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以修正过的Morris-Lecar神经元模型为例,讨论了"Hopf/homoclinic"簇放电和"SubHopf/homoclinic" 簇放电之间的同步行为.首先,分别考察了同一拓扑类型的两个耦合簇放电神经元的同步行为, 发现"Hopf/homoclinic"簇放电比"SubHopf/homoclinic"簇放电达到膜电位完全同步所需要的耦合强度小, 即前者比后者更容易达到膜电位完全同步.其次,对这两个不同拓扑类型的簇放电神经元的耦合同步行为 进行了讨论.通过数值分析发现随着耦合强度的增加,两种不同类型的簇放电首先达到簇放电同步, 然后当耦合强度足够大时甚至可以达到膜电位完全同步,并且同步后的放电类型更接近容易同步的 簇放电类型,即"Hopf/homoclinic"簇放电.然而令人奇怪的是此时慢变量并没有达到完全同步, 而是相位同步;慢变量之间呈现为一种线性关系.这一点和现有文献的结果截然不同.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 52 1112]
[6] Yu H J, Tong W J 2009 Acta Phys. Sin. 58 2977 (in Chinese) [于洪结, 童伟君 2009 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 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 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 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 52 1112]
[6] Yu H J, Tong W J 2009 Acta Phys. Sin. 58 2977 (in Chinese) [于洪结, 童伟君 2009 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 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 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 19 69]
[23] Wang H X, Lu Q S, Shi X 2010 Chin. Phys. B 19 06059
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