Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme

Previous Article
Bursting and two-parameter bifurcation in the Chay neuronal model
DCDS-B Home
This Issue
Next Article
Converting a general 3-D autonomous quadratic system to an extended Lorenz-type system

September 2011, 16(2): 457-474. doi: 10.3934/dcdsb.2011.16.457

Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme

Fang Han ^1, , Bin Zhen ^2, , Ying Du ^3, , Yanhong Zheng ^4, and Marian Wiercigroch ^5,

1.	School of Information Science and Technology, Donghua University, Shanghai 201620, China
2.	School of Civil and Architecture Engineering, Wuhan University of Technology, Wuhan 430070, China
3.	Institute for Cognitive Neurodynamics,School of Science, East China University of Science and Technology, Shanghai, 200237, China
4.	School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, China
5.	Centre for Applied Dynamics Research, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom

Received March 2010 Revised December 2010 Published June 2011

Abstract
Related Papers

Global Hopf bifurcation analysis is carried out on a six-dimensional FitzHugh-Nagumo (FHN) neural network with a time delay. First, the existence of local Hopf bifurcations of the system is investigated and the explicit formulae which can determine the direction of the bifurcations and the stability of the periodic solutions are derived using the normal form method and the center manifold theory. Then the sufficient conditions for the system to have multiple periodic solutions when the delay is far away from the critical values of Hopf bifurcations are obtained by using the Wu's global Hopf bifurcation theory and the Bendixson's criterion. Especially, a synchronized scheme is used during the analysis to reduce the dimension of the system. Finally, example numerical simulations are given to support the theoretical analysis.

Keywords: delay, FitzHugh-Nagumo model, synchronize., Global Hopf bifurcation, neural network.

Mathematics Subject Classification: Primary: 34C23; Secondary: 34C2.

Citation: Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457

References:

[1]	J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons,, Proc. Natl. Acad. Sci. USA, 81 (1984), 3088. doi: 10.1073/pnas.81.10.3088. Google Scholar
[2]	Q. S. Lu, Z. Q. Yang, L. X. Duan, H. G. Gu and W. Ren, Dynamics and transitions of firing patterns in deterministic and stochastic neuronal systems,, Chaos, 40 (2009), 377. doi: 10.1016/j.chaos.2007.08.040. Google Scholar
[3]	J. Xu, K. W. Chung and C. L. Chan, An efficient method for studying weak resonant double Hopf bifurcation in nonlinear systems with delayed feedbacks,, SIAM J. Applied Dynamical Systems, 6 (2007), 29. doi: 10.1137/040614207. Google Scholar
[4]	H. Kitajima and H. Kawakami, Bifurcations in synaptically coupled neurons with external impulsive forces,, IEIC Technical Report, 183 (2003), 33. Google Scholar
[5]	B. Nikola and T. Dragana, Dynamics of Fitzugh-Nagumo excitable systems with delayed coupling,, Phys. Rev. E, 67 (2003). doi: 10.1103/PhysRevE.67.066222. Google Scholar
[6]	B. Nikola, I. Grozdanovi and N. Vasovi, Type I vs. type II excitable systems with delayed coupling,, Chaos, 4 (2005), 1221. Google Scholar
[7]	Q. Y. Wang, Q. S. Lu and G. R. Chen, Bifurcation and synchronization of synaptically coupled FHN models with time delay,, Chaos, 39 (2009), 918. doi: 10.1016/j.chaos.2007.01.061. Google Scholar
[8]	J. J. Wei and M. Y. Li, Global existence of periodic solutions in a tri-neuron network model with delays,, Physica D, 198 (2004), 106. doi: 10.1016/j.physd.2004.08.023. Google Scholar
[9]	C. J. Sun, M. A. Han and X. M. Pang, Global Hopf bifurcation analysis on a BAM neural network with delays,, Physics Letters A, 360 (2007), 689. doi: 10.1016/j.physleta.2006.08.078. Google Scholar
[10]	X. P. Yan, Hopf bifurcation and stability for a delayed tri-neuron network model,, J. Comp. Appl. Math., 196 (2006), 579. doi: 10.1016/j.cam.2005.10.012. Google Scholar
[11]	B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Application of Hopf Bifurcation,", Cambridge University Press, (1981). Google Scholar
[12]	J. Wu, Symmetric functional differential equations and neural networks with memory,, Trans. Amer. Math. Soc., 350 (1998), 4799. doi: 10.1090/S0002-9947-98-02083-2. Google Scholar
[13]	M. Y. Li and J. Muldowney, On Bendixson's criterion,, J. Differential Equations, 106 (1994), 27. doi: 10.1006/jdeq.1993.1097. Google Scholar
[14]	R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophys. J., 1 (1961), 445. doi: 10.1016/S0006-3495(61)86902-6. Google Scholar
[15]	J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon,, Proc. IRE, 50 (1962), 2061. Google Scholar
[16]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Academic Press, (1993). Google Scholar
[17]	J. Hale, "Theory of Functional Differential Equations,", Springer, (1977). Google Scholar

show all references

References:

[1]	J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons,, Proc. Natl. Acad. Sci. USA, 81 (1984), 3088. doi: 10.1073/pnas.81.10.3088. Google Scholar
[2]	Q. S. Lu, Z. Q. Yang, L. X. Duan, H. G. Gu and W. Ren, Dynamics and transitions of firing patterns in deterministic and stochastic neuronal systems,, Chaos, 40 (2009), 377. doi: 10.1016/j.chaos.2007.08.040. Google Scholar
[3]	J. Xu, K. W. Chung and C. L. Chan, An efficient method for studying weak resonant double Hopf bifurcation in nonlinear systems with delayed feedbacks,, SIAM J. Applied Dynamical Systems, 6 (2007), 29. doi: 10.1137/040614207. Google Scholar
[4]	H. Kitajima and H. Kawakami, Bifurcations in synaptically coupled neurons with external impulsive forces,, IEIC Technical Report, 183 (2003), 33. Google Scholar
[5]	B. Nikola and T. Dragana, Dynamics of Fitzugh-Nagumo excitable systems with delayed coupling,, Phys. Rev. E, 67 (2003). doi: 10.1103/PhysRevE.67.066222. Google Scholar
[6]	B. Nikola, I. Grozdanovi and N. Vasovi, Type I vs. type II excitable systems with delayed coupling,, Chaos, 4 (2005), 1221. Google Scholar
[7]	Q. Y. Wang, Q. S. Lu and G. R. Chen, Bifurcation and synchronization of synaptically coupled FHN models with time delay,, Chaos, 39 (2009), 918. doi: 10.1016/j.chaos.2007.01.061. Google Scholar
[8]	J. J. Wei and M. Y. Li, Global existence of periodic solutions in a tri-neuron network model with delays,, Physica D, 198 (2004), 106. doi: 10.1016/j.physd.2004.08.023. Google Scholar
[9]	C. J. Sun, M. A. Han and X. M. Pang, Global Hopf bifurcation analysis on a BAM neural network with delays,, Physics Letters A, 360 (2007), 689. doi: 10.1016/j.physleta.2006.08.078. Google Scholar
[10]	X. P. Yan, Hopf bifurcation and stability for a delayed tri-neuron network model,, J. Comp. Appl. Math., 196 (2006), 579. doi: 10.1016/j.cam.2005.10.012. Google Scholar
[11]	B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Application of Hopf Bifurcation,", Cambridge University Press, (1981). Google Scholar
[12]	J. Wu, Symmetric functional differential equations and neural networks with memory,, Trans. Amer. Math. Soc., 350 (1998), 4799. doi: 10.1090/S0002-9947-98-02083-2. Google Scholar
[13]	M. Y. Li and J. Muldowney, On Bendixson's criterion,, J. Differential Equations, 106 (1994), 27. doi: 10.1006/jdeq.1993.1097. Google Scholar
[14]	R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophys. J., 1 (1961), 445. doi: 10.1016/S0006-3495(61)86902-6. Google Scholar
[15]	J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon,, Proc. IRE, 50 (1962), 2061. Google Scholar
[16]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Academic Press, (1993). Google Scholar
[17]	J. Hale, "Theory of Functional Differential Equations,", Springer, (1977). Google Scholar

[1]	Francesco Cordoni, Luca Di Persio. Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable. Evolution Equations & Control Theory, 2018, 7 (4) : 571-585. doi: 10.3934/eect.2018027
[2]	Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
[3]	Arnold Dikansky. Fitzhugh-Nagumo equations in a nonhomogeneous medium. Conference Publications, 2005, 2005 (Special) : 216-224. doi: 10.3934/proc.2005.2005.216
[4]	Anna Cattani. FitzHugh-Nagumo equations with generalized diffusive coupling. Mathematical Biosciences & Engineering, 2014, 11 (2) : 203-215. doi: 10.3934/mbe.2014.11.203
[5]	B. Ambrosio, M. A. Aziz-Alaoui, V. L. E. Phan. Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3787-3797. doi: 10.3934/dcdsb.2018077
[6]	Vyacheslav Maksimov. Some problems of guaranteed control of the Schlögl and FitzHugh-Nagumo systems. Evolution Equations & Control Theory, 2017, 6 (4) : 559-586. doi: 10.3934/eect.2017028
[7]	John Guckenheimer, Christian Kuehn. Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 851-872. doi: 10.3934/dcdss.2009.2.851
[8]	Anhui Gu, Bixiang Wang. Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1689-1720. doi: 10.3934/dcdsb.2018072
[9]	Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHugh-Nagumo system on the time-varying domains. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150
[10]	Yiqiu Mao. Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3935-3947. doi: 10.3934/dcdsb.2018118
[11]	Jyoti Mishra. Analysis of the Fitzhugh Nagumo model with a new numerical scheme. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 781-795. doi: 10.3934/dcdss.2020044
[12]	Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026
[13]	Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101
[14]	Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106
[15]	Yangrong Li, Jinyan Yin. A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1203-1223. doi: 10.3934/dcdsb.2016.21.1203
[16]	Abiti Adili, Bixiang Wang. Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 643-666. doi: 10.3934/dcdsb.2013.18.643
[17]	Bao Quoc Tang. Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 441-466. doi: 10.3934/dcds.2015.35.441
[18]	Matthieu Alfaro, Hiroshi Matano. On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1639-1649. doi: 10.3934/dcdsb.2012.17.1639
[19]	Abiti Adili, Bixiang Wang. Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise. Conference Publications, 2013, 2013 (special) : 1-10. doi: 10.3934/proc.2013.2013.1
[20]	Willem M. Schouten-Straatman, Hermen Jan Hupkes. Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5017-5083. doi: 10.3934/dcds.2019205

2018 Impact Factor: 1.008

American Institute of Mathematical Sciences