtseva, L. B. Ryashko. Analysis of excitability for the FitzHugh-Nagumo model via a stochastic sensitivity function technique // Phys. Rev. E. — 2011. — V. 83, no. 6. — 8 p. — 061109. — DOI: 10.1103/PhysRevE.83.061109. — ads: 2011PhRvE..83f1109B.
  11. I. A. Bashkirtseva, L. B. Ryashko. Sensitivity analysis of stochastic attractors and noise-induced transitions for population model with Allee effect // Chaos. — 2011. — V. 21, no. 4. — 4 p. — 047514. — DOI: 10.1063/1.3647316. — ads: 2011Chaos..21d7514B.
  12. I. A. Bashkirtseva, L. B. Ryashko. Sensitivity and chaos control for the forced nonlinear oscillations // Chaos, Solitons and Fractals. — 2005. — no. 26. — P. 1437–1451. — DOI: 10.1016/j.chaos.2005.03.029. — MathSciNet: MR2149327. — ads: 2005CSF....26.1437B.
  13. I. A. Bashkirtseva, L. B. Ryashko. Stochastic sensitivity of 3D-cycles // Mathematics and Computers in Simulation. — 2004. — V. 66, no. 1. — P. 55–67. — DOI: 10.1016/j.matcom.2004.02.021. — MathSciNet: MR2064727.
  14. I. A. Bashkirtseva, L. B. Ryashko, E. Slepukhina. Noise-induced oscillation bistability and transition to chaos in FitzHugh-Nagumo model // Fluctuation and noise letters. — 2014. — V. 13, no. 1. — 16 p. — 1450004. — DOI: 10.1142/S0219477514500047.
  15. M. Dembo, O. Zeitouni. Large deviations techniques and applications. — Boston: Jones and Bartlett Publishers, 1995. — 346 p. — MathSciNet: MR1202429.
  16. M. Desroches, T. Kaper, M. Krupa. Mixed-mode bursting oscillations: Dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster // Chaos. — 2013. — V. 23, no. 4. — 13 p. — 046106. — DOI: 10.1063/1.4827026. — MathSciNet: MR3389775. — ads: 2013Chaos..23d6106D.
  17. R. FitzHugh. Impulses and physiological states in theoretical models of nerve membrane // Biophys. J. — 1961. — no. 1. — P. 445–466. — DOI: 10.1016/S0006-3495(61)86902-6.
  18. H. Gu, M. Yang, L. Li, Z. Liu, W. Ren. Experimental observation of the stochastic bursting caused by coherence resonance in a neural pacemaker // Neuroreport. — 2002. — V. 13, no. 13. — P. 1657–1660. — DOI: 10.1097/00001756-200209160-00018.
  19. J. L. Hindmarsh, R. M. Rose. A model of neuronal bursting using three coupled first order differential equations // Proc R Soc Lond B Biol Sci. — 1984. — V. 221, no. 1222. — P. 87–102. — DOI: 10.1098/rspb.1984.0024. — ads: 1984RSPSB.221...87H.
  20. A. L. Hodgkin. The local electric changes associated with repetitive action in a non-medullated axon // J Physiol. — 1948. — V. 107, no. 2. — P. 165–181. — DOI: 10.1113/jphysiol.1948.sp004260.
  21. G. Innocenti, A. Morelli, R. Genesio, A. Torcini. Dynamical phases of the Hindmarsh–Rose neuronal model: Studies of the transition from bursting to spiking chaos // Chaos. — 2007. — V. 17, no. 4. — 11 p. — 043128. — DOI: 10.1063/1.2818153. — MathSciNet: MR2380043. — ads: 2007Chaos..17d3128I.
  22. E. M. Izhikevich. Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. — Cambridge: MIT Press, 2007. — 521 p. — MathSciNet: MR2263523.
  23. E. M. Izhikevich. Neural Excitability, Spiking, and Bursting // Int. J. Bifurcation Chaos. — 2000. — V. 10, no. 6. — P. 1171–1266. — DOI: 10.1142/S0218127400000840. — MathSciNet: MR1779667.
  24. C. Kurrer, K. Schulten. Effect of noise and perturbations on limit cycle systems // Phys. D. — 1991. — V. 50, no. 3. — P. 311–320. — DOI: 10.1016/0167-2789(91)90001-P. — MathSciNet: MR1119029. — ads: 1991ZPhyB..85..311K.
  25. B. Lindner, L. Schimansky-Geier. Analytical approach to the stochastic FitzHugh-Nagumo system and coherence resonance // Phys. Rev. E. — 1999. — V. 60, no. 6. — P. 7270–7276. — DOI: 10.1103/PhysRevE.60.7270. — ads: 1999PhRvE..60.7270L.
  26. B. Lindner, J. Garcia-Ojalvo, A. Neiman, L. Schimansky-Geier. Effects of noise in excitable systems // Physics Reports. — 2004. — V. 392. — P. 321–424. — DOI: 10.1016/j.physrep.2003.10.015. — ads: 2004PhR...392..321L.
  27. D. Li-Xia, L. Qi-Shao. Codimension-Two Bifurcation Analysis in Hindmarsh–Rose Model with Two Parameters // Chin. Phys. Rev. — 2005. — V. 22, no. 6. — P. 1325–1328. — ads: 2005ChPhL..22.1318L.
  28. A. Longtin. Autonomous stochastic resonance in bursting neurons // Phys. Rev. E. — 1997. — V. 55, no. 1. — P. 868–876. — DOI: 10.1103/PhysRevE.55.868. — ads: 1997PhRvE..55..868L.
  29. V. V. Osipov, E. V. Ponizovskaya. Multivalued stochastic resonance in a model of an excitable neuron // Phys. Lett. A. — 2000. — V. 271, no. 3. — P. 191–197. — DOI: 10.1016/S0375-9601(00)00356-X. — MathSciNet: MR1767488. — ads: 2000PhLA..271..191O.
  30. A. S. Pikovsky, J. Kurths. Coherence resonance in a noise-driven excitable system // Phys. Rev. Lett. — 1997. — V. 78, no. 5. — P. 775–778. — DOI: 10.1103/PhysRevLett.78.775. — MathSciNet: MR1429571. — ads: 1997PhRvL..78..775P.
  31. S. Reinker, E. Puil, R. M. Miura. Resonances and Noise in a Stochastic Hindmarsh–Rose Model of Thalamic Neurons // Bull Math Biol. — 2003. — V. 65, no. 4. — P. 641–663. — DOI: 10.1016/S0092-8240(03)00026-0.
  32. A. Shilnikov, M. Kolomiets. Methods of the qualitative theory for the Hindmarsh–Rose Model: A case study – A Tutorial // Int. J. Bifurcation Chaos. — 2008. — V. 18, no. 8. — P. 2141–2168. — DOI: 10.1142/S0218127408021634. — MathSciNet: MR2463856.
  33. M. Storace, D. Linaro, E. de Lange. The Hindmarsh–Rose neuron model: bifurcation analysis and piecewise-linear approximations // Chaos. — 2008. — V. 18, no. 3. — 10 p. — 033128. — DOI: 10.1063/1.2975967. — MathSciNet: MR2464307. — ads: 2008Chaos..18c3128S.
  34. Y. Wang, Z. D. Wang, W. Wang. Dynamical Behaviors of Periodically Forced Hindmarsh–Rose Neural Model: The Role of Excitability and ‘Intrinsic’ Stochastic Resonance // J. Phys. Soc. Jpn. — 2000. — V. 69, no. 1. — P. 276–283. — DOI: 10.1143/JPSJ.69.276. — ads: 2000JPSJ...69..276W.
  35. X.-J. Wang. Genesis of bursting oscillations in the Hindmarsh–Rose model and homoclinicity to a chaotic saddle // Physica D. — 1993. — V. 63, no. 1–4. — P. 263–274. — DOI: 10.1016/0167-2789(93)90286-A. — MathSciNet: MR1207426. — ads: 1993PhyD...62..263W.
  36. Xia Shi, Lu. Qi-Shao. Coherence resonance and synchronization of Hindmarsh–Rose neurons with noise // Chinese Physics. — 2005. — V. 14, no. 6. — P. 1088–1094. — DOI: 10.1088/1009-1963/14/6/006.
  37. J. Ying, B. Qin-Sheng. SubHopf/Fold-Cycle Bursting in the Hindmarsh–Rose Neuronal Model with Periodic Stimulation // Chin. Phys. Lett. — 2011. — V. 28, no. 9. — 3 p. — 090201. — DOI: 10.1088/0256-307X/28/9/090201.

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