随机共振及其在神经动力学模型中的应用
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摘要
随机共振是指由于系统、系统的噪声以及系统的输入信号三者之间的协同作用，当输入噪声达到某个强度时，系统的微弱输入信号得到放大，从而使系统的输出信噪比达到最大值这样一种非线性现象。广义随机共振指系统的输出信号(信噪比、输出平均幅度等)是噪声、系统或信号的某个参数的非单调函数。相干共振指受到噪声作用的可激发系统中，存在最优噪声强度使放电序列更规整这样一种非线性现象。
本文研究色噪声作用下线性系统和非线性双稳态系统的随机共振现象，以及随机共振和相干共振在整合放电模型和FitHugh-Nagumo模型中的应用。
本文的主要成果和创新为：
1．研究了加性双值噪声和乘性双值噪声作用下一阶线性系统的随机共振现象。根据噪声特性和线性系统理论，得到了系统输出信噪比的表达式。研究表明，系统输出信噪比是加性双值噪声和乘性双值噪声的相关时间、乘性双值噪声的强度、激励周期信号频率的非单调函数。
2．研究了双值噪声作用下的二阶过阻尼线性系统的随机共振现象。研究表明，系统的输出幅度增益是双值噪声的强度和相关率、系统固有频率、系统阻尼系数，以及激励信号的频率的非单调函数，适当的噪声参数可以使系统的输出幅度增益大于无噪声时的输出幅度增益。
3．研究了双稳系统受双值噪声作用下的随机共振现象。研究表明，信噪比是双值噪声相关时间的非单调函数，且出现两个极大值，功率谱增益是双值噪声强度的非单调函数。
4．研究了非对称双稳势中受周期力和乘性色噪声作用的过阻尼布朗粒子的随机共振现象。得到了系统的准稳态概率分布函数。研究表明，系统从噪声中吸取的能量(输入能量)出现随机共振现象，其吸取的能量是噪声强度、相关时间以及系统的非对称势的非单调函数。
5．研究了色噪声作用下FitHugh-Nagumo模型的相干共振和随机共振现象。研究表明，变差系数和信噪比是色噪声相关时间、色噪声强度的非单调函数。
6．研究了神经递质以泊松点序列释放和电压门控离子通道噪声共同作用下线性整合放电模型的相干共振现象。研究表明，神经元放电的峰峰间隔是神经递质的达到强度、离子通道噪声强度的非单调函数。
7．研究了电压门控离子通道噪声和突触噪声共同作用下非线性整合放电模型的相干共振现象。研究表明，首次放电概率分布、放电率是离子通道噪声和突触噪声强度的非单调函数，适当的噪声强度可以使神经元自发放电。
Because of the cooperation between a system, the input noise and input signal of the system, at some mediate noise intensity, the input signal is amplified, thus the output signal-to-noise ratio (SNR) reaches a maximum. This nonlinear phenomenon is called stochastic resonance (SR). Stochastic resonance in a broad sense means the output signal of the system is a non-monotonic function of some parameter of the noise, of the system or parameter of the input signal. Coherence resonance (CR) characterizes a nonlinear phenomenon that a noisy excitable system responds with an excitation sequence called a "spike train", at an optimum noise intensity the spike sequence is much more regular.
In this paper, the SR phenomena of some linear and nonlinear systems are investigated. The application of the SR and CR in integral-and-fire model and FitHugh-Nagumo model is researched. The main contributions are listed below:
1. The SR in a first-order linear system subject to additive and multiplicative dichotomous-noise (DN) is investigated. By using the properties of the DN and linear-response theory, the expressions have been found for the output SNR of the system. It is shown that the SNR is a non-monotonic function of the correlation time of the additive and multiplicative DN, and it varies non-monotonously with the intensity of the multiplicative DN as well as with the external field frequency.
2. The SR phenomenon of an over-damped second-order linear system subject to the DN is investigated. Result shows that the output amplitude gain (OAG) is a non-monotonic function of the strength, the correlation rate of the DN, as well as the damping coefficient and frequency of the driving signal. In addition, by choosing appropriate parameters of the noise, the OAG of the noisy system can be larger than that of the noise-free system.
3. The SR in a bistable system subject to the DN is investigated. It is found that the SNR is a non-monotonic function of the correlation time of the DN, two peaks can occur on the curve of the SNR. The spectral power amplification (SPA) is a non-monotonic function of the strength of the DN.
4. The SR phenomenon of an overdamped Brown particle in an asymmetric bistable potential, driven by external periodical signal and multiplicative noise is investigated. The expressions have been obtained for the quasi-steady-state probability distribution function. It is found that the input energy (IE) pumped into the system by the external driving shows a SR-like behavior as a function of the strength and correlation time of the noise and as a function of the asymmetry potential of the system.
5. The CR and the SR phenomenon in the FitzHugh-Nagumo model induced by colored noise are investigated. It is shown that the coefficient of variation (CV) and the SNR are non-monotonic functions of the correlation time of and intensity of the colored noise.
6. The CR phenomenon of a linear integral-and-fire model subject to a neurotransmitter Poisson point trains and a voltage-gated noise is investigated. It is shown that the interval-spike-interval (ISI) is a non-monotonic function of the intensity of neurotransmitter arrivals trains and the strength of the voltage-gated channel noise.
7. The CR phenomenon of a nonlinear integral-and-fire model subject to a voltage-gated channel noise and a synaptic noise is investigated. It is shown that the probability distribution of the first fire (FPD) and the firing rate (FR) are non-monotonic functions of the strength of the voltage-gated channel noise and the synaptic noise. Furthermore, by choosing appropriate noise parameter, spontaneous fire of neuron can occur.
本文研究色噪声作用下线性系统和非线性双稳态系统的随机共振现象，以及随机共振和相干共振在整合放电模型和FitHugh-Nagumo模型中的应用。
本文的主要成果和创新为：
1．研究了加性双值噪声和乘性双值噪声作用下一阶线性系统的随机共振现象。根据噪声特性和线性系统理论，得到了系统输出信噪比的表达式。研究表明，系统输出信噪比是加性双值噪声和乘性双值噪声的相关时间、乘性双值噪声的强度、激励周期信号频率的非单调函数。
2．研究了双值噪声作用下的二阶过阻尼线性系统的随机共振现象。研究表明，系统的输出幅度增益是双值噪声的强度和相关率、系统固有频率、系统阻尼系数，以及激励信号的频率的非单调函数，适当的噪声参数可以使系统的输出幅度增益大于无噪声时的输出幅度增益。
3．研究了双稳系统受双值噪声作用下的随机共振现象。研究表明，信噪比是双值噪声相关时间的非单调函数，且出现两个极大值，功率谱增益是双值噪声强度的非单调函数。
4．研究了非对称双稳势中受周期力和乘性色噪声作用的过阻尼布朗粒子的随机共振现象。得到了系统的准稳态概率分布函数。研究表明，系统从噪声中吸取的能量(输入能量)出现随机共振现象，其吸取的能量是噪声强度、相关时间以及系统的非对称势的非单调函数。
5．研究了色噪声作用下FitHugh-Nagumo模型的相干共振和随机共振现象。研究表明，变差系数和信噪比是色噪声相关时间、色噪声强度的非单调函数。
6．研究了神经递质以泊松点序列释放和电压门控离子通道噪声共同作用下线性整合放电模型的相干共振现象。研究表明，神经元放电的峰峰间隔是神经递质的达到强度、离子通道噪声强度的非单调函数。
7．研究了电压门控离子通道噪声和突触噪声共同作用下非线性整合放电模型的相干共振现象。研究表明，首次放电概率分布、放电率是离子通道噪声和突触噪声强度的非单调函数，适当的噪声强度可以使神经元自发放电。
Because of the cooperation between a system, the input noise and input signal of the system, at some mediate noise intensity, the input signal is amplified, thus the output signal-to-noise ratio (SNR) reaches a maximum. This nonlinear phenomenon is called stochastic resonance (SR). Stochastic resonance in a broad sense means the output signal of the system is a non-monotonic function of some parameter of the noise, of the system or parameter of the input signal. Coherence resonance (CR) characterizes a nonlinear phenomenon that a noisy excitable system responds with an excitation sequence called a "spike train", at an optimum noise intensity the spike sequence is much more regular.
In this paper, the SR phenomena of some linear and nonlinear systems are investigated. The application of the SR and CR in integral-and-fire model and FitHugh-Nagumo model is researched. The main contributions are listed below:
1. The SR in a first-order linear system subject to additive and multiplicative dichotomous-noise (DN) is investigated. By using the properties of the DN and linear-response theory, the expressions have been found for the output SNR of the system. It is shown that the SNR is a non-monotonic function of the correlation time of the additive and multiplicative DN, and it varies non-monotonously with the intensity of the multiplicative DN as well as with the external field frequency.
2. The SR phenomenon of an over-damped second-order linear system subject to the DN is investigated. Result shows that the output amplitude gain (OAG) is a non-monotonic function of the strength, the correlation rate of the DN, as well as the damping coefficient and frequency of the driving signal. In addition, by choosing appropriate parameters of the noise, the OAG of the noisy system can be larger than that of the noise-free system.
3. The SR in a bistable system subject to the DN is investigated. It is found that the SNR is a non-monotonic function of the correlation time of the DN, two peaks can occur on the curve of the SNR. The spectral power amplification (SPA) is a non-monotonic function of the strength of the DN.
4. The SR phenomenon of an overdamped Brown particle in an asymmetric bistable potential, driven by external periodical signal and multiplicative noise is investigated. The expressions have been obtained for the quasi-steady-state probability distribution function. It is found that the input energy (IE) pumped into the system by the external driving shows a SR-like behavior as a function of the strength and correlation time of the noise and as a function of the asymmetry potential of the system.
5. The CR and the SR phenomenon in the FitzHugh-Nagumo model induced by colored noise are investigated. It is shown that the coefficient of variation (CV) and the SNR are non-monotonic functions of the correlation time of and intensity of the colored noise.
6. The CR phenomenon of a linear integral-and-fire model subject to a neurotransmitter Poisson point trains and a voltage-gated noise is investigated. It is shown that the interval-spike-interval (ISI) is a non-monotonic function of the intensity of neurotransmitter arrivals trains and the strength of the voltage-gated channel noise.
7. The CR phenomenon of a nonlinear integral-and-fire model subject to a voltage-gated channel noise and a synaptic noise is investigated. It is shown that the probability distribution of the first fire (FPD) and the firing rate (FR) are non-monotonic functions of the strength of the voltage-gated channel noise and the synaptic noise. Furthermore, by choosing appropriate noise parameter, spontaneous fire of neuron can occur.
引文
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