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http://dx.doi.org/10.5831/HMJ.2022.44.3.402

BIFURCATIONS OF STOCHASTIC IZHIKEVICH-FITZHUGH MODEL  

Nia, Mehdi Fatehi (Department of Mathematics, Yazd University)
Mirzavand, Elaheh (Department of Mathematics, Yazd University)
Publication Information
Honam Mathematical Journal / v.44, no.3, 2022 , pp. 402-418 More about this Journal
Abstract
Noise is a fundamental factor to increased validity and regularity of spike propagation and neuronal firing in the nervous system. In this paper, we examine the stochastic version of the Izhikevich-FitzHugh neuron dynamical model. This approach is based on techniques presented by Luo and Guo, which provide a general framework for the bifurcation and stability analysis of two dimensional stochastic dynamical system as an Itô averaging diffusion system. By using largest lyapunov exponent, local and global stability of the stochastic system at the equilibrium point are investigated. We focus on the two kinds of stochastic bifurcations: the P-bifurcation and the D-bifurcations. By use of polar coordinate, Taylor expansion and stochastic averaging method, it is shown that there exists choices of diffusion and drift parameters such that these bifurcations occurs. Finally, numerical simulations in various viewpoints, including phase portrait, evolution in time and probability density, are presented to show the effects of the diffusion and drift coefficients that illustrate our theoretical results.
Keywords
Stochastic systems; Izhikevich-FitzHugh model; Lyapunov exponent; Stability; P-bifurcation;
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1 R. Z. Khas'minskii, Necessary and Sufficient Conditions for the Asymptotic Stability of Linear Stochastic Systems, Theory of Probability and its Applications 12 (1967), no. 1, 144-147.   DOI
2 C. Luo and S. Guo, Stability and Bifurcation of Two-dimensional Stochastic Differential Equations with Multiplicative Excitations, Bull. Malays. Math. Sci. Soc. 40 (2017), no. 2, 795-817.   DOI
3 X. Mao, Stochastic differential equations and applications, Woodhead Publishing, UK, 2007.
4 C. Rocsoreanu, A. Georgescu, and N. Giurgiteanu, The FitzHugh Nagumo model: Bifurcations and Dynamics, Kluver Academic Publishers Boston, 2000.
5 M. E. Yamakou, T. D. Tran, L. H. Duc, and J. M. Jost, Stochastic FitzHugh-Nagumo neuron model in excitable regime embeds a leaky integrate-and-fire model, Journal of Mathematical Biology 79 (2019) 509-532.   DOI
6 U. Wagner and W. V. Weding, On the calculation of stationary solutions of multidimensional Fokker-Planck equation by orthogonal function, Nonlinear Dynamics 29 (2000), 283-306.
7 N. Berglund and D. Landon, Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh-Nagumo model, Nonlinearity 25 (2012), no. 8, 2303-2335.   DOI
8 D. Brown, J. Feng, and S. Feerick, Variability of firing of Hodgkin-Huxley and FitzHughNagumo neurons with stochastic synaptic input, Phys. Rev. Lett. 82 (1999), no. 7, 4731-4734.   DOI
9 T. Caraballo, J. A. Langa, and J. C. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation, Proceedings Mathematical Physical and Engineering Sciences 457 (2013), no. 2013, 2041-2061.
10 L. Arnold, Random Dynamical Systems. Berlin, Springer, 2007.
11 S. R. D. Dtchetgnia, R. Yamapi, T. C. Kofane, and M. A. Aziz-Alaoui, Deterministic and Stochastic Bifurcations in the Hindmarsh-Rose neuronal Model, Chaos 23 (2013), no. 3, 033125.   DOI
12 B. Ermentrout, Type I membranes, phase resetting curves, and synchrony, Neural Comput. 8 (1996), no. 5, 979-1001.   DOI
13 A. Hodgkin and A. Huxley, A quantitative description of membrane current and application to conduction and excitation in nerve, The Journal of Physiology 117 (1952), no. 4, 500-544.   DOI
14 E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, The MIT Press, 2007.
15 C. Laing and G. J. Lord, Stochastic Methods in neuroscience. Clarendon Press. Oxford, 2008.
16 Y. Liang and N. S. Namachchivaya, P-Bifurcations in the Noisy Duffing-van der Pol Equation, Stochastic Dynamics. (Bremen, 1997), 49-70, Springer, New York, 1999.
17 M. Ringqvist, On Dynamical Behaviour of FitzHugh-Nagumo Systems, Filosofie licentiatavhandling, 2006.
18 J. H. Yang, Miguel A.F. Sanju'an, H. G. Liu, and X. Li, Stochastic P-bifurcation and stochastic resonance in a noisy bistable fractional-order system, Commun Nonlinear Sci Numer Simulat. 41 (2016), 104-117 .   DOI
19 M. H. Akrami and M. Fatehi Nia, Stochastic Stability and Bifurcation for the Selkov Model with Noise, Iranian Journal of Mathematical Chemistry 12 (2021), no.1, 39-55.
20 M. Fatehi Nia and M. H. Akrami, Stability and bifurcation in a stochastic vocal folds model, Communications in Nonlinear Science and Numerical Simula- tion 79, (2019), 104898.   DOI
21 N. Berglund and B. Gentz, Pathwise description of dynamic pitchfork bifurcations with additive noise, Probability Theory and Related Fields 122 (2002), 341-388.   DOI
22 S. Bonaccorsi and E. Mastrogiacomo, Analysis of the stochastic FitzHugh-Nagumo system, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11 (2008), no. 3, 427-446.   DOI
23 S. Chakraborty, S. Pal, and N. Bairagi, Predator-prey fishery model under deterministic and stochastic environments: a mathematical perspective, International Journal of Dynamical Systems and Differential Equations 4 (2012), no. 3, 215-241.   DOI