哺乳动物皮层神经元模型的放电模式研究
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摘要
针对一类三维哺乳动物新皮层神经元模型进行分析。该模型是结合Wilson模型和Hindmarsh-Rose模型而建立的,得到的快变量遵守欧姆定律。快慢动力学对时间尺度相差很大的系统的分析是非常有效的,因此本研究使用快慢动力学方法,并结合理论计算和数值模拟来探究该模型的动力学行为。首先,随着系统控制参数的改变,发现了3类簇放电模式;然后,计算一阶Lyapunov系数,以此确定Hopf分岔是超临界或次临界的;最后,将模型与经典的Morris-Lecar模型耦合,分析并讨论了耦合强度及外界交流刺激对耦合系统的影响。
A three-dimensional neuron model which express the characteristics of mammals new cortex is analyzed in this study.The neuron model is obtained by combining Wilson model and Hindmarsh-Rose model, and its fast variables obey Ohm's law.The fast-slow dynamical methods is effective for systematic analysis with large differences in time scales. Hence the fast-slow dynamics method is adopted and combined with theoretical calculation and numerical simulation to explore the dynamic activity of the model. As the control parameters change, 3 types of bursting patterns are demonstrated. Furthermore, the first Lyapunov coefficient of the Hopf bifurcation is calculated, thereby determining whether the Hopf bifurcation is subcritical or supercritical.Finally, the model is coupled with the classic Morris-Lecar model for analyzing and discussing the effects of coupling strength and external alternating current stimulation on the coupled system.
A three-dimensional neuron model which express the characteristics of mammals new cortex is analyzed in this study.The neuron model is obtained by combining Wilson model and Hindmarsh-Rose model, and its fast variables obey Ohm's law.The fast-slow dynamical methods is effective for systematic analysis with large differences in time scales. Hence the fast-slow dynamics method is adopted and combined with theoretical calculation and numerical simulation to explore the dynamic activity of the model. As the control parameters change, 3 types of bursting patterns are demonstrated. Furthermore, the first Lyapunov coefficient of the Hopf bifurcation is calculated, thereby determining whether the Hopf bifurcation is subcritical or supercritical.Finally, the model is coupled with the classic Morris-Lecar model for analyzing and discussing the effects of coupling strength and external alternating current stimulation on the coupled system.
引文
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[11]杨卓琴,陆启韶.神经元Chay模型中不同类型的簇放电模式[J].中国科学(G辑), 2007, 37(4):440-450.YANG Z Q, LU Q S. Various kinds of bursting pattern in the Chay neuron model[J]. Science China:Series G, 2007, 37(4):440-450.
[12]WANG J, LIU S Q, LIU X L, et al. Bifurcation and firing patterns of the pancreaticβ-cell[J]. Int J Bifurcat Chaos, 2015, 25(9):15300244-153002414.
[13]WANG J, LU B, LIU S Q, et al. Bursting types and bifurcation analysis in the pre-B?tzinger complex respiratory rhythm neuron[J]. Int J Bifurcat Chaos, 2017, 27(1):17500109-175001025.
[14]ZHAN F B, LIU S Q, WANG J, et al. Bursting patterns and mixedmode oscillations in reduced purkinje model[J]. Int J Modern Phys B, 2018, 32(5):18500431-185004315.
[15]ZHAN F B, LIU S Q, ZHANG X H, et al. Mixed-mode oscillations and bifurcation analysis in a pituitary model[J]. Nonlinear Dynam,2018, 94:807-826.
[16]ZHAO X, KIM J W, ROBINSON P A, et al. Low dimensional model of bursting neurons[J]. J Comput Neurosci, 2014, 36(1):81-95.
[2] MCCORMIC K, DAVID A. Membrane properties and neurotransmitter actions[M]. New York:Oxford University Press, 2004:39-79.
[3] MORRIS C, LECAR H. Voltage oscillations in the barnacle giant muscle-fiber[J]. Biophys J, 1981, 35(1):193-213.
[4] HINDMARSH J L, ROSE R M. A model of neuronal bursting using three coupled first order differential equations[J]. Proc R Soc Lond B Biol Sci, 1984, 221(1222):87-102.
[5] WILSON H R. Simplified dynamics of human and mammalian neocortical neurons[J]. J Theor Biol, 1999, 200(4):375-388.
[6] RINZEL J. Bursting oscillations in an excitable membrane model[J].Lect Notes Math, 1985, 1151:304-316.
[7] CHAY T R, FAN Y S, LEE S. Bursting, spiking, chaos, fractals and universality in biological rhythms[J]. Int J Bifurcat Chaos, 2000, 5(3):595-635.
[8] SHILNIKOV A. Complete dynamical analysis of a neuron model[J].Nonlinear Dynam, 2012, 68(3):305-328.
[9] IZHIKEVICH E M. Neural excitability, spiking and bursting[J]. Int J Bifurcat Chaos, 2000, 10(6):1171-1266.
[10]IZHIKEVICH E M. Dynamical systems in neuroscience[M].Massachusetts:the MIT Press, 2007.
[11]杨卓琴,陆启韶.神经元Chay模型中不同类型的簇放电模式[J].中国科学(G辑), 2007, 37(4):440-450.YANG Z Q, LU Q S. Various kinds of bursting pattern in the Chay neuron model[J]. Science China:Series G, 2007, 37(4):440-450.
[12]WANG J, LIU S Q, LIU X L, et al. Bifurcation and firing patterns of the pancreaticβ-cell[J]. Int J Bifurcat Chaos, 2015, 25(9):15300244-153002414.
[13]WANG J, LU B, LIU S Q, et al. Bursting types and bifurcation analysis in the pre-B?tzinger complex respiratory rhythm neuron[J]. Int J Bifurcat Chaos, 2017, 27(1):17500109-175001025.
[14]ZHAN F B, LIU S Q, WANG J, et al. Bursting patterns and mixedmode oscillations in reduced purkinje model[J]. Int J Modern Phys B, 2018, 32(5):18500431-185004315.
[15]ZHAN F B, LIU S Q, ZHANG X H, et al. Mixed-mode oscillations and bifurcation analysis in a pituitary model[J]. Nonlinear Dynam,2018, 94:807-826.
[16]ZHAO X, KIM J W, ROBINSON P A, et al. Low dimensional model of bursting neurons[J]. J Comput Neurosci, 2014, 36(1):81-95.