全耦合Rulkov神经系统的动态转迁
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摘要
神经元在中枢神经系统信息处理过程中起着关键的作用,神经元信息的产生和传输体现了丰富的非线性特征.单个神经元的不动点的稳定性可能会因为不同参数取值而改变,神经元的活动状态和静息状态也会因为外部刺激而产生明显的变化,这主要是根据神经元数学模型的不同分岔值来确定.神经元脉冲的传递至少要两个以上的神经元通过耦合的方式来完成,因此耦合的神经元系统是一个非常复杂的高维非线性动力系统,其中有些振子活动的振幅幅度非常小或者几乎为零,那么大尺度耦合的非线性振子可能会因为Aging而导致非自震荡元素部分数目增加.研究神经元系统的Aging Transition及其非线性特征可以为生理和医学试验提供理论参考和试验依据.
本文首先在参考全耦合的Stuart-Landau方程组的基础上,研究了全耦合的Rulkov神经元系统,整个系统因为参数σ的不同而分为两组.假设用p表示非活动振子所占比率,则在第一部分中,重点分析了在p=O,k=0时整个系统的神经元动态,并用中心流形的方法讨论了单个神经元不动点的稳定性,尤其详细的分析了Neimark-Sacker分岔的性质,给出了单个神经元活动与静止的分界线.同时也分析了其它分岔,根据这些分岔可以明确判定单个神经元是处于方形簇放电,锥形簇放电,峰放电,还是静息态.
其次,讨论了当p≠0时的耦合Rulkov神经元系统,取定两个参数σ1,σ2,分别使单个Rulkov神经元模型处于活动状态和静息状态,从而将整个系统因为参数σ的不同而分为两个集群,即活动振幅较大的神经元组和活动振幅非常小(认为是非活动)的神经元组,随着不活跃元素所占比率p的增加,全耦合神经元系统震荡行为会在某一临界值终止.在这部分中,重点讨论了全耦合神经元系统活动状态和非活动状态之间的变迁,研究了全耦合的基于映射的Rulkov神经元系统中(k,p)的结构相图,其中k是耦合强度,p是非活动元素所占比率.
最后,对本文的研究内容进行了总结,本文的研究成果为更全面完整的讨论离散神经元模型的动态转迁性质奠定了基础.
Neurons in human brain play a key role in the processing, generation, and transmission of information. The stability of fixed point in a single neuron can change corresponding to different parameter values, and the active and rest state of neurons can also take place a obvious transition due to the external stimulus. The coupl-ing neuron network is a very complex high dimensional nonlinear dynamical system, its amplitude of oscillation may go to a small range under some circumstances. Similarly, a population of coupled nonlinear oscillators may age when the fraction of non-self-oscillatory elements increases. Likewise, a global oscillation state will turn into a quiescence state as the proportion of inactive elements exceeds a critical value.
In this paper, we at first introduce the coupling Rulkov neuron network, and divide the entire network into two clusters with regard to different values of σ. We discuss the single Rulkov neuron model when p=0, k=0, here p is the ratio of inactive oscillators, and k is the coupling strength, and take into account the stability of fixed point in a single Rulkov neuron by using the method of center manifold. We also analyze the Neimark-Sacker bifurcation, and other bifurcations of single Rulkov neuron, gives a boundary between the activity and the inactivity of a single neuron.
Then we discuss the coupling Rulkov neuron model when p≠O, here we choose two different values of σ1, σ2. We divide the entire system into two clusters, active and inactive neuron groups. Our study demonstrates that the global oscillation state will turn into a quiescence state at a critical value as the proportion p of inactive elements increases. In particular, we discuss the Aging Transitions between the active and the resting state, and one of very important results is to present the phase diagrams of structure (k, p) in the map-based global coupling neurons.
Finally, we summarize the whole contents of this paper. What we have done provides a basic mathematical foundation for a more comprehensive and complete discussion of the rich dynamic characteristic of the discrete neuron models, which could be useful for the biological and medical experiments in the future.
本文首先在参考全耦合的Stuart-Landau方程组的基础上,研究了全耦合的Rulkov神经元系统,整个系统因为参数σ的不同而分为两组.假设用p表示非活动振子所占比率,则在第一部分中,重点分析了在p=O,k=0时整个系统的神经元动态,并用中心流形的方法讨论了单个神经元不动点的稳定性,尤其详细的分析了Neimark-Sacker分岔的性质,给出了单个神经元活动与静止的分界线.同时也分析了其它分岔,根据这些分岔可以明确判定单个神经元是处于方形簇放电,锥形簇放电,峰放电,还是静息态.
其次,讨论了当p≠0时的耦合Rulkov神经元系统,取定两个参数σ1,σ2,分别使单个Rulkov神经元模型处于活动状态和静息状态,从而将整个系统因为参数σ的不同而分为两个集群,即活动振幅较大的神经元组和活动振幅非常小(认为是非活动)的神经元组,随着不活跃元素所占比率p的增加,全耦合神经元系统震荡行为会在某一临界值终止.在这部分中,重点讨论了全耦合神经元系统活动状态和非活动状态之间的变迁,研究了全耦合的基于映射的Rulkov神经元系统中(k,p)的结构相图,其中k是耦合强度,p是非活动元素所占比率.
最后,对本文的研究内容进行了总结,本文的研究成果为更全面完整的讨论离散神经元模型的动态转迁性质奠定了基础.
Neurons in human brain play a key role in the processing, generation, and transmission of information. The stability of fixed point in a single neuron can change corresponding to different parameter values, and the active and rest state of neurons can also take place a obvious transition due to the external stimulus. The coupl-ing neuron network is a very complex high dimensional nonlinear dynamical system, its amplitude of oscillation may go to a small range under some circumstances. Similarly, a population of coupled nonlinear oscillators may age when the fraction of non-self-oscillatory elements increases. Likewise, a global oscillation state will turn into a quiescence state as the proportion of inactive elements exceeds a critical value.
In this paper, we at first introduce the coupling Rulkov neuron network, and divide the entire network into two clusters with regard to different values of σ. We discuss the single Rulkov neuron model when p=0, k=0, here p is the ratio of inactive oscillators, and k is the coupling strength, and take into account the stability of fixed point in a single Rulkov neuron by using the method of center manifold. We also analyze the Neimark-Sacker bifurcation, and other bifurcations of single Rulkov neuron, gives a boundary between the activity and the inactivity of a single neuron.
Then we discuss the coupling Rulkov neuron model when p≠O, here we choose two different values of σ1, σ2. We divide the entire system into two clusters, active and inactive neuron groups. Our study demonstrates that the global oscillation state will turn into a quiescence state at a critical value as the proportion p of inactive elements increases. In particular, we discuss the Aging Transitions between the active and the resting state, and one of very important results is to present the phase diagrams of structure (k, p) in the map-based global coupling neurons.
Finally, we summarize the whole contents of this paper. What we have done provides a basic mathematical foundation for a more comprehensive and complete discussion of the rich dynamic characteristic of the discrete neuron models, which could be useful for the biological and medical experiments in the future.
引文
[1]寿天德,神经生物学,高等教育出版社,2002.
[2]Izhikevich E. M., Neural excitability,spiking and bursting, Int.J.Bifurcation and Chaos,10(2000),1171-1266.
[3]Izhikevich E. M., Dynamical systems in neuroscience:the geometry of excitability and bursting, The MIT press, 2007.
[4]Izhikevich E. M., Synchronization of elliptic bursters, SIAM Rev.,43(2001),315-344.
[5]Izhikevich E. M., Simple model of spiking neurons, IEEE Trans. Neural Networks, 14(2003),1569-1572.
[6]范少光等.人体生理学,北京：北京医科大学出版社,2000.
[7]朗斯塔夫.神经科学,韩济生等译,北京：科学出版社,2000.
[8]Rinzel J., A formal classification of bursting mechanisms in excitable sys-tems. In: Ma-thematical topics in population biology, morphogenesis and neu-rosciences, eds. Tera-moto,E. & Yamaguti,M.,Lecture Notes in Biomathematics, (Springer-Verlag, Berlin) Vol.71,(1987).
[9]Rinzel J. Bursting oscillation in an excitable membrane model. In:ordinary and part-ial differential equations.eds.Sleeman B D,Jarvis R J(Springer-Verlag,Berlin),1985. 304-316.
[10]H. Daido and K. Nakanishi. Aging Transition and Universal Scaling in Oscillator Neet-works. Phys. Rev. Lett.,93(10):104101,2004.
[11]H. Daido and K. Nakanishi, Phys. Rev. Lett.96, 054101(2006).
[12]H. Daido and K. Nakanishi. Aging and clustering in globallycoupled oscillators. Phys. Rev. E, 75:056206,2007.
[13]Izhikevich, E. M., Desai, N., Walcott, E. & Hoppen-steadt, F. C. [2003]\Bursts as a unit of neural in-formation: Selective communication via resonance,"Trends Neurosci.26,161-167.
[14]Izhikevich, E. M. [2004] \Which model to use for cortical spiking neurons?" IEEE Trans. Neural Networks 15,1063-1070.
[15]FitzHugh R., Impulses and physiological states in theoretical models of nerve me-mbrane, Biphysical Journal, 1961,1; 445-466.
[16]N. F. Rulkov. Regularization of synchronized chaotic bursts. Phys. Rev.Lett.,86(1): 183-186,2001.
[17]N. F. Rulkov. Modeling of spiking-bursting neural behavior using two-dimensional map. Phys. Rev. E, 65:041922, 2002.
[18]N. F. Rulkov, I. Timofeev, and M. Bazhenov. Oscillationsin large-scale cortical networks:Map-based model. J. Comp. Neurosci.,17:203-223,2004.
[19]N. F. Rulkov and M. Bazhenov. Oscillatins and synchrony in large-scale cortical network models. J. Biol. Phys.,34:279-299,2008.
[20]Aihara K., Takabe T., and Toyoda M., Chaotic neural networks, Phys. Lett. A, 1990,144(6,7),333-340.
[21]Chialvo D. R., Generic excitable dynamics on a two-dimensional map, Chaos, So-litons and Fractals,1995,5,3-4,461-479.
[22]Cazelles B., Courbage M., and Rabinovich M., Anti-phase regularization of coupl-ed chaotic maps modelling bursting neurons, Europhys. Lett.,2001,56(4),504-509.
[24]Ibarz B.,Hongjun Cao,Miguel A.F.Sanjuan,[2008] Bursting regimes in map-based neu-ron models coupled through fast threshold modulation, Physical Re-view. E77(2008), 051918.
[25]Hongjun Cao, Borja Ibarz & Miguel A.F.Sanjuan, Hybrid discrete-time neural networks, Phil. Trans. R. Soc. A. 368(2010),5071-5086.
[26]G. Tanaka, Y. Okada, and K. Aihara, Phys. Rev. E 82, 035202(R)(2010).
[27]G. Tanaka and K. Aihara. Collective Skipping: A periodic Phase Locking in of Ensembles Bursting Oscillators.Europhys. Lett,78(1):10003,2007.
[28]Hongjun Cao, Miguel A. F. Sanjuan. A mechanism for elliptic-like bursting and synchronization of bursts in a map-based neuron network, Cogn Process,2008.
[29]陈予恕.非线性振动系统的分岔和混沌理论,北京：高等教育出版社,1993.
[30]Wiggins S. Introduction to applied nonlinear dynamical systems and chaos. New York: Springer-Verlag;1990.
[31]K. Nakanishi and H. Daido, Prog. Theor. Phys. Suppl.161,173 (2006).
[32]H. Daido, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 807(1997).
[33]D. Pazo and E. Montbrio, Phys. Rev. E 73, 055202(R)(2006).
[34]S. H. Strogatz, Physica D 143,1 (2000).
[35]J. A. Acebron, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, Rev. Mod. Phys. 77,137(2005).
[36]A. T. Winfree, The Geometry of Biological Time (Springer, New York, 2001).
[37]Borja Ibarz, Hongjun Cao, and Miguel A. F. Sanjuan. Bursting regimes in map-based neuron models coupled through fast threshold modulation, Physical Review E 77, 2008,051918.
[38]N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, H. D. I. Abarbanel. Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E 1995,51:980-984.
[39]Yuri A. Kuznetsov-Elements of Applied BifurcationTheory.Springer-Verlag;1998. ISBN:03879-83821.
[40]Huijing Sun, Hongjun Cao, Chaos control and synchronization of a modified chaotic system, Chaos, Solitons & Fractals, Vol.37, Issue 5, September 2008, Pages1442-1455.
[41]H. Cao and B. Ibarz. Hybrid discrete-time neural networks. Phil. Trans. R. Soc. A., 368:5071-5086,2010.
[42]Shilnikov, A. L. & Rulkov, N. F. [2004] "Subthreshold oscillations in a map-based neron model," Phys. Lett. A 328,177-184.
[43]Tanaka, G., Ibarz, B., Sanjuan, A. F. & Aihara, K. [2006] "Synchronization and propagation of bursts in networks of coupled map neurons," Chaos 16, 013113.
[44]Hopfield JJ. Neural networks and physical systems with emergent collective computational abilities. Proc Natl Acad Sci USA, 1982,79:2554~2558.
[45]I. Leyva, I. S. Nadal, J. A. Almendral, and M. A. F. Sanjua'n, Phys. Rev. E. 74, 05-6112(2006).
[46]E.M. Izhikevich, Int. J. Bifurcation Chaos Appl. Sci. Eng. 10,1171 (2000).
[47]J. Aguirre, E. Mosekilde, M. A. F. Sanju'an, Analysis of the noise-induced bursting-spiking transition in a pancreatic β-cell model, Phys. Rev. E 69 (2004) 041910.
[48]E. M. Izhikevich, Bursting mappings, Int. J. Bifurcation Chaos 14 (2004).3847-3854.
[49]王青云,石霞,陆启韶.神经元耦合系统的同步动力学,北京：科学出版社,2008.
[50]杜艳梅,彭建华,刘延柱.FitzHugh-Nagumo神经元网络的同步振荡与联想记忆,力学季刊,2005,26(1)：66-71.
[2]Izhikevich E. M., Neural excitability,spiking and bursting, Int.J.Bifurcation and Chaos,10(2000),1171-1266.
[3]Izhikevich E. M., Dynamical systems in neuroscience:the geometry of excitability and bursting, The MIT press, 2007.
[4]Izhikevich E. M., Synchronization of elliptic bursters, SIAM Rev.,43(2001),315-344.
[5]Izhikevich E. M., Simple model of spiking neurons, IEEE Trans. Neural Networks, 14(2003),1569-1572.
[6]范少光等.人体生理学,北京：北京医科大学出版社,2000.
[7]朗斯塔夫.神经科学,韩济生等译,北京：科学出版社,2000.
[8]Rinzel J., A formal classification of bursting mechanisms in excitable sys-tems. In: Ma-thematical topics in population biology, morphogenesis and neu-rosciences, eds. Tera-moto,E. & Yamaguti,M.,Lecture Notes in Biomathematics, (Springer-Verlag, Berlin) Vol.71,(1987).
[9]Rinzel J. Bursting oscillation in an excitable membrane model. In:ordinary and part-ial differential equations.eds.Sleeman B D,Jarvis R J(Springer-Verlag,Berlin),1985. 304-316.
[10]H. Daido and K. Nakanishi. Aging Transition and Universal Scaling in Oscillator Neet-works. Phys. Rev. Lett.,93(10):104101,2004.
[11]H. Daido and K. Nakanishi, Phys. Rev. Lett.96, 054101(2006).
[12]H. Daido and K. Nakanishi. Aging and clustering in globallycoupled oscillators. Phys. Rev. E, 75:056206,2007.
[13]Izhikevich, E. M., Desai, N., Walcott, E. & Hoppen-steadt, F. C. [2003]\Bursts as a unit of neural in-formation: Selective communication via resonance,"Trends Neurosci.26,161-167.
[14]Izhikevich, E. M. [2004] \Which model to use for cortical spiking neurons?" IEEE Trans. Neural Networks 15,1063-1070.
[15]FitzHugh R., Impulses and physiological states in theoretical models of nerve me-mbrane, Biphysical Journal, 1961,1; 445-466.
[16]N. F. Rulkov. Regularization of synchronized chaotic bursts. Phys. Rev.Lett.,86(1): 183-186,2001.
[17]N. F. Rulkov. Modeling of spiking-bursting neural behavior using two-dimensional map. Phys. Rev. E, 65:041922, 2002.
[18]N. F. Rulkov, I. Timofeev, and M. Bazhenov. Oscillationsin large-scale cortical networks:Map-based model. J. Comp. Neurosci.,17:203-223,2004.
[19]N. F. Rulkov and M. Bazhenov. Oscillatins and synchrony in large-scale cortical network models. J. Biol. Phys.,34:279-299,2008.
[20]Aihara K., Takabe T., and Toyoda M., Chaotic neural networks, Phys. Lett. A, 1990,144(6,7),333-340.
[21]Chialvo D. R., Generic excitable dynamics on a two-dimensional map, Chaos, So-litons and Fractals,1995,5,3-4,461-479.
[22]Cazelles B., Courbage M., and Rabinovich M., Anti-phase regularization of coupl-ed chaotic maps modelling bursting neurons, Europhys. Lett.,2001,56(4),504-509.
[24]Ibarz B.,Hongjun Cao,Miguel A.F.Sanjuan,[2008] Bursting regimes in map-based neu-ron models coupled through fast threshold modulation, Physical Re-view. E77(2008), 051918.
[25]Hongjun Cao, Borja Ibarz & Miguel A.F.Sanjuan, Hybrid discrete-time neural networks, Phil. Trans. R. Soc. A. 368(2010),5071-5086.
[26]G. Tanaka, Y. Okada, and K. Aihara, Phys. Rev. E 82, 035202(R)(2010).
[27]G. Tanaka and K. Aihara. Collective Skipping: A periodic Phase Locking in of Ensembles Bursting Oscillators.Europhys. Lett,78(1):10003,2007.
[28]Hongjun Cao, Miguel A. F. Sanjuan. A mechanism for elliptic-like bursting and synchronization of bursts in a map-based neuron network, Cogn Process,2008.
[29]陈予恕.非线性振动系统的分岔和混沌理论,北京：高等教育出版社,1993.
[30]Wiggins S. Introduction to applied nonlinear dynamical systems and chaos. New York: Springer-Verlag;1990.
[31]K. Nakanishi and H. Daido, Prog. Theor. Phys. Suppl.161,173 (2006).
[32]H. Daido, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 807(1997).
[33]D. Pazo and E. Montbrio, Phys. Rev. E 73, 055202(R)(2006).
[34]S. H. Strogatz, Physica D 143,1 (2000).
[35]J. A. Acebron, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, Rev. Mod. Phys. 77,137(2005).
[36]A. T. Winfree, The Geometry of Biological Time (Springer, New York, 2001).
[37]Borja Ibarz, Hongjun Cao, and Miguel A. F. Sanjuan. Bursting regimes in map-based neuron models coupled through fast threshold modulation, Physical Review E 77, 2008,051918.
[38]N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, H. D. I. Abarbanel. Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E 1995,51:980-984.
[39]Yuri A. Kuznetsov-Elements of Applied BifurcationTheory.Springer-Verlag;1998. ISBN:03879-83821.
[40]Huijing Sun, Hongjun Cao, Chaos control and synchronization of a modified chaotic system, Chaos, Solitons & Fractals, Vol.37, Issue 5, September 2008, Pages1442-1455.
[41]H. Cao and B. Ibarz. Hybrid discrete-time neural networks. Phil. Trans. R. Soc. A., 368:5071-5086,2010.
[42]Shilnikov, A. L. & Rulkov, N. F. [2004] "Subthreshold oscillations in a map-based neron model," Phys. Lett. A 328,177-184.
[43]Tanaka, G., Ibarz, B., Sanjuan, A. F. & Aihara, K. [2006] "Synchronization and propagation of bursts in networks of coupled map neurons," Chaos 16, 013113.
[44]Hopfield JJ. Neural networks and physical systems with emergent collective computational abilities. Proc Natl Acad Sci USA, 1982,79:2554~2558.
[45]I. Leyva, I. S. Nadal, J. A. Almendral, and M. A. F. Sanjua'n, Phys. Rev. E. 74, 05-6112(2006).
[46]E.M. Izhikevich, Int. J. Bifurcation Chaos Appl. Sci. Eng. 10,1171 (2000).
[47]J. Aguirre, E. Mosekilde, M. A. F. Sanju'an, Analysis of the noise-induced bursting-spiking transition in a pancreatic β-cell model, Phys. Rev. E 69 (2004) 041910.
[48]E. M. Izhikevich, Bursting mappings, Int. J. Bifurcation Chaos 14 (2004).3847-3854.
[49]王青云,石霞,陆启韶.神经元耦合系统的同步动力学,北京：科学出版社,2008.
[50]杜艳梅,彭建华,刘延柱.FitzHugh-Nagumo神经元网络的同步振荡与联想记忆,力学季刊,2005,26(1)：66-71.