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

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.
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]

Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the Fitzhugh-Nagumo system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 775-794. doi: 10.3934/dcdsb.2020134

[2]

Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1549-1563. doi: 10.3934/dcdsb.2020172

[3]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344

[4]

Ziang Long, Penghang Yin, Jack Xin. Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data. Inverse Problems & Imaging, 2021, 15 (1) : 41-62. doi: 10.3934/ipi.2020077

[5]

Editorial Office. Retraction: Honggang Yu, An efficient face recognition algorithm using the improved convolutional neural network. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 901-901. doi: 10.3934/dcdss.2019060

[6]

Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342

[7]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084

[8]

Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021013

[9]

Simone Göttlich, Elisa Iacomini, Thomas Jung. Properties of the LWR model with time delay. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020032

[10]

Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147

[11]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003

[12]

Mohamed Dellal, Bachir Bar. Global analysis of a model of competition in the chemostat with internal inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1129-1148. doi: 10.3934/dcdsb.2020156

[13]

Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042

[14]

Guihong Fan, Gail S. K. Wolkowicz. Chaotic dynamics in a simple predator-prey model with discrete delay. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 191-216. doi: 10.3934/dcdsb.2020263

[15]

Hui Zhao, Zhengrong Liu, Yiren Chen. Global dynamics of a chemotaxis model with signal-dependent diffusion and sensitivity. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021011

[16]

Shujing Shi, Jicai Huang, Yang Kuang. Global dynamics in a tumor-immune model with an immune checkpoint inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1149-1170. doi: 10.3934/dcdsb.2020157

[17]

Yanhong Zhang. Global attractors of two layer baroclinic quasi-geostrophic model. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021023

[18]

Tin Phan, Bruce Pell, Amy E. Kendig, Elizabeth T. Borer, Yang Kuang. Rich dynamics of a simple delay host-pathogen model of cell-to-cell infection for plant virus. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 515-539. doi: 10.3934/dcdsb.2020261

[19]

Thazin Aye, Guanyu Shang, Ying Su. On a stage-structured population model in discrete periodic habitat: III. unimodal growth and delay effect. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021005

[20]

Kengo Nakai, Yoshitaka Saiki. Machine-learning construction of a model for a macroscopic fluid variable using the delay-coordinate of a scalar observable. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1079-1092. doi: 10.3934/dcdss.2020352

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (1)

[Back to Top]