P-Bifurcation of Stochastic van der Pol Model as a Dynamical System in Neuroscience

doi:10.1007/s42967-021-00176-9

摘要/Abstract

摘要： This study aims to determine the phenomenological bifurcation (P-bifurcation) occurring in the van der Pol (VDP) neuronal model of burst neurons with a random signal. We observe the P-bifurcation under an intense noise stimulus which would become chaotic transitions. Bursting and spiking simulations are used to describe the causes of chaotic transitions between two periodic phases that are the reason for the neuronal activities. Randomness plays a crucial role in detecting the P-bifurcation. To determine the equilibrium points, stability or instability of the stochastic VDP equation, and bifurcation, we use the stochastic averaging method and some related theorems. Apart from theoretical methods, we also examine numerical simulations in the particular case of that stochastic equation that illustrates the outcome of theorems for various noise types. The most striking part of our theoretical findings is that these results are also valid for the Izhikevich-FitzHugh model, Bonhoeffer-van der Pol oscillator in dynamical systems of neuroscience. Finally, we will discuss some applications of the VDP equation in neuronal activity.

关键词: Averaging diffusion system, P-bifurcation, Stability, Neuroscience, VDP stochastic equation

Abstract: This study aims to determine the phenomenological bifurcation (P-bifurcation) occurring in the van der Pol (VDP) neuronal model of burst neurons with a random signal. We observe the P-bifurcation under an intense noise stimulus which would become chaotic transitions. Bursting and spiking simulations are used to describe the causes of chaotic transitions between two periodic phases that are the reason for the neuronal activities. Randomness plays a crucial role in detecting the P-bifurcation. To determine the equilibrium points, stability or instability of the stochastic VDP equation, and bifurcation, we use the stochastic averaging method and some related theorems. Apart from theoretical methods, we also examine numerical simulations in the particular case of that stochastic equation that illustrates the outcome of theorems for various noise types. The most striking part of our theoretical findings is that these results are also valid for the Izhikevich-FitzHugh model, Bonhoeffer-van der Pol oscillator in dynamical systems of neuroscience. Finally, we will discuss some applications of the VDP equation in neuronal activity.

Key words: Averaging diffusion system, P-bifurcation, Stability, Neuroscience, VDP stochastic equation

中图分类号:

F. S. Mousavinejad, M. FatehiNia, A. Ebrahimi. P-Bifurcation of Stochastic van der Pol Model as a Dynamical System in Neuroscience[J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1293-1312.

参考文献

1. Abrevaya, G., Aravkin, A., Cecchi, G., Rish, I., Polosecki, P., Zheng, P., Dawson S.P.:Learning nonlinear brain dynamics:van der Pol meets LSTM. (2018). arXiv:1805.09874
2. Akman, O.:Analysis of a nonlinear dynamics model of the saccadic system. PhD thesis, University of Manchester Institute of Science and Technology (UMIST), Manchester, UK (2003)
3. Arnold, L.:Trends and open problems in the theory of random dynamical systems. In:Accardi, L., Heyde, C.C. (eds) Probability Towards 2000. Lecture Notes in Statistics, vol. 128, pp. 34-46. Springer, New York, NY (1998)
4. Bala, B.K., Arshad, F.M., Noh, K.M.:System dynamics:modelling and simulation. In:Springer Texts in Business and Economics. Springer (2017)
5. Bashkirtseva, I., Ryashko, L., Schurz, H.:Analysis of noise-induced transitions for Hopf system with additive and multiplicative random disturbances. Chaos Solit. Fract. 39(1), 72-82 (2009)
6. Dtchetgnia, S.R., Yamapi, R., Kofane, T.C., Aziz-Alaoui, M.A.:Deterministic and stochastic bifurcations in the hindmarsh-rose neuronal model. Chaos Interdiscipl. J. Nonlinear Sci. 23(3), 033125 (2013)
7. Gaudreault, M., Drolet, F., Vinals, J.:Bifurcation threshold of the delayed van der Pol oscillator under stochastic modulation. Phys. Rev. E 85(5), 056214 (2012)
8. Gilani, T., Hövel, P.:Dynamical systems in neuroscience. In:Comput Neurosci. The MIT Press (2012)
9. Ginoux, J.-M., Letellier, C.:Van der Pol and the history of relaxation oscillations:toward the emergence of a concept. Chaos Interdiscip. J. Nonlinear Sci. 22(2), 023120 (2012)
10. Huang, Z., Yang, Q.-G., Cao, J.:Stochastic stability and bifurcation analysis on Hopfield neural networks with noise. Exp. Syst. Appl. 38(8), 10437-10445 (2011)
11. Huang, Z.L., Zhu, W.Q.:Stochastic averaging of quasi-generalized Hamiltonian systems. Internat. J. Non-Linear Mech. 44(1), 71-80 (2009)
12. Izhikevich, E.M.:Dynamical systems in neuroscience. The geometry of excitability and bursting. In:Sejnowski, T.J., Poggio, T.A. (eds) Computational Neuroscience. The MIT Press, Cambridge, Massachusetts, London, England (2007)
13. Khas' minskii, R. Z.:The averaging principle for stochastic differential equations. Probl. Peredachi Inf. 4(2), 86-87 (1968)
14. Li, W., Wei, X., Zhao, J., Jin, Y.:Stochastic stability and bifurcation in a macroeconomic model. Chaos Solit. Fract. 31(3), 702-711 (2007)
15. Lin, Y.K., Cai, G.Q.:Probabilistic Structural Dynamics:Advanced Theory and Applications. McGraw-Hill, New York (1995)
16. Luo, C., Guo, S.:Stability and bifurcation of two-dimensional stochastic differential equations with multiplicative excitations. Bull. Malay. Math. Sci. Soc. 40(2), 795-817 (2017)
17. Mao, L.:A new understanding of particles by G-flow interpretation of differential equation. Prog. Phys. 11(3), 193 (2015)
18. Rosli, M.H.B.:Development of an integrated spatial distributed travel time method using GIS to model rainfall runoff in Bentong catchment in Peninsular Malaysia. Doctoral dissertation, Universiti putra malaysia, Malaysia (2018)
19. Tsatsos, M.:Theoretical and numerical study of the van der Pol equation. Doctoral desertation, Aristotle University of Thessaloniki, Thessaloniki (2006)
20. Wang, L.:Global well-posedness and stability of the mild solutions for a class of stochastic partial functional differential equations. Sci. Sinica Math. 47(3), 371-382 (2016)
21. Yang, Q., Zeng, C., Wang, C.:Fractional noise destroys or induces a stochastic bifurcation. Chaos Interdiscip. J. Nonlinear Sci. 23(4), 043120 (2013)
22. Zduniak, B., Bodnar, M., Foryś, U.:A modified van der Pol equation with delay in a description of the heart action. Internat. J. Appl. Math. Comput. Sci. 24(4), 853-863 (2014)
23. Zeng, C., Yang, Q., Chen, Y.Q.:Bifurcation dynamics of the tempered fractional Langevin equation. Chaos Interdiscip. J. Nonlinear Sci. 26(8), 084310 (2016)
24. Zhang, X., Wu, Z.:Dynamics of a horizontal saccadic oculomotor system with colored noise. Chinese J. Phys. 56(5), 2052-2060 (2018)
25. Zhang, Y., Wei, X., Fang, T.:Stochastic Hopf bifurcation and chaos of stochastic Bonhoeffer-van der Pol system via Chebyshev polynomial approximation. Appl. Math. Comput. 190(2), 1225-1236 (2007)

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