神经系统的非线性动力学分析与控制
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摘要
神经系统是复杂的非线性系统,本文从非线性角度研究其丰富的特性。
HH模型采用四阶非线性微分方程组的形式,定量描述了细胞跨膜电压与离子电流的变化过程,并准确反映了实验现象。本文以HHM模型为研究对象,选取漏电导和钠离子通道反电动势作为分岔参数,采用高维方程的代数判据进行多参数分岔分析。证明了当漏电导和钠离子通道反电动势变化时,HHM模型将产生动态Hopf分岔,即细胞膜受刺激后产生连续的动作电位,这符合医学实验中观察到的病变现象。通过对外电场下改进HH模型的多参数分岔分析,证明外电场参数的变化将引发模型的分岔,这可能是电磁辐射致病的原因。
类似HH模型的高阶非线性系统,具有丰富的物理意义,但很难得到其解析解;并且数值分析十分复杂,渐近方法为解决这一问题提供了一种途径。本文利用Tikhonov渐进系统理论及Zeeman模型的分析方法,研究了改进HH模型的渐近特性,证明了HH模型仍具有可兴奋等复杂特性。
本文从峰峰时间间隔,研究改进HH模型在极低频外加电场作用下神经元的放电模式。通过数值仿真,出现了神经元的周期n放电、快/慢簇放电、“调幅调制”放电、“多正弦调制”放电、混沌放电等多种放电模式;及一种神经元通往混沌放电的路径——音叉分岔;通过对外电场作用下的HH模型引入不同的时间常数,将其四阶非线性系统按照快慢系统分析,得到在hopf分岔点附近出现的一种特殊峰放电模式——慢峰放电模式,证实这种模式是混沌放电。
基于功能电刺激的思想,本文使用自适应模糊控制实现对神经元HH模型和FHN模型的混沌同步控制,这种方法综合了自适应控制和模糊控制的优点,实现未知非线性部分和干扰的逼近。仿真结果证实了方法的有效性。
基于计算机数字控制技术,本文使用离散变论域模糊控制理论实现离散混沌系统的控制。这一方法避免了采用传统的T-S模糊模型需要在线调节全部的后件参数等缺点。本文使用的控制算法通过在线调节伸缩因子一个参数实现连续变换论域,动态调节模糊规则,提高了控制性能,仿真结果实现了Lorenz混沌系统的控制。
本文对针刺获得的神经系统的电信号利用连续小波变换、离散小波变换、功率谱和非线性动力学等理论对针灸信号进行全面的分析,证明不同手法具有不同的量化特征,为针刺电信息的量化研究进行初步尝试。
The neural system is a complex nonlinear system and its characteristics are studied from a nonlinear theory perspective in this dissertation.
The Hodgkin-Huxley (HH) model is a four-dimensional differential equation which quantitatively describes the dynamical process of trans-membrane voltage and ionic current that observed in experiments. In this dissertation, the algebra criterion in high-dimensional equations is employed to perform the bifurcation analysis of HHM model in which the leakage conductance and the anti-electromotive of sodium ion channel are chosen to be the bifurcation parameters. The conclusion reveals that the Hopf bifurcation will occur in the HHM model when the leakage conductance and the anti-electromotive sodium ion channel changes. That is, the membrane produces continuous action potentials in response to external stimulation, which is consistent with the pathological phenomena observed in medical experiments. The multi-parameters bifurcation analysis of HHM model exposed to external electrical field shows that the change of parameters of external electrical field will cause the bifurcation of the model, which might be the reason that electromagnetism radiation leads to disease.
Analytical solution to high-dimensional nonlinear system like Hodgkin-Huxley model is impossible to be obtained, and the numerical analysis is also very complex. The asymptotic methods can solve this problem through describing the fast and slow process of the system in the same time. This dissertation investigates the modified HH model by employing the Tikhonov asymptotic theory and the analysis method of Zeeman model, and proves that important characteristics of HH model such as excitability are still preserved.
In this dissertation, the firing patterns of HH model exposed to ELF external electric field are studied from the view of ISI (interspikes interval). Various firing patterns such as period-n spiking, fast/slow burst, amplitude modulation firing, multi-sinusoidal firing and chaotic burst are found through numerical simulation. The tuning fork bifurcation, a route to chaos of the neuron, has also been found. By introducing different time constants into the HH model exposed to external electrical field, the four-dimensional nonlinear system is analyzed in terms of fast/slow system. A special firing pattern, slow spiking, is obtained near the Hopf bifurcation point, and it is proved that this firing pattern is chaotic.
An adaptive fuzzy control method based on functional electric stimulation is employed to realize the synchronization in HH models and FHN models. This control method which combines the advantages of adaptive control and fuzzy control can approach the unknown nonlinear part as well as the disturbances. The results of simulation demonstrate the availability of the design. This new method provides a route to treatment of diseases such as epilepsy.
In this dissertation, based on the computer digital control technique, a discrete variable universe fuzzy control strategy, which avoids the disadvantage that all the consequent parameters must be tuned on-line in the traditional fuzzy controller based on T-S model, is employed to control the discrete chaos system. This control strategy realizes continuous changes of the universe and dynamical fuzzy rules by on-line tuning a single parameter -- the contraction and expansion factor. Thus the performance of control is improved. The simulation results give the perfect control of Lorenz chaos system.
Theories such as continuous wavelet transform, discrete wavelet transform, power spectrum and nonlinear dynamics are employed to investigate the electrical signals of neural system recorded in the experiment of acupuncture and it is demonstrated that the signals induced by different acupuncture method have different quantitative characteristics. This attempt gives elementary study on quantifying of the electrical information of acupuncture.
HH模型采用四阶非线性微分方程组的形式,定量描述了细胞跨膜电压与离子电流的变化过程,并准确反映了实验现象。本文以HHM模型为研究对象,选取漏电导和钠离子通道反电动势作为分岔参数,采用高维方程的代数判据进行多参数分岔分析。证明了当漏电导和钠离子通道反电动势变化时,HHM模型将产生动态Hopf分岔,即细胞膜受刺激后产生连续的动作电位,这符合医学实验中观察到的病变现象。通过对外电场下改进HH模型的多参数分岔分析,证明外电场参数的变化将引发模型的分岔,这可能是电磁辐射致病的原因。
类似HH模型的高阶非线性系统,具有丰富的物理意义,但很难得到其解析解;并且数值分析十分复杂,渐近方法为解决这一问题提供了一种途径。本文利用Tikhonov渐进系统理论及Zeeman模型的分析方法,研究了改进HH模型的渐近特性,证明了HH模型仍具有可兴奋等复杂特性。
本文从峰峰时间间隔,研究改进HH模型在极低频外加电场作用下神经元的放电模式。通过数值仿真,出现了神经元的周期n放电、快/慢簇放电、“调幅调制”放电、“多正弦调制”放电、混沌放电等多种放电模式;及一种神经元通往混沌放电的路径——音叉分岔;通过对外电场作用下的HH模型引入不同的时间常数,将其四阶非线性系统按照快慢系统分析,得到在hopf分岔点附近出现的一种特殊峰放电模式——慢峰放电模式,证实这种模式是混沌放电。
基于功能电刺激的思想,本文使用自适应模糊控制实现对神经元HH模型和FHN模型的混沌同步控制,这种方法综合了自适应控制和模糊控制的优点,实现未知非线性部分和干扰的逼近。仿真结果证实了方法的有效性。
基于计算机数字控制技术,本文使用离散变论域模糊控制理论实现离散混沌系统的控制。这一方法避免了采用传统的T-S模糊模型需要在线调节全部的后件参数等缺点。本文使用的控制算法通过在线调节伸缩因子一个参数实现连续变换论域,动态调节模糊规则,提高了控制性能,仿真结果实现了Lorenz混沌系统的控制。
本文对针刺获得的神经系统的电信号利用连续小波变换、离散小波变换、功率谱和非线性动力学等理论对针灸信号进行全面的分析,证明不同手法具有不同的量化特征,为针刺电信息的量化研究进行初步尝试。
The neural system is a complex nonlinear system and its characteristics are studied from a nonlinear theory perspective in this dissertation.
The Hodgkin-Huxley (HH) model is a four-dimensional differential equation which quantitatively describes the dynamical process of trans-membrane voltage and ionic current that observed in experiments. In this dissertation, the algebra criterion in high-dimensional equations is employed to perform the bifurcation analysis of HHM model in which the leakage conductance and the anti-electromotive of sodium ion channel are chosen to be the bifurcation parameters. The conclusion reveals that the Hopf bifurcation will occur in the HHM model when the leakage conductance and the anti-electromotive sodium ion channel changes. That is, the membrane produces continuous action potentials in response to external stimulation, which is consistent with the pathological phenomena observed in medical experiments. The multi-parameters bifurcation analysis of HHM model exposed to external electrical field shows that the change of parameters of external electrical field will cause the bifurcation of the model, which might be the reason that electromagnetism radiation leads to disease.
Analytical solution to high-dimensional nonlinear system like Hodgkin-Huxley model is impossible to be obtained, and the numerical analysis is also very complex. The asymptotic methods can solve this problem through describing the fast and slow process of the system in the same time. This dissertation investigates the modified HH model by employing the Tikhonov asymptotic theory and the analysis method of Zeeman model, and proves that important characteristics of HH model such as excitability are still preserved.
In this dissertation, the firing patterns of HH model exposed to ELF external electric field are studied from the view of ISI (interspikes interval). Various firing patterns such as period-n spiking, fast/slow burst, amplitude modulation firing, multi-sinusoidal firing and chaotic burst are found through numerical simulation. The tuning fork bifurcation, a route to chaos of the neuron, has also been found. By introducing different time constants into the HH model exposed to external electrical field, the four-dimensional nonlinear system is analyzed in terms of fast/slow system. A special firing pattern, slow spiking, is obtained near the Hopf bifurcation point, and it is proved that this firing pattern is chaotic.
An adaptive fuzzy control method based on functional electric stimulation is employed to realize the synchronization in HH models and FHN models. This control method which combines the advantages of adaptive control and fuzzy control can approach the unknown nonlinear part as well as the disturbances. The results of simulation demonstrate the availability of the design. This new method provides a route to treatment of diseases such as epilepsy.
In this dissertation, based on the computer digital control technique, a discrete variable universe fuzzy control strategy, which avoids the disadvantage that all the consequent parameters must be tuned on-line in the traditional fuzzy controller based on T-S model, is employed to control the discrete chaos system. This control strategy realizes continuous changes of the universe and dynamical fuzzy rules by on-line tuning a single parameter -- the contraction and expansion factor. Thus the performance of control is improved. The simulation results give the perfect control of Lorenz chaos system.
Theories such as continuous wavelet transform, discrete wavelet transform, power spectrum and nonlinear dynamics are employed to investigate the electrical signals of neural system recorded in the experiment of acupuncture and it is demonstrated that the signals induced by different acupuncture method have different quantitative characteristics. This attempt gives elementary study on quantifying of the electrical information of acupuncture.
引文
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