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Recently, the dynamics problems of nonlinear systems driven by noises have attracted considerable attention. The researches indicate that the noise plays a determinative role in system evolution. This irregular random interference does not always play a negative role in the macro order. Sometimes it can play a positive role. The various effects of noise are found in physics, biology, chemistry and other fields, such as noise-induced non-equilibrium phase transition, noise-enhanced system stability, stochastic resonance, etc. Especially, in the field of biology, the effects of noise on life process are significant. At present, a large number of researchers have studied the kinetic properties of the neuron system subjected to noises. However, these studies focus on the Gaussian noise, while the researches about non-Gaussian noise are less. In fact, it is found that all the noise sources among neuronal systems, physical systems and biological systems tend to non-Gaussian distribution. So it is reasonable to consider the effects of the non-Gaussian noise on systems, and it is closer to the actual process. Therefore, it has some practical significance to study the FHN system driven by the non-Gaussian noise and analyze the kinetic properties of this system. In this work, we study the stationary probability distribution (SPD) in FitzHugh-Nagumo (FHN) neural system driven by correlated multiplicative non-Gaussian noise and additive Gaussian white noise. Using the path integral approach and the unified colored approximation, the analytical expression of the stationary probability distribution is first derived, and then the change regulations of the SPD with the strength and relevance between multiplicative noise and additive noise are analyzed. After that, the simulation results show that the intensity of multiplicative noise, the intensity of additive noise, the correlation time of the non-Gaussian noise and the cross-correlation strength between noises can induce non-equilibrium phase transition. This means that the effect of the non-Gaussian noise intensity on SPD is the same as that of the Gaussian noise intensity. However, the non-Gaussian noise deviation parameter cannot induce non-equilibrium phase transition. Moreover, we also find that the increases of the multiplicative noise intensity and the cross-correlation strength between noises are conducive to the conversion of the exciting state into the resting state. And with the additive noise intensity and the correlation time increasing, the conversion of the resting state into the exciting state becomes obvious. Meanwhile, the increase of non-Gaussian noise deviation parameter increases the probability of staying in the resting state.
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Keywords:
- FHN neural system /
- non-Gaussian noise /
- stationary probability distribution /
- non-equilibrium phase transition
[1] Mangioni S, Deza R 2000 Phys. Rev. E 61 223
[2] Van den Broeck C, Parrondo J M R, Toral R 1994 Phys. Rev. Lett. 73 3395
[3] Hnggi P, Jung P, Zerbe C, Moss F 1993 J. Stat. Phys. 70 25
[4] He M J, Xu W, Sun Z K, Du L 2015 Commun. Nonlinear Sci. Numer. Simul. 28 39
[5] Sun Z K, Yang X L, Xu W 2012 Phys. Rev. E 85 061125
[6] Sun Z K, Yang X L, Xiao Y Z, Xu W 2014 Chaos 24 023126
[7] Sun Z K, Wu Y Z, Du L, Xu W 2016 Nonlinear Dyn. 84 1011
[8] Sun Z K, Yang X L 2011 Chaos 21 033114
[9] Sun Z K, Fu J, Xiao Y Z, Xu W 2015 Chaos 25 083102
[10] Sun Z K, Yang X L, Xu W 2016 Sci. China Technol. Sci. 59 403
[11] Yang X L, Senthilkumar D V, Sun Z K, Kurths J 2011 Chaos 21 047522
[12] Sun Z K, Lu P J, Xu W 2014 Acta Phys. Sin. 63 220503 (in Chinese) [孙中奎, 鲁捧菊, 徐伟 2014 63 220503]
[13] Bezrnkov S M, Vodyanoy I 1997 Nature 385 319
[14] Goychuk I, Hnggi P 2000 Phys. Rev. E 61 4272
[15] Hodgkin A L, Huxley A F 1952 Physiology 117 500
[16] Wiesenfeld K, Pierson D, Pantazelou E, Dames C, Moss F 1994 Phys. Rev. Lett 72 2125
[17] Tuckwell H C, Rodriguez R, Wan F Y M 2003 Neural Comput. 15 143159
[18] Acebron J A, Bulsara A R, Rappel W J 2004 Phys. Rev. E 69 026202
[19] Kitajima H, Kurths J 2005 Chaos 15 023704
[20] Fitzhhugh R 1960 J. Gen. Physiol. 43 867
[21] Alarcon T, Perez-Madrid A, Rubi J M 1998 Phys. Rev. E 57 4979
[22] Wang Z Q, Xu Y, Yang H 2016 Sci. China: Technol. Sci. 59 371
[23] Xiao Y Z, Tang S F, Sun Z K 2014 Eur. Phys. J. B 87 134
[24] Zhang J J, Jin Y F 2012 Acta Phys. Sin. 61 130502 (in Chinese) [张静静, 靳艳飞 2012 61 130502]
[25] Zhao Y, Xu W, Zou S C 2009 Acta Phys. Sin. 58 1396 (in Chinese) [赵燕, 徐伟, 邹少存 2009 58 1396]
[26] Gardiner C W 1985 Handbook of Stochastic Methods (Berlin: Springer-Verlag) pp80-115
[27] Bouzat S, Wio H S 2005 Physica A 351 69
[28] Fuentes M A, Wio H S, Toral R 2002 Physica A 303 91
[29] Wio H S, Colet P, San Miguel M, Pesquera L, Rodrguez M A 1989 Phys. Rev. A 40 7312
[30] Wu D, Zhu S Q 2007 Phys. Lett. A 363 202
[31] Jung P, Hnggi P 1987 Phys. Rev. A 35 4464
[32] Cao L, Wu D J, Ke S Z 1995 Phys. Rev. E 52 3228
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[1] Mangioni S, Deza R 2000 Phys. Rev. E 61 223
[2] Van den Broeck C, Parrondo J M R, Toral R 1994 Phys. Rev. Lett. 73 3395
[3] Hnggi P, Jung P, Zerbe C, Moss F 1993 J. Stat. Phys. 70 25
[4] He M J, Xu W, Sun Z K, Du L 2015 Commun. Nonlinear Sci. Numer. Simul. 28 39
[5] Sun Z K, Yang X L, Xu W 2012 Phys. Rev. E 85 061125
[6] Sun Z K, Yang X L, Xiao Y Z, Xu W 2014 Chaos 24 023126
[7] Sun Z K, Wu Y Z, Du L, Xu W 2016 Nonlinear Dyn. 84 1011
[8] Sun Z K, Yang X L 2011 Chaos 21 033114
[9] Sun Z K, Fu J, Xiao Y Z, Xu W 2015 Chaos 25 083102
[10] Sun Z K, Yang X L, Xu W 2016 Sci. China Technol. Sci. 59 403
[11] Yang X L, Senthilkumar D V, Sun Z K, Kurths J 2011 Chaos 21 047522
[12] Sun Z K, Lu P J, Xu W 2014 Acta Phys. Sin. 63 220503 (in Chinese) [孙中奎, 鲁捧菊, 徐伟 2014 63 220503]
[13] Bezrnkov S M, Vodyanoy I 1997 Nature 385 319
[14] Goychuk I, Hnggi P 2000 Phys. Rev. E 61 4272
[15] Hodgkin A L, Huxley A F 1952 Physiology 117 500
[16] Wiesenfeld K, Pierson D, Pantazelou E, Dames C, Moss F 1994 Phys. Rev. Lett 72 2125
[17] Tuckwell H C, Rodriguez R, Wan F Y M 2003 Neural Comput. 15 143159
[18] Acebron J A, Bulsara A R, Rappel W J 2004 Phys. Rev. E 69 026202
[19] Kitajima H, Kurths J 2005 Chaos 15 023704
[20] Fitzhhugh R 1960 J. Gen. Physiol. 43 867
[21] Alarcon T, Perez-Madrid A, Rubi J M 1998 Phys. Rev. E 57 4979
[22] Wang Z Q, Xu Y, Yang H 2016 Sci. China: Technol. Sci. 59 371
[23] Xiao Y Z, Tang S F, Sun Z K 2014 Eur. Phys. J. B 87 134
[24] Zhang J J, Jin Y F 2012 Acta Phys. Sin. 61 130502 (in Chinese) [张静静, 靳艳飞 2012 61 130502]
[25] Zhao Y, Xu W, Zou S C 2009 Acta Phys. Sin. 58 1396 (in Chinese) [赵燕, 徐伟, 邹少存 2009 58 1396]
[26] Gardiner C W 1985 Handbook of Stochastic Methods (Berlin: Springer-Verlag) pp80-115
[27] Bouzat S, Wio H S 2005 Physica A 351 69
[28] Fuentes M A, Wio H S, Toral R 2002 Physica A 303 91
[29] Wio H S, Colet P, San Miguel M, Pesquera L, Rodrguez M A 1989 Phys. Rev. A 40 7312
[30] Wu D, Zhu S Q 2007 Phys. Lett. A 363 202
[31] Jung P, Hnggi P 1987 Phys. Rev. A 35 4464
[32] Cao L, Wu D J, Ke S Z 1995 Phys. Rev. E 52 3228
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