Geodesic
Stochastic generation of high amplitude oscillations in two-dimensional Hindmarsh–Rose model
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 2 (2014), pp. 76-85 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the stochastic dynamics of the two-dimensional Hindmarsh–Rose model. In the deterministic Hindmarsh–Rose model the parametric zones of coexistence of different stable attractors (equilibria and limit cycles) are possible. The emergence of high amplitude oscillations under the influence of random disturbances on the system in these zones is due to the presence of a limit cycle. However, the stochastic generation of high amplitude oscillations is possible in a parametric zone where the deterministic system has the only stable equilibrium. This article discusses this case. For a sufficiently low noise intensity values, random states concentrate near the stable equilibrium. With the increasing of the noise intensity, trajectories go far from the equilibrium making high amplitude oscillations in the neighborhood of the unstable equilibrium. This phenomenon is confirmed by changing of the probability distribution of random trajectories. This effect is analyzed using the stochastic sensitivity function technique. A method of estimation of critical values for noise intensity is proposed.
Keywords: Hindmarsh–Rose model, excitability, stochastic sensitivity, stochastic generation of high amplitude oscillations. 
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     title = {Stochastic generation of high amplitude oscillations in two-dimensional {Hindmarsh{\textendash}Rose} model},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
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L. B. Ryashko; E. S. Slepukhina. Stochastic generation of high amplitude oscillations in two-dimensional Hindmarsh–Rose model. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 2 (2014), pp. 76-85. http://geodesic.mathdoc.fr/item/VUU_2014_2_a4/

[1] Hindmarsh J. L., Rose R. M., “A model of neuronal bursting using three coupled first order differential equations”, Proc. R. Soc. Lond. B. Biol. Sci., 221:1222 (1984), 87–102 | DOI

[2] Innocenti G., Morelli A., Genesio R., Torcini A., “Dynamical phases of the Hindmarsh–Rose neuronal model: Studies of the transition from bursting to spiking chaos”, Chaos, 17:4 (2007), 043128, 11 pp. | DOI | MR | Zbl

[3] Shilnikov A., Kolomiets M., “Methods of the qualitative theory for the Hindmarsh–Rose model: A case study — A Tutorial”, Int. J. Bifurcation Chaos, 18:8 (2008), 2141–2168 | DOI | MR | Zbl

[4] FitzHugh R., “Impulses and physiological states in theoretical models of nerve membrane”, Biophys. J., 1961, no. 1, 445–466 | DOI | MR

[5] Pikovsky A. S., Kurths J., “Coherence resonance in a noise-driven excitable system”, Phys. Rev. Lett., 78:5 (1997), 775–778 | DOI | MR | Zbl

[6] Lindner B., Schimansky-Geier L., “Analytical approach to the stochastic FitzHugh–Nagumo system and coherence resonance”, Phys. Rev. E, 60:6 (1999), 7270–7276 | DOI | MR

[7] Lindner B., Garcia-Ojalvo J., Neiman A., Schimansky-Geier L., “Effects of noise in excitable systems”, Physics Reports, 392 (2004), 321–424 | DOI

[8] Bashkirtseva I., Ryashko L., “Analysis of excitability for the FitzHugh–Nagumo model via a stochastic sensitivity function technique”, Phys. Rev. E, 83:6 (2011), 061109, 8 pp. | DOI | MR

[9] Bashkirtseva I. A., Ryashko L. B., Slepukhina E. S., “Splitting bifurcation of stochastic cycles in the FitzHugh–Nagumo model”, Rus. J. Nonlin. Dyn., 9:2 (2013), 295–307 (in Russian)

[10] Bashkirtseva I., Ryashko L., Slepukhina E., “Noise-induced oscillation bistability and transition to chaos in FitzHugh–Nagumo model”, Fluctuation and Noise Letters, 13:1 (2014), 1450004, 16 pp. | DOI

[11] Gikhman I. I., Skorokhod A. V., Stochastic differential equations and its applications, Naukova dumka, Kiev, 1982, 612 pp. | MR

[12] Gardiner K. V., Stochastic methods in the natural sciences, Mir, M., 1986, 538 pp. | MR | Zbl

[13] Kurrer C., Schulten K., “Effect of noise and perturbations on limit cycle systems”, Phys. D, 50:3 (1991), 311–320 | DOI | MR | Zbl

[14] Mil'shtein G. N., Ryashko L. B., “The first approximation of the quasipotential in problems of the stability of systems with random nonsingular perturbations”, Prikl. Mat. Mekh., 59:1 (1995), 53–63 (in Russian) | MR

[15] Ventsel' A. D., Freidlin M. I., Fluctuations in dynamical systems under the small random perturbations, Nauka, M., 1979, 424 pp. | MR | Zbl

[16] Dembo A., Zeitouni O., Large deviations techniques and applications, Johnes and Bartlett Publishers, Boston–London, 1995, 346 pp. | MR

[17] Bashkirtseva I. A., Ryashko L. B., “Quasipotential method in the study of local stability of limit cycles to random perturbations”, Izv. Vyssh. Uchebn. Zaved., Prikl. Nelin. Din., 9:6 (2001), 104–113 (in Russian) | MR

[18] Bashkirtseva I., Ryashko L., “Sensitivity and chaos control for the forced nonlinear oscillations”, Chaos, Solitons and Fractals, 2005, no. 26, 1437–1451 | DOI | MR | Zbl

[19] Bashkirtseva I. A., Ryashko L. B., “Stochastic sensitivity of 3D-cycles”, Mathematics and Computers in Simulation, 66:1 (2004), 55–67 | DOI | MR | Zbl

[20] Bashkirtseva I., Ryashko L., “Sensitivity analysis of stochastic attractors and noise-induced transitions for population model with Allee effect”, Chaos, 21:4 (2011), 047514, 4 pp. | DOI