American Institute of Mathematical Sciences

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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

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

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