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Taking the modified Morris-Lecar neuron model for example, we consider the synchronous behaviour between "Hopf/homoclinic" bursting and "SubHopf/homoclinic" bursting. Firstly, the synchronization between two coupled bursting neurons with the same topological type is investigated numerically, and the results show that the coupling strength reaching the synchronization of the membrane potential of "Hopf/homoclinic" bursting is smaller than that of "SubHopf/homoclinic" bursting, that is to say, the former can reach complete synchrony of the membrane potential more easily than the latter. Secondly, we study the synchronous behavior of two coupled bursting neurons with different topological types by numerical analysis, and find that with the increase of the coupling strength the two different types of bursting neurons reach the bursting-synchrony first, and then they can reach complete synchrony of the membrane potential when the coupling strength is strong enough, and the type of synchronous state is inclined to the type of easy synchronization, namely, "Hopf/homoclinic" bursting. To our surprise, the slow variables exhibit phase synchronization instead of complete synchronization. Moreover, there is a linear relationship between the both slow variables. This point is distinctly different from the results of the existing documents.
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Keywords:
- bursting /
- synchronization /
- bifurcation /
- neuron
[1] Lisman J 1997 Trends in Neuroscience 20 38
[2] Wang Q Y, Shi X, Lu Q S 2008 Synchronization dynamics in the coupled system of neurons (1st Ed.) (Beijing: Science Press ) p46 (in Chinese) [王青云, 石霞, 陆启韶 2008 神经元耦合系统的同步动力学(第一版)(北京:科学出版社) 第46页]
[3] Wang W T, Hu S J, H D 2005 Progress in Physiological Sciences 36 137 (in Chinese) [王文挺, 胡三觉, 韩丹 2005 生物力学进展 36 137]
[4] Xu J, Clancy C E 2008 PloS ONE 3 e2056
[5] Xie Y, Xu J X, Kang Y M, Hu S J, Duan Y B 2003 Acta Phys. Sin. 52 1112 (in Chinese) [谢勇, 徐建学, 康艳梅, 胡三觉, 段玉斌 2003 物理学报 52 1112]
[6] Yu H J, Tong W J 2009 Acta Phys. Sin. 58 2977 (in Chinese) [于洪结, 童伟君 2009 物理学报 58 2977]
[7] L L, Li G, Zhang M, Li Y S, Wen L L, Yu M 2011 Acta Phys. Sin. 60 090505 (in Chinese) [吕翎, 李刚, 张檬, 李雨珊, 韦琳玲, 于淼 2011 物理学报 60 090505]
[8] Wu Y, Xu J X, He D H, Jin W Y 2005 Acta Phys. Sin. 54 3457 (in Chinese) [吴瑛, 徐建学, 何岱海, 靳伍银 2005 物理学报 54 3457]
[9] Sleeman B D, Jarvis R J 1985 Ordinary and Partial Differential Equations (Berlin: Springer-Verlag) p304
[10] Teramoto E, Yamaguti M 1987 Mathematical Topics in Population Biology, Morphogenesis and Neurosciences (Berlin: Springer-Verlag) p267
[11] Izhikevich E M 2000 Int. J. Bifurcation and Chaos 10 1171
[12] Izhikevich E M 2007 Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (London: The MIT Press) p325
[13] Dhamala M, Jirsa V K, Ding M Z 2004 Phys. Rev. Lett. 92 028101
[14] Su J Z, Perez-Gonzalez H, He M 2007 Discrete and Continuous Dynamical Systems, Suppl 946
[15] Yang Z Q, Lu Q S 2007 Sci. China Ser. G: Physics, Mechanics Astronomy 37 440 (in Chinese) [杨卓琴, 陆启韶 2007 中国科学G辑: 物理学、力学、天文学 37 440]
[16] Shi X 2010 Chinese Quarterly of Mechanics 1 52 (in Chinese) [石霞 2010 力学季刊 1 52]
[17] Wu Y, Xu J X, Jin W Y 2005 Lecture Notes in Computer Science 3496 302
[18] Wu Y, Xu J X, He M 2005 Lecture Notes in Computer Science 3610 508
[19] Shen Y, Hou Z H, Xin H W 2008 Phys. Rev. E 77 031920
[20] Wang H X, Lu Q S, Wang Q Y 2008 Communications in Nonlinear Science and Numerical Simulation 13 1668
[21] Izhikevich E M 2001 SIAM Review 43 315
[22] Gu H G, Li L, Yang M H, Liu Z Q, Ren W 2003 Acta Biophysica Sinica 19 69 (in Chinese) [古华光, 李莉, 杨明浩, 刘志强, 任维 2003 生物物理学报 19 69]
[23] Wang H X, Lu Q S, Shi X 2010 Chin. Phys. B 19 06059
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[1] Lisman J 1997 Trends in Neuroscience 20 38
[2] Wang Q Y, Shi X, Lu Q S 2008 Synchronization dynamics in the coupled system of neurons (1st Ed.) (Beijing: Science Press ) p46 (in Chinese) [王青云, 石霞, 陆启韶 2008 神经元耦合系统的同步动力学(第一版)(北京:科学出版社) 第46页]
[3] Wang W T, Hu S J, H D 2005 Progress in Physiological Sciences 36 137 (in Chinese) [王文挺, 胡三觉, 韩丹 2005 生物力学进展 36 137]
[4] Xu J, Clancy C E 2008 PloS ONE 3 e2056
[5] Xie Y, Xu J X, Kang Y M, Hu S J, Duan Y B 2003 Acta Phys. Sin. 52 1112 (in Chinese) [谢勇, 徐建学, 康艳梅, 胡三觉, 段玉斌 2003 物理学报 52 1112]
[6] Yu H J, Tong W J 2009 Acta Phys. Sin. 58 2977 (in Chinese) [于洪结, 童伟君 2009 物理学报 58 2977]
[7] L L, Li G, Zhang M, Li Y S, Wen L L, Yu M 2011 Acta Phys. Sin. 60 090505 (in Chinese) [吕翎, 李刚, 张檬, 李雨珊, 韦琳玲, 于淼 2011 物理学报 60 090505]
[8] Wu Y, Xu J X, He D H, Jin W Y 2005 Acta Phys. Sin. 54 3457 (in Chinese) [吴瑛, 徐建学, 何岱海, 靳伍银 2005 物理学报 54 3457]
[9] Sleeman B D, Jarvis R J 1985 Ordinary and Partial Differential Equations (Berlin: Springer-Verlag) p304
[10] Teramoto E, Yamaguti M 1987 Mathematical Topics in Population Biology, Morphogenesis and Neurosciences (Berlin: Springer-Verlag) p267
[11] Izhikevich E M 2000 Int. J. Bifurcation and Chaos 10 1171
[12] Izhikevich E M 2007 Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (London: The MIT Press) p325
[13] Dhamala M, Jirsa V K, Ding M Z 2004 Phys. Rev. Lett. 92 028101
[14] Su J Z, Perez-Gonzalez H, He M 2007 Discrete and Continuous Dynamical Systems, Suppl 946
[15] Yang Z Q, Lu Q S 2007 Sci. China Ser. G: Physics, Mechanics Astronomy 37 440 (in Chinese) [杨卓琴, 陆启韶 2007 中国科学G辑: 物理学、力学、天文学 37 440]
[16] Shi X 2010 Chinese Quarterly of Mechanics 1 52 (in Chinese) [石霞 2010 力学季刊 1 52]
[17] Wu Y, Xu J X, Jin W Y 2005 Lecture Notes in Computer Science 3496 302
[18] Wu Y, Xu J X, He M 2005 Lecture Notes in Computer Science 3610 508
[19] Shen Y, Hou Z H, Xin H W 2008 Phys. Rev. E 77 031920
[20] Wang H X, Lu Q S, Wang Q Y 2008 Communications in Nonlinear Science and Numerical Simulation 13 1668
[21] Izhikevich E M 2001 SIAM Review 43 315
[22] Gu H G, Li L, Yang M H, Liu Z Q, Ren W 2003 Acta Biophysica Sinica 19 69 (in Chinese) [古华光, 李莉, 杨明浩, 刘志强, 任维 2003 生物物理学报 19 69]
[23] Wang H X, Lu Q S, Shi X 2010 Chin. Phys. B 19 06059
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