阈值神经元模型的随机共振
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摘要
随机共振的本质是在噪声和周期信号的共同驱动下,随机系统产生的一种协同效应。在神经信息处理方面,由于神经元的电活动具有非线性阈值特性,并且在噪声环境中能感受各种外界刺激,因此神经系统具有产生随机共振现象的条件,从而研究神经元的随机共振现象吸引了众多学者的关注。从广义的角度来看,随机共振是指在有噪声的系统中,系统输出的信噪比、平均峰电位时间间隔或幅值增益随输入信号频率、噪声强度及相关时间等参量的变化曲线呈现非单调性。
本文主要从两个方面来讨论阈值神经元模型的随机共振。一方面基于带阈值的积分放电模型研究了神经元在背景噪声和周期信号驱动下的随机共振现象。利用镜像法得到峰电位时间间隔(interspike interval,简记为ISI)的概率密度,通过数值积分得到平均ISI,数值模拟表明ISI概率密度函数的峰值与噪声强度的关系曲线是非单调的,平均ISI关于信号频率、噪声强度的演化分别呈现出非单调性,这些都表明在背景噪声和周期信号驱动下的神经发放确实存在随机共振现象。另一方面,已有的相关研究通常将背景噪声视为离子通道噪声,在此情况下研究噪声对系统响应的影响,而突触递质噪声的影响并未做深入研究。本文基于带阈值的积分放电模型研究了神经元在突触递质噪声和周期信号驱动下的随机共振现象。利用平均法得到系统输出幅值增益的精确表达式,考察了输出幅值增益与信号频率、噪声强度、相关时间及非对称度的关系,发现输出幅值增益随着这些参量的演化曲线在一定条件下呈非单调的,这些都表明在突触递质噪声和周期信号驱动下的神经发放确实存在随机共振现象。
Stochastic resonance(SR) is a phenomenon, where the response of a system to a weak periodic signal is enhanced at a nonzero level of noise. In neuronal information processing, SR of neuronal system with threshold driven by external period input and background noise are of great interest. In general, SR occurs during neuronal spiking driven by noise and period input which is illustrated by the signal-noise ratio(SNR), the mean interspike interval(ISI) and the amplitude gain versus the input frequency, the intensity of noise,the correlation time and so on are nonmonotomous.
This article mainly discussed two aspects of SR of neuron models with threshold. On one hand, SR of an integrate-and-fire neuronal model with threshold subject to white background noise and input signal is investigated.The probability density function of ISI and the mean ISI are obtained by the method of mirror image. Numerical simulation shows that the evolution of the peak height of probability density function with regard to the noise intensity and the mean ISI versus the input frequency is nonmonotomous. Those show that SR occurs during neuronal spiking driven by background noise and periodic input. On the other hand, most researches only have taken into account the channel noise and they didn't consider the synaptic noise. In our work, stochastic resonance of an integrate-and-fire neuron model with threshold subject to synaptic noise and input signal is investigated. The amplitude gain of the output signal is obtained by the method of average. Numerical simulation shows that the evolution of the amplitude gain with regard to the input frequency is nonmonotonous in different intensity, correlation time and asymmetry of the synaptic noise. Those show that SR occurs during neuronal spiking driven by synaptic noise and period input.
本文主要从两个方面来讨论阈值神经元模型的随机共振。一方面基于带阈值的积分放电模型研究了神经元在背景噪声和周期信号驱动下的随机共振现象。利用镜像法得到峰电位时间间隔(interspike interval,简记为ISI)的概率密度,通过数值积分得到平均ISI,数值模拟表明ISI概率密度函数的峰值与噪声强度的关系曲线是非单调的,平均ISI关于信号频率、噪声强度的演化分别呈现出非单调性,这些都表明在背景噪声和周期信号驱动下的神经发放确实存在随机共振现象。另一方面,已有的相关研究通常将背景噪声视为离子通道噪声,在此情况下研究噪声对系统响应的影响,而突触递质噪声的影响并未做深入研究。本文基于带阈值的积分放电模型研究了神经元在突触递质噪声和周期信号驱动下的随机共振现象。利用平均法得到系统输出幅值增益的精确表达式,考察了输出幅值增益与信号频率、噪声强度、相关时间及非对称度的关系,发现输出幅值增益随着这些参量的演化曲线在一定条件下呈非单调的,这些都表明在突触递质噪声和周期信号驱动下的神经发放确实存在随机共振现象。
Stochastic resonance(SR) is a phenomenon, where the response of a system to a weak periodic signal is enhanced at a nonzero level of noise. In neuronal information processing, SR of neuronal system with threshold driven by external period input and background noise are of great interest. In general, SR occurs during neuronal spiking driven by noise and period input which is illustrated by the signal-noise ratio(SNR), the mean interspike interval(ISI) and the amplitude gain versus the input frequency, the intensity of noise,the correlation time and so on are nonmonotomous.
This article mainly discussed two aspects of SR of neuron models with threshold. On one hand, SR of an integrate-and-fire neuronal model with threshold subject to white background noise and input signal is investigated.The probability density function of ISI and the mean ISI are obtained by the method of mirror image. Numerical simulation shows that the evolution of the peak height of probability density function with regard to the noise intensity and the mean ISI versus the input frequency is nonmonotomous. Those show that SR occurs during neuronal spiking driven by background noise and periodic input. On the other hand, most researches only have taken into account the channel noise and they didn't consider the synaptic noise. In our work, stochastic resonance of an integrate-and-fire neuron model with threshold subject to synaptic noise and input signal is investigated. The amplitude gain of the output signal is obtained by the method of average. Numerical simulation shows that the evolution of the amplitude gain with regard to the input frequency is nonmonotonous in different intensity, correlation time and asymmetry of the synaptic noise. Those show that SR occurs during neuronal spiking driven by synaptic noise and period input.
引文
[1]R. Benzi, A. Sutera, A. Vulpiana. The mechanism of stochastic resonance[J]. J. Phys.A,1981,14:453-457.
[2]S. Fauve and F. Helslot. Stochastic resonance in a bistable system[J]. Phys. Lett. A,1983,97:5-7.
[3]B.McNamara and K. Wiesenfeld, Theory of stochastic resonance[J]. Phys. Rev. A,1989,39:4854-4869.
[4]Frank Moss, Lawrence M. Ward, Walter G. Sannita. Stochastic resonance and sensory information processing:a turorial and review of application[J]. Clinical Neurophysiology,2004,115:267-281.
[5]Simonotto E, Raini M. Visual perception of stochastic resonance[J]. Phys Rev Lett,1997,78:1186-1189.
[6]Douglass JK, Wilkens L, Pantazelou E, Moss F. Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance[J]. Nature,1993,365:337-340.
[7]Levin JE, Mille JP. Broadband neural encoding in the cricket cercal sensory system enhanced by stochastic resonance[J]. Nature,1996,380:165-168.
[8]Russell DF, Wilkens LA, Moss F. Use of behavioral stochastic resonance by paddle fish for feeding[J]. Nature,1999,402:291294.
[9]Hu Gang, T Ditzinger, CZ Ning, H Haken. Stochastic resonance without external periodic force[J]. Phys. Rev. Lett,1993,71:807-810.
[10]A Longtin. Autonomous stochastic resonance in bursting neurons[J]. Phys. Rev. E,1997,55:868-876.
[11]Gu HG, Ren W, Lu QS, et al. Integer multiple spiking in the neuronal pacemaker without external stimulation[J]. Phys. Lett. A,2001,285:63-68.
[12]Srebro R, Malladi P. Stochastic resonance of the visually evoked potential[J]. Phys. Rev. E,1999,59:2566-2570.
[13]Mori T, Kai S. Noise-induced entrainment and stochastic resonance in human brain waves[J]. Phys. Rev. Lett,2002,88:1-4.
[14]E. Manjarrez, O. Diez-Martinez, I. Mendez, A. Flores. Stochastic resonance in human electroenphalographic activity elicited by mechanical tactile stimuli[J]. Neuroscience Letters.2002,324:213-216.
[15]Priplata A, Niemi J, Salen M, et al. Noise-enhanced human balance control[J]. Phys. Rev. Lett,2002,89:238101.
[16]Gammaitoni L, Hanggi P, Jung P, et al. Stochastic resonance[J]. Revs of Mod Phys,1998,70:223-283.
[17]Jaramillo F, Wiesenfeld K. Physiological noise level enhances mechanoelectr ical transduction in hair cells[J]. Chaos, Solutions and Fractals,2000,11:1869-1874.
[18]Bulsara A, Lowen S, Rees C. Cooperative behavior in the periodically modulated Wiener process:Noise-induced complexity in a model neutron[J]. Phys. Rev. E,1994,49:4989-5000.
[19]Bulsara A, Elston T, Doering C,et al. Cooperative behavior in periodically driven noisy integrate-and-fire models of neuronal dynamics[J]. Phys.Rev.E, 1996,53:3958-3969.
[20]Silva M, Lyra M, Vermelho M. Analog study of the first passage time problem driven by power-law distributed noise[J]. Physics A,2005,348:85-96.
[21]Verechtchaguina T,Sokolov I, Geier L. Interspike interval densities of resonate and fire neurons[J].BioSystems,2007,89:63-68.
[22]Elston T C, Bulsara A R. First passage statistics of periodically driven models for neural dynamics[J]. Biosystems,1997,40:37-43.
[23]Shimokawa T, Pakdaman K, Sato S. Time-scale matching in the response of a leaky integrate-and-fire neuron model to periodic stimulus with additive noise[J]. Phys. Rev. E,1999,59:3427-3443.
[24]Yang YM, Xu JX, Xie Y. A further insight into stochastic resonance in an integrate-and-fire neuron with noisy periodic input[J]. Chaos, Solutions and Fractals,2005,25:165-170.
[25]B.Lindner, J. Garcia-Ojalvo, A. Neiman, L. Schimansky-Geier. Effects of noise in excitable systems[J]. Physics Reports,2004,392,321.
[26]Long ZC, Qin YG. Stochastic resonance driven by time-modulated neuro-transmitter random point trains[J]. Phys.Rev.Lett,2003,91:2081031-208 1034.
[27]Shimokawa T, Pakdaman K, Sato S. Time-scale matching in the response of a leaky integrate-and-fire neuron model to periodic stimulus with additive noise[J]. Phys. Rev. E,1999,59:3427-3443.
[28]Lindner B, Garcia-Ojalbo J, Neiman A, et al. Effects of noise in excitable systems[J]. Physics Reports,2004,392:321-424.
[29]Brunel B,Chance F S,Fourcaud N,et al. Effects of synaptic noise and filtering on the frequency response of spiking neurons[J]. Phys.Rev.Let,2001,86:2186-2189.
[30]Li JH, Han YX. Phenomenon of stochastic resonance caused by multiplicative asymmetric dichotomous noise[J]. Phys.Rev.E,2006,74:051151-051159.
[31]徐科.神经生物学纲要[M].科学出版社,2001.
[32]L.B.莱维坦,L.K.卡茨玛克.神经元：细胞和分子生物学[M].科学出版社,2001.
[33]Tass P. A. Resetting biological oscillators a stochastic approach[J]. J. Biol. Phy,1996,22:122-155.
[34]Tass P. A. Phase resetting in medicine and biology[M]. Springer Verlag,1984.
[35]Kuramoto. Y. Chemical oscillations, waves, and turbulence[M]. Spring Verlag ,1984.
[36]R.L. Stratonovich. Some Markov methods in the theory of stochastic processes in nonlinear dynamical systems[M]. In:Noise in nonlinear dynamical systems, ed. By F. Moss, P.V.E. McClintock. Cambridge University Press, Cambridge, 1990, p.16.
[37]V.I. Tikhonov, M.A. Mironov. Markovian processes[M]. Soviet. Radio, Moscow,1979.
[38]C.W. Gardiner. Handbook of stochastic methods[M]. Springer Series in Synergetics, Springer,Berlin,1982, Vol.13.
[39]张太荣.统计动力学[M].北京：金工业出版社,2007.
[40]E. Wong. Stochastic processes in information and dynamical systems [M]. McGraw Series in system Science, New York,1971.
[41]L. Arnold. Stochastic differential equations, theory and applications[M]. Wileyinterscience,1974.
[42]Berdichevsky V, Gitterman M. Stochastic resonance in linear systems subject to multiplicative and additive noise[J]. Phys. Rev. E,1999,60:1494-1499.
[43]Li JH and Han YX. Phenomenon of stochastic resonance caused by multiplicative asymmetric dichotomous noise[J]. Phys. Rev. E,2006,74: 051115.
[44]Gerstner W, Kistler W M. Spiking neuron models-signal Neurons, populations, plasticity[M]. Cambridge:Cambridge University Press,2002.
[45]Yu YG, Wang W, Wang JF, Liu F. Resonance-enhanced signal detection and transduction in the Hodgkin-Huxley neuronal systems[J]. Phys Rev E,2001, 63(2):021907.
[46]Guo F, Zhou YR, Jiang SQ, Gu TX. Stochastic resonance in a bias linear system with multiplicative and additive noise[J]. Chinese Physics,2006, 15(05):5027-5033.
[47]Gitterman M. Underdamped oscillator with fluctuating damping[J]. Physics A, 2004,37(22):5729-5736.
[48]张季谦,辛厚文.噪声在非线性化学体系中作用的研究现状[J].化学进展.2001,13(4)：241-250.
[49]White JA, Rubinstein JT, Kay AR. Channel noise in neurons[J]. Trends Neuroscience,2000,23:131-137.
[50]Bark K, Sanya M. Robust stochastic resonance for simple threshold neurons[J]. Phys Rev E,2004,79:031911.
[2]S. Fauve and F. Helslot. Stochastic resonance in a bistable system[J]. Phys. Lett. A,1983,97:5-7.
[3]B.McNamara and K. Wiesenfeld, Theory of stochastic resonance[J]. Phys. Rev. A,1989,39:4854-4869.
[4]Frank Moss, Lawrence M. Ward, Walter G. Sannita. Stochastic resonance and sensory information processing:a turorial and review of application[J]. Clinical Neurophysiology,2004,115:267-281.
[5]Simonotto E, Raini M. Visual perception of stochastic resonance[J]. Phys Rev Lett,1997,78:1186-1189.
[6]Douglass JK, Wilkens L, Pantazelou E, Moss F. Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance[J]. Nature,1993,365:337-340.
[7]Levin JE, Mille JP. Broadband neural encoding in the cricket cercal sensory system enhanced by stochastic resonance[J]. Nature,1996,380:165-168.
[8]Russell DF, Wilkens LA, Moss F. Use of behavioral stochastic resonance by paddle fish for feeding[J]. Nature,1999,402:291294.
[9]Hu Gang, T Ditzinger, CZ Ning, H Haken. Stochastic resonance without external periodic force[J]. Phys. Rev. Lett,1993,71:807-810.
[10]A Longtin. Autonomous stochastic resonance in bursting neurons[J]. Phys. Rev. E,1997,55:868-876.
[11]Gu HG, Ren W, Lu QS, et al. Integer multiple spiking in the neuronal pacemaker without external stimulation[J]. Phys. Lett. A,2001,285:63-68.
[12]Srebro R, Malladi P. Stochastic resonance of the visually evoked potential[J]. Phys. Rev. E,1999,59:2566-2570.
[13]Mori T, Kai S. Noise-induced entrainment and stochastic resonance in human brain waves[J]. Phys. Rev. Lett,2002,88:1-4.
[14]E. Manjarrez, O. Diez-Martinez, I. Mendez, A. Flores. Stochastic resonance in human electroenphalographic activity elicited by mechanical tactile stimuli[J]. Neuroscience Letters.2002,324:213-216.
[15]Priplata A, Niemi J, Salen M, et al. Noise-enhanced human balance control[J]. Phys. Rev. Lett,2002,89:238101.
[16]Gammaitoni L, Hanggi P, Jung P, et al. Stochastic resonance[J]. Revs of Mod Phys,1998,70:223-283.
[17]Jaramillo F, Wiesenfeld K. Physiological noise level enhances mechanoelectr ical transduction in hair cells[J]. Chaos, Solutions and Fractals,2000,11:1869-1874.
[18]Bulsara A, Lowen S, Rees C. Cooperative behavior in the periodically modulated Wiener process:Noise-induced complexity in a model neutron[J]. Phys. Rev. E,1994,49:4989-5000.
[19]Bulsara A, Elston T, Doering C,et al. Cooperative behavior in periodically driven noisy integrate-and-fire models of neuronal dynamics[J]. Phys.Rev.E, 1996,53:3958-3969.
[20]Silva M, Lyra M, Vermelho M. Analog study of the first passage time problem driven by power-law distributed noise[J]. Physics A,2005,348:85-96.
[21]Verechtchaguina T,Sokolov I, Geier L. Interspike interval densities of resonate and fire neurons[J].BioSystems,2007,89:63-68.
[22]Elston T C, Bulsara A R. First passage statistics of periodically driven models for neural dynamics[J]. Biosystems,1997,40:37-43.
[23]Shimokawa T, Pakdaman K, Sato S. Time-scale matching in the response of a leaky integrate-and-fire neuron model to periodic stimulus with additive noise[J]. Phys. Rev. E,1999,59:3427-3443.
[24]Yang YM, Xu JX, Xie Y. A further insight into stochastic resonance in an integrate-and-fire neuron with noisy periodic input[J]. Chaos, Solutions and Fractals,2005,25:165-170.
[25]B.Lindner, J. Garcia-Ojalvo, A. Neiman, L. Schimansky-Geier. Effects of noise in excitable systems[J]. Physics Reports,2004,392,321.
[26]Long ZC, Qin YG. Stochastic resonance driven by time-modulated neuro-transmitter random point trains[J]. Phys.Rev.Lett,2003,91:2081031-208 1034.
[27]Shimokawa T, Pakdaman K, Sato S. Time-scale matching in the response of a leaky integrate-and-fire neuron model to periodic stimulus with additive noise[J]. Phys. Rev. E,1999,59:3427-3443.
[28]Lindner B, Garcia-Ojalbo J, Neiman A, et al. Effects of noise in excitable systems[J]. Physics Reports,2004,392:321-424.
[29]Brunel B,Chance F S,Fourcaud N,et al. Effects of synaptic noise and filtering on the frequency response of spiking neurons[J]. Phys.Rev.Let,2001,86:2186-2189.
[30]Li JH, Han YX. Phenomenon of stochastic resonance caused by multiplicative asymmetric dichotomous noise[J]. Phys.Rev.E,2006,74:051151-051159.
[31]徐科.神经生物学纲要[M].科学出版社,2001.
[32]L.B.莱维坦,L.K.卡茨玛克.神经元：细胞和分子生物学[M].科学出版社,2001.
[33]Tass P. A. Resetting biological oscillators a stochastic approach[J]. J. Biol. Phy,1996,22:122-155.
[34]Tass P. A. Phase resetting in medicine and biology[M]. Springer Verlag,1984.
[35]Kuramoto. Y. Chemical oscillations, waves, and turbulence[M]. Spring Verlag ,1984.
[36]R.L. Stratonovich. Some Markov methods in the theory of stochastic processes in nonlinear dynamical systems[M]. In:Noise in nonlinear dynamical systems, ed. By F. Moss, P.V.E. McClintock. Cambridge University Press, Cambridge, 1990, p.16.
[37]V.I. Tikhonov, M.A. Mironov. Markovian processes[M]. Soviet. Radio, Moscow,1979.
[38]C.W. Gardiner. Handbook of stochastic methods[M]. Springer Series in Synergetics, Springer,Berlin,1982, Vol.13.
[39]张太荣.统计动力学[M].北京：金工业出版社,2007.
[40]E. Wong. Stochastic processes in information and dynamical systems [M]. McGraw Series in system Science, New York,1971.
[41]L. Arnold. Stochastic differential equations, theory and applications[M]. Wileyinterscience,1974.
[42]Berdichevsky V, Gitterman M. Stochastic resonance in linear systems subject to multiplicative and additive noise[J]. Phys. Rev. E,1999,60:1494-1499.
[43]Li JH and Han YX. Phenomenon of stochastic resonance caused by multiplicative asymmetric dichotomous noise[J]. Phys. Rev. E,2006,74: 051115.
[44]Gerstner W, Kistler W M. Spiking neuron models-signal Neurons, populations, plasticity[M]. Cambridge:Cambridge University Press,2002.
[45]Yu YG, Wang W, Wang JF, Liu F. Resonance-enhanced signal detection and transduction in the Hodgkin-Huxley neuronal systems[J]. Phys Rev E,2001, 63(2):021907.
[46]Guo F, Zhou YR, Jiang SQ, Gu TX. Stochastic resonance in a bias linear system with multiplicative and additive noise[J]. Chinese Physics,2006, 15(05):5027-5033.
[47]Gitterman M. Underdamped oscillator with fluctuating damping[J]. Physics A, 2004,37(22):5729-5736.
[48]张季谦,辛厚文.噪声在非线性化学体系中作用的研究现状[J].化学进展.2001,13(4)：241-250.
[49]White JA, Rubinstein JT, Kay AR. Channel noise in neurons[J]. Trends Neuroscience,2000,23:131-137.
[50]Bark K, Sanya M. Robust stochastic resonance for simple threshold neurons[J]. Phys Rev E,2004,79:031911.