phone +7 (3412) 91 60 92

Archive of Issues

Russia Yekaterinburg
Section Mathematics
Title Noise-induced intermittency and transition to chaos in the neuron Rulkov model
Author(-s) Bashkirtseva I.A.a, Nasyrova V.M.a, Ryashko L.B.a, Tsvetkov I.N.a
Affiliations Ural Federal Universitya
Abstract A discrete neuron model proposed by Rulkov is studied. In the deterministic version, this system simulates different modes of neural activity, such as quiescence, tonic and chaotic spiking. In the presence of random disturbances, another important mode of bursting characterized by the alternation of quiescence and excitement regimes can be observed. We study the probabilistic mechanisms of noise-induced transitions from quiescence to bursting in the zone of the tangent bifurcation. It is shown that such transitions are accompanied by a transformation of the system dynamics from regular to chaotic. For the analysis of these bifurcation phenomena, the stochastic sensitivity functions technique and method of confidence intervals are used.
Keywords Rulkov model of neural activity, random perturbations, stochastic sensitivity function, tangent bifurcation, noise-induced transitions, stochastic bifurcations
UDC 51-76, 519.216
MSC 39A50
DOI 10.20537/vm160401
Received 27 September 2016
Language Russian
Citation Bashkirtseva I.A., Nasyrova V.M., Ryashko L.B., Tsvetkov I.N. Noise-induced intermittency and transition to chaos in the neuron Rulkov model, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 4, pp. 453-462.
  1. Izhikevich E.M. Dynamical systems in neuroscience: the geometry of excitability and bursting, Cambridge: MIT Press, 2007, 497 p.
  2. Kuznetsov Yu.A. Elements of applied bifurcation theory, New York: Springer, 2004, 632 p. DOI: 10.1007/978-1-4757-3978-7
  3. Ibarz B., Casado J.M., Sanjuan M.A.F. Map-based models in neuronal dynamics, Physics Reports, 2011, vol. 501, issues 1-2, pp. 1-74. DOI: 10.1016/j.physrep.2010.12.003
  4. Girardi-Schappo M., Tragtenberg M.H.R., Kinouchi O. A brief history of excitable map-based neurons and neural networks, Journal of Neuroscience Methods, 2013, vol. 220, issue 2, pp. 116-130. DOI: 10.1016/j.jneumeth.2013.07.014
  5. Rulkov N.F. Regularization of synchronized chaotic bursts, Physical Review Letters, 2001, vol. 86, issue 1, pp. 183-186. DOI: 10.1103/PhysRevLett.86.183
  6. Pomeau Y., Manneville P. Intermittent transition to turbulence in dissipative dynamical systems, Communications in Mathematical Physics, 1980, vol. 74, issue 2, pp. 189-197. DOI: 10.1007/BF01197757
  7. Manneville P., Pomeau Y. Different ways to turbulence in dissipative dynamical systems, Physica D: Nonlinear Phenomena, 1980, vol. 1, issue 2, pp. 219-226. DOI: 10.1016/0167-2789(80)90013-5
  8. Neimark Yu.I., Landa P.S. Stokhasticheskie i khaoticheskie kolebaniya (Stochastic and chaotic oscillations), Moscow: Nauka, 1987, 424 p.
  9. Crutchfield J.P., Farmer J.D., Huberman B.A. Fluctuations and simple chaotic dynamics, Physics Reports, 1982, vol. 92, issue 2, pp. 45-82. DOI: 10.1016/0370-1573(82)90089-8
  10. Lasota A., Mackey M.C. Chaos, fractals, and noise: stochastic aspects of dynamic, New York: Springer-Verlag, 1994, 474 p. DOI: 10.1007/978-1-4612-4286-4
  11. Bashkirtseva I., Ryashko L., Tsvetkov I. Stochastic sensitivity of equilibrium and cycles for 1d discrete maps, Izvestiya Vysshikh Uchebnykh Zavedenii. Prikladnaya Nelineinaya Dinamika, 2009, vol. 17, no. 6, pp. 74-85 (in Russian).
  12. Bashkirtseva I., Ryashko L., Tsvetkov I. Sensitivity analysis of stochastic equilibria and cycles for the discrete dynamic systems, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis, 2010, vol. 17, pp. 501-515.
  13. Grebogi C., Ott E., Yorke J.A. Crises, sudden changes in chaotic attractors, and transient chaos, Physica D: Nonlinear Phenomena, 1983, vol. 7, issues 1-3, pp. 181-200. DOI: 10.1016/0167-2789(83)90126-4
  14. Arnold V.I., Afrajmovich V.S., Il'yashenko Yu.S., Shil'nikov L.P. Dynamical systems. V. Bifurcation theory and catastrophe theory, Berlin-Heidelberg: Springer, 1994. DOI: 10.1007/978-3-642-57884-7
  15. Bashkirtseva I., Ryashko L. Stochastic sensitivity of the closed invariant curves for discrete-time systems, Physica A: Statistical Mechanics and its Application, 2014, vol. 410, pp. 236-243. DOI: 10.1016/j.physa.2014.05.037
  16. Bashkirtseva I., Ryashko L. Sensitivity analysis of stochastic attractors and noise-induced transitions for population model with Allee effect, Chaos, 2011, vol. 21, issue 4, 047514. DOI: 10.1063/1.3647316
  17. Bashkirtseva I., Ryashko L., Slepukhina E. Noise-induced oscillation bistability and transition to chaos in FitzHugh-Nagumo model, Fluctuation and Noise Letters, 2014, vol. 13, issue 01, 1450004, 16 p. DOI: 10.1142/S0219477514500047
  18. Ryashko L.B., Sysolyatina A.A. Analysis of stochastic dynamics in discrete-time macroeconomic Kaldor's model, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 1, pp. 60-70. DOI: 10.20537/vm150107
Full text
Next article >>