# American Institute of Mathematical Sciences

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 and 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), 3088-3092. doi: 10.1073/pnas.81.10.3088. [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, Solitons & Fractals, 40 (2009), 377-397. doi: 10.1016/j.chaos.2007.08.040. [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), 29-60. doi: 10.1137/040614207. [4] H. Kitajima and H. Kawakami, Bifurcations in synaptically coupled neurons with external impulsive forces, IEIC Technical Report, 183 (2003), 33-36. [5] B. Nikola and T. Dragana, Dynamics of Fitzugh-Nagumo excitable systems with delayed coupling, Phys. Rev. E, 67 (2003), 066222. doi: 10.1103/PhysRevE.67.066222. [6] B. Nikola, I. Grozdanovi and N. Vasovi, Type I vs. type II excitable systems with delayed coupling, Chaos, Solitons & Fractals, 4 (2005), 1221-1233. [7] Q. Y. Wang, Q. S. Lu and G. R. Chen, Bifurcation and synchronization of synaptically coupled FHN models with time delay, Chaos, Solitons & Fractals, 39 (2009), 918-925. doi: 10.1016/j.chaos.2007.01.061. [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), 106-119. doi: 10.1016/j.physd.2004.08.023. [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), 689-695. doi: 10.1016/j.physleta.2006.08.078. [10] X. P. Yan, Hopf bifurcation and stability for a delayed tri-neuron network model, J. Comp. Appl. Math., 196 (2006), 579-595. doi: 10.1016/j.cam.2005.10.012. [11] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Application of Hopf Bifurcation," Cambridge University Press, Cambridge, 1981. [12] J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838. doi: 10.1090/S0002-9947-98-02083-2. [13] M. Y. Li and J. Muldowney, On Bendixson's criterion, J. Differential Equations, 106 (1994), 27-39. doi: 10.1006/jdeq.1993.1097. [14] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6. [15] J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070. [16] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, INC, 1993. [17] J. Hale, "Theory of Functional Differential Equations," Springer, New York, 1977.

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), 3088-3092. doi: 10.1073/pnas.81.10.3088. [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, Solitons & Fractals, 40 (2009), 377-397. doi: 10.1016/j.chaos.2007.08.040. [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), 29-60. doi: 10.1137/040614207. [4] H. Kitajima and H. Kawakami, Bifurcations in synaptically coupled neurons with external impulsive forces, IEIC Technical Report, 183 (2003), 33-36. [5] B. Nikola and T. Dragana, Dynamics of Fitzugh-Nagumo excitable systems with delayed coupling, Phys. Rev. E, 67 (2003), 066222. doi: 10.1103/PhysRevE.67.066222. [6] B. Nikola, I. Grozdanovi and N. Vasovi, Type I vs. type II excitable systems with delayed coupling, Chaos, Solitons & Fractals, 4 (2005), 1221-1233. [7] Q. Y. Wang, Q. S. Lu and G. R. Chen, Bifurcation and synchronization of synaptically coupled FHN models with time delay, Chaos, Solitons & Fractals, 39 (2009), 918-925. doi: 10.1016/j.chaos.2007.01.061. [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), 106-119. doi: 10.1016/j.physd.2004.08.023. [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), 689-695. doi: 10.1016/j.physleta.2006.08.078. [10] X. P. Yan, Hopf bifurcation and stability for a delayed tri-neuron network model, J. Comp. Appl. Math., 196 (2006), 579-595. doi: 10.1016/j.cam.2005.10.012. [11] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Application of Hopf Bifurcation," Cambridge University Press, Cambridge, 1981. [12] J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838. doi: 10.1090/S0002-9947-98-02083-2. [13] M. Y. Li and J. Muldowney, On Bendixson's criterion, J. Differential Equations, 106 (1994), 27-39. doi: 10.1006/jdeq.1993.1097. [14] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6. [15] J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070. [16] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, INC, 1993. [17] J. Hale, "Theory of Functional Differential Equations," Springer, New York, 1977.
 [1] Joachim Crevat. Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network. Kinetic and Related Models, 2019, 12 (6) : 1329-1358. doi: 10.3934/krm.2019052 [2] Francesco Cordoni, Luca Di Persio. Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable. Evolution Equations and Control Theory, 2018, 7 (4) : 571-585. doi: 10.3934/eect.2018027 [3] 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 [4] Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the Fitzhugh-Nagumo system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 775-794. doi: 10.3934/dcdsb.2020134 [5] Arnold Dikansky. Fitzhugh-Nagumo equations in a nonhomogeneous medium. Conference Publications, 2005, 2005 (Special) : 216-224. doi: 10.3934/proc.2005.2005.216 [6] Anna Cattani. FitzHugh-Nagumo equations with generalized diffusive coupling. Mathematical Biosciences & Engineering, 2014, 11 (2) : 203-215. doi: 10.3934/mbe.2014.11.203 [7] B. Ambrosio, M. A. Aziz-Alaoui, V. L. E. Phan. Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3787-3797. doi: 10.3934/dcdsb.2018077 [8] Gaetana Gambino, Valeria Giunta, Maria Carmela Lombardo, Gianfranco Rubino. Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022063 [9] Yiqiu Mao. Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3935-3947. doi: 10.3934/dcdsb.2018118 [10] Vyacheslav Maksimov. Some problems of guaranteed control of the Schlögl and FitzHugh-Nagumo systems. Evolution Equations and Control Theory, 2017, 6 (4) : 559-586. doi: 10.3934/eect.2017028 [11] John Guckenheimer, Christian Kuehn. Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 851-872. doi: 10.3934/dcdss.2009.2.851 [12] Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1549-1563. doi: 10.3934/dcdsb.2020172 [13] Anhui Gu, Bixiang Wang. Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1689-1720. doi: 10.3934/dcdsb.2018072 [14] Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHugh-Nagumo system on the time-varying domains. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150 [15] Jyoti Mishra. Analysis of the Fitzhugh Nagumo model with a new numerical scheme. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 781-795. doi: 10.3934/dcdss.2020044 [16] Boris Anicet Guimfack, Conrad Bertrand Tabi, Alidou Mohamadou, Timoléon Crépin Kofané. Stochastic dynamics of the FitzHugh-Nagumo neuron model through a modified Van der Pol equation with fractional-order term and Gaussian white noise excitation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2229-2243. doi: 10.3934/dcdss.2020397 [17] Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026 [18] Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106 [19] Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101 [20] Willem M. Schouten-Straatman, Hermen Jan Hupkes. Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5017-5083. doi: 10.3934/dcds.2019205

2021 Impact Factor: 1.497