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Through the research on the fractional-order FitzHugh-Nagumo model neuron，it is found that the Hopf bifurcation point of the fractional-order model，where the state of the model neuron changes from quiescence to periodic spiking，is different from that of the corresponding integer-order model when the externally applied current is considered as the bifurcation parameter. We further demonstrate that the range of the strength of the externally applied current in the fractional-order model neuron，which can make the model neuron exhibit periodic spiking，is smaller than that in the corresponding integer-order model neuron. However，the firing frequency of the fractional-order model neuron is higher than that of the integer-order counterpart. Meanwhile，we show that the synchronization rate of two electrically coupled fractional-order FitzHugh-Nagumo model neurons is greater than that of the integer-order counterpart. The Adomian decomposition method is employed to calculate fractional-order differential equations numerically because of its rapid convergence and high accuracy.
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Keywords:
- fractional-order /
- Hopf bifurcation /
- FitzHugh-Nagumo model /
- synchronization
[1] ［1］ Zhu C X，ZOU Y 2009 Control and Decision 24 161 （in Chinese）［朱呈祥、邹云 2009 控制与决策 24 161］
[2] ［2］ Bagley R L Calico R A 1999 Journal of Guidance，Control，and Dynamics 14 304
[3] ［3］ Koeller R C 1984 Acta Mechanica 58 299
[4] ［4］ Koeller R C 1984 Journal of Applied Mechanics 51 299
[5] ［5］ Skaar S B，Michel A N，Miller R K 1988 IEEE Transactionson on Automatic Control 3（4） 348
[6] ［6］ Ichise M，Nagayanay Y，Kojima T 1971 Journal of Electroa-nalytical Chemistry Interfacial Electrochemisry 33 253［7］ Sun H H，Onaral B，Tsao Y Y 1984 IEEE Transactions on Ciomedical Engineering 31 664
[7] ［8］ Sun H H，Abdelwahed A A，Onaral B 1984 IEEE Tranactions on Automatic Control 29 441
[8] ［9］ Mark R J，Hall M W 1981 IEEE Trans.Acoust，Speech Signal Processing 29 872［10］Podlubny I 1999 Fractional Differential Equations（San Diego.CA：Academic Press）
[9] ］Wheatcraft S W，Benson D A，Meerschaert M M 2000 Water Resour.Res. 36 1403
[10] ］Wheatcraft S W，Benson D A，Meerschaert M M 2000 Water Resour.Res. 36 1413
[11] ］Mandelbrot B 1967 IEEE Transactions on Information Theory 13 289
[12] ］Lin R 2007 Ph. D. Dissertation（Xiamen Univ.） （in Chinese）［林然 2007 博士学位论文（厦门大学）］
[13] ］Deng W H 2007 Ph. D. Dissertation （Shanghai Univ.） （in Chinese）［邓伟华 2007 博士学位论文（上海大学）］
[14] ］Che Y Q 2005 Thesis for the Masters Degree （Tianjin Univ.） （in Chinese）［车艳秋 2005 硕士学位论文（天津大学）］
[15] ］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］
[16] ］Lu Q S，Liu S Q，Liu F，Wang Q Y，Hou Z H，Zheng Y H 2008 Advances in Mechanics 38 766 （in Chinese）［陆启韶、刘深泉、刘峰、王青云、侯中怀、郑艳红 2008 力学进展 38 766］
[17] ］Yu H J，Tong W J 2009 Acta Phys. Sin. 58 2977 （in Chinese）［于洪洁、童伟君 2009 58 2977］
[18] ］Zhang H，Fang L P，Tong Q Y 2007 Acta Phys. Sin. 56 7339 （in Chinese）［张宏、方路平、童勤业 2007 56 7339］
[19] ］Naundorf B，Wolf F，Volgushev M 2006 Nature 440 1060
[20] ］Auastasio T J 1994 Biol.Cybern. 72 69［23］Lundstrom B N，Higgs M H，Spain W J，Fairhall A L 2008 Nature Neuroscience 11 1335
[21] ］Malek M H，Hashemi S R IFMBE Proceedings 21 825
[22] ］Xie Y，Chen L，Kang Y M，Aihara K 2008 Phys. Rev. E 77 061921
[23] ］Xie Y，Aihara K，Kang Y M 2008 Phys. Rev. E 77 021917
[24] ］Wang Q Y，Shi X，Lu Q S 2008 Synchronization dynamics in the coupled system of neurons （Beijing：Science Press）p4 （in Chinese）［王青云、石霞、陆启韶 2008 神经元耦合系统的同步动力学（北京：科学出版社）第4页］
[25] ］Wen T，Tan X K，Yin S J 2003 Acta Physiologica Sinica 55 663 （in Chinese）［文挺、覃星奎、尹世金、韩丹 2003 生理学报 55 663］
[26] ］Wang Q Y，Lu S Q 2008 Journal of Dynamics and Control 6 35 （in Chinese）［王青云、陆启韶 2008 动力学与控制学报 6 35］
[27] ］Li H Q，Xu W，Wang C Q，Liu X D 2008 Acta Biophysica Sinica 24 29 （in Chinese）［李海泉、徐伟、王朝庆、刘小丹 2008生物 24 29］
[28] ］Wang H X，Lu Q S 2006 Journal of Beijing University of Aeronautics and Astronautics 32 320 （in Chinese）［王海霞、陆启韶 2006 北京航空航天大学学报 32 320］
[29] ］Wu Y，Xu J X，Jin W Y 2005 Journal of Xian Jiaotong University 39 544 （in chinese）［吴莹、徐建学、靳伍银 2005 西安交通大学学报 39 544］
[30] ］Wu Y，Xu J X，He D H，Jin W Y 2005 Acta Phys. Sin. 54 3457 （in Chinese）［吴莹 、徐建学、何岱海、靳伍银 2005 54 3457］
[31] ］Zhao P D，Zhang X D 2008 Acta Phys. Sin. 57 2791 （in Chinese）［赵品栋、张晓丹 2008 57 2791］
[32] ］Matignon D 1996 IMACS, IEEE-SMC，Lille，France 963
[33] ］Ahmed E，El2Sayed A M A，Elsaka H A A 2007 J. Math. Anal. Appl. 325 542
[34] ］Zhang G J，Xu J X 2005 Acta Phys. Sin. 54 0557 （in Chinese）［张广军、徐建学 2005 54 0557］
[35] ］Zhang Y J 2009 Journal of Dynamics and Control 7 19 （in Chinese）［张艳娇、李美生、陆启韶 2009 动力学与控制报 7 19］
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[1] ［1］ Zhu C X，ZOU Y 2009 Control and Decision 24 161 （in Chinese）［朱呈祥、邹云 2009 控制与决策 24 161］
[2] ［2］ Bagley R L Calico R A 1999 Journal of Guidance，Control，and Dynamics 14 304
[3] ［3］ Koeller R C 1984 Acta Mechanica 58 299
[4] ［4］ Koeller R C 1984 Journal of Applied Mechanics 51 299
[5] ［5］ Skaar S B，Michel A N，Miller R K 1988 IEEE Transactionson on Automatic Control 3（4） 348
[6] ［6］ Ichise M，Nagayanay Y，Kojima T 1971 Journal of Electroa-nalytical Chemistry Interfacial Electrochemisry 33 253［7］ Sun H H，Onaral B，Tsao Y Y 1984 IEEE Transactions on Ciomedical Engineering 31 664
[7] ［8］ Sun H H，Abdelwahed A A，Onaral B 1984 IEEE Tranactions on Automatic Control 29 441
[8] ［9］ Mark R J，Hall M W 1981 IEEE Trans.Acoust，Speech Signal Processing 29 872［10］Podlubny I 1999 Fractional Differential Equations（San Diego.CA：Academic Press）
[9] ］Wheatcraft S W，Benson D A，Meerschaert M M 2000 Water Resour.Res. 36 1403
[10] ］Wheatcraft S W，Benson D A，Meerschaert M M 2000 Water Resour.Res. 36 1413
[11] ］Mandelbrot B 1967 IEEE Transactions on Information Theory 13 289
[12] ］Lin R 2007 Ph. D. Dissertation（Xiamen Univ.） （in Chinese）［林然 2007 博士学位论文（厦门大学）］
[13] ］Deng W H 2007 Ph. D. Dissertation （Shanghai Univ.） （in Chinese）［邓伟华 2007 博士学位论文（上海大学）］
[14] ］Che Y Q 2005 Thesis for the Masters Degree （Tianjin Univ.） （in Chinese）［车艳秋 2005 硕士学位论文（天津大学）］
[15] ］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］
[16] ］Lu Q S，Liu S Q，Liu F，Wang Q Y，Hou Z H，Zheng Y H 2008 Advances in Mechanics 38 766 （in Chinese）［陆启韶、刘深泉、刘峰、王青云、侯中怀、郑艳红 2008 力学进展 38 766］
[17] ］Yu H J，Tong W J 2009 Acta Phys. Sin. 58 2977 （in Chinese）［于洪洁、童伟君 2009 58 2977］
[18] ］Zhang H，Fang L P，Tong Q Y 2007 Acta Phys. Sin. 56 7339 （in Chinese）［张宏、方路平、童勤业 2007 56 7339］
[19] ］Naundorf B，Wolf F，Volgushev M 2006 Nature 440 1060
[20] ］Auastasio T J 1994 Biol.Cybern. 72 69［23］Lundstrom B N，Higgs M H，Spain W J，Fairhall A L 2008 Nature Neuroscience 11 1335
[21] ］Malek M H，Hashemi S R IFMBE Proceedings 21 825
[22] ］Xie Y，Chen L，Kang Y M，Aihara K 2008 Phys. Rev. E 77 061921
[23] ］Xie Y，Aihara K，Kang Y M 2008 Phys. Rev. E 77 021917
[24] ］Wang Q Y，Shi X，Lu Q S 2008 Synchronization dynamics in the coupled system of neurons （Beijing：Science Press）p4 （in Chinese）［王青云、石霞、陆启韶 2008 神经元耦合系统的同步动力学（北京：科学出版社）第4页］
[25] ］Wen T，Tan X K，Yin S J 2003 Acta Physiologica Sinica 55 663 （in Chinese）［文挺、覃星奎、尹世金、韩丹 2003 生理学报 55 663］
[26] ］Wang Q Y，Lu S Q 2008 Journal of Dynamics and Control 6 35 （in Chinese）［王青云、陆启韶 2008 动力学与控制学报 6 35］
[27] ］Li H Q，Xu W，Wang C Q，Liu X D 2008 Acta Biophysica Sinica 24 29 （in Chinese）［李海泉、徐伟、王朝庆、刘小丹 2008生物 24 29］
[28] ］Wang H X，Lu Q S 2006 Journal of Beijing University of Aeronautics and Astronautics 32 320 （in Chinese）［王海霞、陆启韶 2006 北京航空航天大学学报 32 320］
[29] ］Wu Y，Xu J X，Jin W Y 2005 Journal of Xian Jiaotong University 39 544 （in chinese）［吴莹、徐建学、靳伍银 2005 西安交通大学学报 39 544］
[30] ］Wu Y，Xu J X，He D H，Jin W Y 2005 Acta Phys. Sin. 54 3457 （in Chinese）［吴莹 、徐建学、何岱海、靳伍银 2005 54 3457］
[31] ］Zhao P D，Zhang X D 2008 Acta Phys. Sin. 57 2791 （in Chinese）［赵品栋、张晓丹 2008 57 2791］
[32] ］Matignon D 1996 IMACS, IEEE-SMC，Lille，France 963
[33] ］Ahmed E，El2Sayed A M A，Elsaka H A A 2007 J. Math. Anal. Appl. 325 542
[34] ］Zhang G J，Xu J X 2005 Acta Phys. Sin. 54 0557 （in Chinese）［张广军、徐建学 2005 54 0557］
[35] ］Zhang Y J 2009 Journal of Dynamics and Control 7 19 （in Chinese）［张艳娇、李美生、陆启韶 2009 动力学与控制报 7 19］
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