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June  2017, 10(3): 523-542. doi: 10.3934/dcdss.2017026

## Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay

 1 Department of mathematics, Yunnan Normal University, Kunming, Yunnan 650500, China 2 School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, TX 78539, USA

* Corresponding author

Received  December 2015 Revised  November 2016 Published  February 2017

Fund Project: The authors are supported by NNSFC (11562021/11572278/11526182) and the Science Foundations (2014FB138/2015FB140/YJG2014-B07) of Yunnan Province

In this paper, we study a coupled FitzHugh-Nagumo (FHN) neurons model with time delay. The existence conditions on Hopf-pitchfork singularity are given. By selecting the coupling strength and time delay as the bifurcation parameters, and by means of the center manifold reduction and normal form theory, the normal form for this singularity is found to analyze the behaviors of the system. We perform the bifurcation analysis and numerical simulations, and present the bifurcation diagrams. Some interesting phenomena are observed, such as the existence of a stable fixed point, a stable periodic solution, a pair of stable fixed points, and the coexistence of a pair of stable fixed points and a stable periodic solution near the Hopf-pitchfork critical point.

Citation: 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
##### References:
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Magalhaes, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations, 122 (1995), 201-224. doi: 10.1006/jdeq.1995.1145. [13] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurca-tions of Vector Fields, New York, Springer-Verlag, 1990. doi: 10.1007/978-1-4612-1140-2. [14] A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane and its application to conduction and excitation in nerve, J. Physiol, 117 (1952), 500-544. [15] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation, Cambridge, Cambridge Univ. Press, 1981. [16] J. Hale, Theory of Functional Differential Equations, Berlin, Springer-Verlag, 1977. [17] J. Jia, H. Liu, C. Xu and F. Yan, Dynamic effects of time delay on a coupled fitzhugh-nagumo neural system, Alexandria Eng. J, 54 (2015), 241-250. doi: 10.1016/j.aej.2015.03.006. [18] W. Jiang and B. Niu, On the coexistence of periodic or quasi-periodic oscillations near a Hopf-pitchfork bifurcation in NFDE, Commun Nonlinear Sci Numer Simulat, 18 (2013), 464-477. doi: 10.1016/j.cnsns.2012.08.004. [19] T. Kunichika, Y. Tetsuya and K. Hiroshi, Bifurcations in synaptically coupled BVP neurons, Int. J. Bifurcat. Chaos, 11 (2001), 1053-1064. [20] Y. A. Kuznetsov, Elements of Applied Nonlinear Dynamical Systems and Chaos, New York, Springer, 2004. [21] H. Li, X. Liao, C. Li, H. Huang and C. Li, Edge detection of noisy images based on cellu-lar neural networks, Communications in Nonlinear Science and Numerical Simulations, 16 (2011), 3746-3759. doi: 10.1016/j.cnsns.2010.12.017. [22] J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070. [23] B. Nikola and T. Dragana, Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling, Phys. Rev. E, 67 (2003), 066222, 15 pp. [24] J. Plaza, A. Plaza, R. Perez and P. Martinez, On the use of small training sets for neu-ral network-based characterization of mixed pixels in remotely sensed hyperspectral images, Pattern Recognition, 42 (2009), 3032-3045. [25] S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delayed differential equations with twodelays, Dyn. Cont. Disc. Impul. Syst. A: Math. Anal, 10 (2003), 863-874. [26] Y. Song, M. Han and J. Wei, Stability and Hopf bifurcation on a simplified BAM neural network with delays, Physica D, 200 (2005), 185-204. doi: 10.1016/j.physd.2004.10.010. [27] U. Tetsushi, M. Hisayo, K. Hiroshi and K. Takuji, Bifurcation and chaos in coupled BVP oscillators, Int. J. Bifurcat. Chaos, 14 (2004), 1305-1324. [28] U. Tetsushi and K. Hiroshi, Bifurcation in asymmetrically coupled BVP oscillators, Int. J. Bifurcat. Chaos, 13 (2003), 1319-1327. [29] Q. Wang, Q. Lu, G. R. Chen, Z. Feng and L. X. Duan, Bifurcation and synchronization of synaptically coupled FHN models with time delay, Chaos Soliton Fract, 39 (2009), 918-25. [30] H. Wang and W. Jiang, Hopf-pitchfork bifurcation in van der Pol's oscillator with nonlinear delayed feedback, J. Math. Anal. Appl, 368 (2010), 9-18. doi: 10.1016/j.jmaa.2010.03.012. [31] Z. Zeng and J. Wang, Analysis and design of associative memories based on recurrent neural networks with linear saturation activation functions and time-varying delays, Neural compu-tation, 19 (2007), 2149-2182. doi: 10.1162/neco.2007.19.8.2149. [32] Z. Zeng, D. S. Huang and Z. Wang, Pattern memory analysis based on stability theory of cellular neural networks, Applied Mathematical Modelling, 32 (2008), 112-121. [33] Z. Zeng and J. Wang, Design and analysis of high-capacity associative memories based on a class of discrete-time recurrent neural networks, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 38 (2008), 1525-1536.

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##### References:
 [1] S. I. Amari and A Cichocki, Adaptive blind signal processing-neural network approaches, Proceedings of the IEEE, 86 (1998), 2026-2048. doi: 10.1109/5.720251. [2] F. Ahmadkhanlou and H. Adeli, Optimum cost design of reinforced concrete slabs using neural dynamics model, Engineering Applications of Artificial Intelligence, 18 (2005), 65-72. doi: 10.1016/j.engappai.2004.08.025. [3] A. N. Bautin, Qualitative investigation of a particular nonlinear system, J. Appl. Math. Mech, 39 (1975), 606-615. doi: 10.1016/0021-8928(75)90061-1. [4] L. O. Chua and H. Kokubu, Normal forms for nonlinear vector fields-part Ⅰ: Theory and algorithm, IEEE. Trans. Circuits Syst, 35 (1988), 863-880. doi: 10.1109/31.1833. [5] L. O. Chua and H. Kokubu, Normal forms for nonlinear vector fields-part Ⅱ: Applications, IEEE. Trans. Circuits Syst, 1988 (36), 51-70. doi: 10.1109/31.16563. [6] J. Carr, Applications of Centre Manifold Theory, New York, Springer-Verlag, 1981. [7] S. N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, New York, Cambridge University Press, 1994. doi: 10.1017/CBO9780511665639. [8] Y. Ding, W. Jiang and P. Yu, Bifurcation analysis in a recurrent neural network model with delays, Commun Nonlinear Sci Numer Simulat, 18 (2013), 351-372. doi: 10.1016/j.cnsns.2012.07.002. [9] R. FitzHugh, Impulses and physiological state in theoretical models of nerve membrane, Biophys J, 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6. [10] T. Faria and L. T. Magalhaes, Normal form for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. Differential Equations, 122 (1995), 181-200. doi: 10.1006/jdeq.1995.1144. [11] D. Fan and L. Hong, Hopf bifurcation analysis in a synaptically coupled FHN neuron model with delays, Commun Nonlinear Sci Numer Simulat, 2010 (15), 1873-1886. doi: 10.1016/j.cnsns.2009.07.025. [12] T. Faria and L. T. Magalhaes, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations, 122 (1995), 201-224. doi: 10.1006/jdeq.1995.1145. [13] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurca-tions of Vector Fields, New York, Springer-Verlag, 1990. doi: 10.1007/978-1-4612-1140-2. [14] A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane and its application to conduction and excitation in nerve, J. Physiol, 117 (1952), 500-544. [15] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation, Cambridge, Cambridge Univ. Press, 1981. [16] J. Hale, Theory of Functional Differential Equations, Berlin, Springer-Verlag, 1977. [17] J. Jia, H. Liu, C. Xu and F. Yan, Dynamic effects of time delay on a coupled fitzhugh-nagumo neural system, Alexandria Eng. J, 54 (2015), 241-250. doi: 10.1016/j.aej.2015.03.006. [18] W. Jiang and B. Niu, On the coexistence of periodic or quasi-periodic oscillations near a Hopf-pitchfork bifurcation in NFDE, Commun Nonlinear Sci Numer Simulat, 18 (2013), 464-477. doi: 10.1016/j.cnsns.2012.08.004. [19] T. Kunichika, Y. Tetsuya and K. Hiroshi, Bifurcations in synaptically coupled BVP neurons, Int. J. Bifurcat. Chaos, 11 (2001), 1053-1064. [20] Y. A. Kuznetsov, Elements of Applied Nonlinear Dynamical Systems and Chaos, New York, Springer, 2004. [21] H. Li, X. Liao, C. Li, H. Huang and C. Li, Edge detection of noisy images based on cellu-lar neural networks, Communications in Nonlinear Science and Numerical Simulations, 16 (2011), 3746-3759. doi: 10.1016/j.cnsns.2010.12.017. [22] J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070. [23] B. Nikola and T. Dragana, Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling, Phys. Rev. E, 67 (2003), 066222, 15 pp. [24] J. Plaza, A. Plaza, R. Perez and P. Martinez, On the use of small training sets for neu-ral network-based characterization of mixed pixels in remotely sensed hyperspectral images, Pattern Recognition, 42 (2009), 3032-3045. [25] S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delayed differential equations with twodelays, Dyn. Cont. Disc. Impul. Syst. A: Math. Anal, 10 (2003), 863-874. [26] Y. Song, M. Han and J. Wei, Stability and Hopf bifurcation on a simplified BAM neural network with delays, Physica D, 200 (2005), 185-204. doi: 10.1016/j.physd.2004.10.010. [27] U. Tetsushi, M. Hisayo, K. Hiroshi and K. Takuji, Bifurcation and chaos in coupled BVP oscillators, Int. J. Bifurcat. Chaos, 14 (2004), 1305-1324. [28] U. Tetsushi and K. Hiroshi, Bifurcation in asymmetrically coupled BVP oscillators, Int. J. Bifurcat. Chaos, 13 (2003), 1319-1327. [29] Q. Wang, Q. Lu, G. R. Chen, Z. Feng and L. X. Duan, Bifurcation and synchronization of synaptically coupled FHN models with time delay, Chaos Soliton Fract, 39 (2009), 918-25. [30] H. Wang and W. Jiang, Hopf-pitchfork bifurcation in van der Pol's oscillator with nonlinear delayed feedback, J. Math. Anal. Appl, 368 (2010), 9-18. doi: 10.1016/j.jmaa.2010.03.012. [31] Z. Zeng and J. Wang, Analysis and design of associative memories based on recurrent neural networks with linear saturation activation functions and time-varying delays, Neural compu-tation, 19 (2007), 2149-2182. doi: 10.1162/neco.2007.19.8.2149. [32] Z. Zeng, D. S. Huang and Z. Wang, Pattern memory analysis based on stability theory of cellular neural networks, Applied Mathematical Modelling, 32 (2008), 112-121. [33] Z. Zeng and J. Wang, Design and analysis of high-capacity associative memories based on a class of discrete-time recurrent neural networks, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 38 (2008), 1525-1536.
The bifurcation diagrams for system (2) with the parameters $(\mu_1,\mu_2)$ around $(0,0)$.
The phase portraits in $D_{1}-D_{6}$
A stable trivial equilibria: $(a)$ waveform diagram for variable $v_1(t)$, $(b)$ waveform diagram for variable $v_2(t)$, $(c)$ waveform diagram for variable $v_3(t)$, $(d)$ waveform diagram for variable $v_4(t)$, and $(e)$ phase diagram for variable $v_1(t)$, $v_2(t)$, $v_3(t)$.
A stable periodic solution: $(a)$ waveform diagram for variable $v_1(t)$, $(b)$ waveform diagram for variable $v_2(t)$, $(c)$ waveform diagram for variable $v_3(t)$, $(d)$ waveform diagram for variable $v_4(t)$, $(e)$ phase diagram for variable $v_1(t)$, $v_2(t)$, $v_3(t)$, and $(f)$ phase diagram for variable $v_1(t)$, $v_3(t)$, $v_4(t)$.
A stable periodic solution: $(a)$ waveform diagram for variable $v_1(t)$, $(b)$ waveform diagram for variable $v_2(t)$, $(c)$ waveform diagram for variable $v_3(t)$, $(d)$ waveform diagram for variable $v_4(t)$, $(e)$ phase diagram for variable $v_1(t)$, $v_2(t)$, $v_3(t)$, and $(f)$ phase diagram for variable $v_1(t)$, $v_3(t)$, $v_4(t)$.
A pair of stable fixed points and a stable periodic solution coexist: $(a)$ waveform diagram for variable $v_1(t)$, $(b)$ waveform diagram for variable $v_2(t)$, $(c)$ waveform diagram for variable $v_3(t)$, $(d)$ waveform diagram for variable $v_4(t)$, $(e)$ phase diagram for variable $v_1(t)$, $v_2(t)$, $v_3(t)$, and $(f)$ phase diagram for variable $v_1(t)$, $v_3(t)$, $v_4(t)$.
A pair of stable fixed points: $(a)$ waveform diagram for variable $v_1(t)$, $(b)$ waveform diagram for variable $v_2(t)$, $(c)$ waveform diagram for variable $v_3(t)$, $(d)$ waveform diagram for variable $v_4(t)$, $(e)$ phase diagram for variable $v_1(t)$, $v_2(t)$, $v_3(t)$, and $(f)$ phase diagram for variable $v_1(t)$, $v_3(t)$, $v_4(t)$.
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