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Bursting and two-parameter bifurcation in the Chay neuronal model
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 |
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. |
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