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Neural mass model (NMM) can generate spontaneous oscillation even in a resting state. However, it remains little known which mechanism is responsible for NMM’s spontaneous oscillation. From dynamical theory, spontaneous oscillation is an intrinsic property of nonlinear system, which means that the sigmoid nonlinear function (S function) of NMM plays a key role in the emergence of its spontaneous oscillation. In this study, describing function approach is employed to analyze the spontaneous oscillation characteristics of a kind of extended NMM. Firstly, the describing function of S function is derived, through which the two S functions in excitatory and inhibitory feedback loop, respectively, are approximated. Secondly, the NMM is transformed into a typical block diagram composed of a nonlinear unit and a linear unit. Thirdly, in the theoretical framework of describing function approach, theoretical analysis of the spontaneous oscillation characteristics of NMM is conducted, and the oscillation frequencies are determined. The simulation results demonstrate that the theoretical results are correct and the employed approach is effective. Since S function exists extensively in neural system, the proposed approach has a potential application in the spontaneous oscillation analysis of other neural model.
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Keywords:
- neural mass model /
- sigmoid nonlinear function /
- describing function /
- spontaneous oscillation
[1] Uhlhaas P J, Singer W 2010 Nat. Rev. Neurosci. 11 100
[2] Schnitzler A, Gross J 2005 Nat. Rev. Neurosci. 6 285
[3] Liu S B, Wu Y, Hao Z W, Li Y J, Jia N 2012 Acta Phys. Sin. 61 020503 (in Chinese) [刘少宝, 吴莹, 郝忠文, 李银军, 贾宁 2012 61 020503]
[4] Gu H G, Hui L, Jia B 2012 Acta Phys. Sin. 61 080504 (in Chinese) [古华光, 惠磊, 贾冰 2012 61 080504]
[5] Jansen B H, Rit V G 1995 Biol. Cybern. 73 357
[6] Deco G, Jirsa V K, Robinson P A, Breakspear M, Friston K 2008 PLOS. Comput. Biol. 4 e1000092
[7] Lopes S F H, Hoeks A, Smits H, Zetterberg L H 1974 Kybernetik 15 27
[8] Destexhe A, Sejnowski T J 2009 Biol. Cybern. 101 1
[9] Zavaglia M, Astolfi L, Babiloni F, Ursino M 2008 Biomed. Eng. 55 69
[10] Wendling F, Bartolomei F, Bellanger J J, Chauvel P 2002 Eur. J. Neurosci 15 1499
[11] Liu X, Ma B W, Liu H J 2013 Acta Phys. Sin. 62 020202 (in Chinese) [刘仙, 马百旺, 刘会军 2013 62 020202]
[12] Zheng Y, Luo J J, Harris S, Kennerley A, Berwick J, Billings S A, Mayhew J 2012 NeuroImage 63 81
[13] Stephan K E, Kasper L, Harrison L M, Daunizeau J, den Ouden H E, Breakspear M, Friston K J 2008 NeuroImage 42 649
[14] Babajani A, Soltanian-Zadeh H 2006 IEEE Trans. Biomed. Eng. 53 1794
[15] David O, Friston K J 2003 NeuroImage 20 1743
[16] Buzsáki G, Draguhn A 2004 Science 304 1926
[17] Cui D, Li X L, Ji X Q, Liu L X 2011 Sci. China: Inf. Sci. 54 1283 (in Chinese) [崔冬, 李小俚, 吉学青, 刘兰祥 2011 中国科学: 信息科学 54 1283]
[18] van Rotterdam A, Lopes da Silva F H, van den Ende J, Viergever M A, Hermans A J 1982 Bull. Math. Biol. 44 283
[19] Abbas B F, Hamid S Z 2010 NeuroImage 52 793
[20] Hu S S 2007 Automatic Control Principle (5th Ed.) (Beijing: Science Press) pp408–422 (in Chinese) [胡寿松 2007 自动控制原理 (第五版) (北京: 科学出版社)第408–422 页]
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[1] Uhlhaas P J, Singer W 2010 Nat. Rev. Neurosci. 11 100
[2] Schnitzler A, Gross J 2005 Nat. Rev. Neurosci. 6 285
[3] Liu S B, Wu Y, Hao Z W, Li Y J, Jia N 2012 Acta Phys. Sin. 61 020503 (in Chinese) [刘少宝, 吴莹, 郝忠文, 李银军, 贾宁 2012 61 020503]
[4] Gu H G, Hui L, Jia B 2012 Acta Phys. Sin. 61 080504 (in Chinese) [古华光, 惠磊, 贾冰 2012 61 080504]
[5] Jansen B H, Rit V G 1995 Biol. Cybern. 73 357
[6] Deco G, Jirsa V K, Robinson P A, Breakspear M, Friston K 2008 PLOS. Comput. Biol. 4 e1000092
[7] Lopes S F H, Hoeks A, Smits H, Zetterberg L H 1974 Kybernetik 15 27
[8] Destexhe A, Sejnowski T J 2009 Biol. Cybern. 101 1
[9] Zavaglia M, Astolfi L, Babiloni F, Ursino M 2008 Biomed. Eng. 55 69
[10] Wendling F, Bartolomei F, Bellanger J J, Chauvel P 2002 Eur. J. Neurosci 15 1499
[11] Liu X, Ma B W, Liu H J 2013 Acta Phys. Sin. 62 020202 (in Chinese) [刘仙, 马百旺, 刘会军 2013 62 020202]
[12] Zheng Y, Luo J J, Harris S, Kennerley A, Berwick J, Billings S A, Mayhew J 2012 NeuroImage 63 81
[13] Stephan K E, Kasper L, Harrison L M, Daunizeau J, den Ouden H E, Breakspear M, Friston K J 2008 NeuroImage 42 649
[14] Babajani A, Soltanian-Zadeh H 2006 IEEE Trans. Biomed. Eng. 53 1794
[15] David O, Friston K J 2003 NeuroImage 20 1743
[16] Buzsáki G, Draguhn A 2004 Science 304 1926
[17] Cui D, Li X L, Ji X Q, Liu L X 2011 Sci. China: Inf. Sci. 54 1283 (in Chinese) [崔冬, 李小俚, 吉学青, 刘兰祥 2011 中国科学: 信息科学 54 1283]
[18] van Rotterdam A, Lopes da Silva F H, van den Ende J, Viergever M A, Hermans A J 1982 Bull. Math. Biol. 44 283
[19] Abbas B F, Hamid S Z 2010 NeuroImage 52 793
[20] Hu S S 2007 Automatic Control Principle (5th Ed.) (Beijing: Science Press) pp408–422 (in Chinese) [胡寿松 2007 自动控制原理 (第五版) (北京: 科学出版社)第408–422 页]
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