非线性神经元电活动的数学模型及其分析方法与计算机仿真研究
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摘要
神经元放电活动的研究是非线性科学和神经科学交叉的前沿课题,受到了数理科学和生命科学等领域学者的重视,是国际科学研究的前沿领域。
非线性动力学的突飞猛进为相关科学的发展创造了契机,在生物和医学领域体现得尤为突出,因为该领域的研究对象和系统一般都具有很强的非线性性质。非线性动力学方法的引入使得人们对过去实验观察和记录到的大量杂乱无规则的生物医学信号现在有了全新的理解和认识,它们不能再像过去那样只被看作是随机过程,而完全可能是由确定性机制所产生的混沌运动。现在,围绕着神经元放电模式、心律变化、脑功能等方面的复杂运动形式引起了科研工作者的极大兴趣和广泛注意。
神经系统是人体生理机能的重要调节系统,是通过神经元感受外界刺激,并以神经放电的方式对外界刺激信息进行编码、传递和解码。不同的放电模式反映了不同的外界刺激,相应的产生不同的生理效应。因此,深入研究外刺激与神经放电方式之间的量化关系,对更好的利用电刺激治疗疾病有重要的理论和实际意义。神经电生理活动具有复杂的非线性动力学行为。
本文以Hodgkin-Huxley(HH)模型、FitzHugh-Nagumo(FHN)模型以及神经传播型方程为研究对象,利用稳定性、定性等理论对模型本身进行理论分析;由于HH模型采用了高维微分方程组的数学表述形式,具有多变量强耦合非线性的特点,给分析工作带来了极大的困难,本文导出HH模型的一种实用合理的简化模型,并对它们进行分析;利用非线性动力学理论和方法,研究矩形脉冲和指数衰减波形脉冲刺激下神经元的放电情况,取得了以下一些主要新结果:
1.导出了HH模型的一种实用合理的简化模型,首次给出它们的孤波解,以及孤波出现的条件。
2.利用常微分方程定性理论对新的简化的HH模型、FHN模型以及神经传播型方程进行定性分析,首次给出了不同情况下,HH简化方程以及FHN方程可以用来描述神经元放电机制的数学证明。
3.构造了Lyapunov函数,对FHN模型以及非线性神经传播型方程进行了稳定性分析,首次证明了一组神经传播过程的稳定性定理,给出了其稳定性存在的充分条件。
4.利用多种方法,首次求出神经元的传播是以扭结孤波的形式传播。并得出波幅很小时出现呼吸子现象,随波幅的增大而趋于稳定状态。
5.首次给出了神经元在外部刺激作用下放电情况的数值仿真。
The research of neuron discharge activity, a some-what new subject intersecting nonlinear science and nerve science, raises many scholars concerns in the fields of mathematical science and life science. It is also a marginal and mutual scientific research field.
The rapid development of nonlinear dynamics creates a chance for the development of related sciences, which is shown especially obviously in the biological and medical science fields, because the research objects and system in the very fields generally have strong nonlinear character. The introduction of the method of nonlinear dynamics makes people have a fully new understanding of large quantities of messy and irregular biological and medical signals observed and recorded in the past researches. They are no longer regarded as random processes, but on the contrary, they may surely be chaos movement caused by qualitative mechanisms. Now, the complex movement forms, surrounding neuron discharge patterns, heart ratio variation and brain functions, arouse great interests and wide attention among scientific workers.
The nerve system is an important modulating system in the human physiological enginery. The external stimulus are taken by the neurons of the nerve system, and coded, propagated and decoded by means of various patterns of neuron discharge. Different patterns of discharge reflect the difference of the external stimulus information and induce diverse physiological effects. Therefore, a deep research into the quantitive relationship between external stimulus and nerve discharge pattern has important theological and practical meanings for the better usage of electric stimulus to cure diseases. The electrophysiological function of nerve system contains very complex nonlinear dynamical behaviors.
This paper, taking Hodgkin-Huxley(HH) model, FitzHugh-Nagumo(FHN) model and nerve propagation equation as the research objects, analyzes theologically the models themselves with the theories such as stability and quality. Many variables and strong coupling nonlinear, due to HH models adopting such math expression as high dimensional differential coefficient equation groups, bring a huge difficulty for analytic work. This paper educes the practical rational simplified model of HH model, makes an analysis of them, studies the neuron discharge conditions under stimulus of rectangle impulse and exponent attenuation wave mode impulse, using nonlinear dynamics theory and method, and gets the following results:
1. We have given a practical rational simplified model of HH model, their solitary wave solutions and the solutions exited conditions.
2. We give a new specific analysis of the simplified HH model, FHN model and nerve propagation equation, by using the qualitative theory as a series of differential equations. We give the mathematical proof which simplified HH equations; FHN equations may describe neuron discharge mechanism under different condition.
3. We have constructed a Lyapunov function and given a stability analysis of FHN model and nerve propagation equation. We proved series stability theorems and given the full conditions for stability existence.
4. We get the solitary wave is of the kink mode waves and its solution of nerve wave propagation equation by using the expansion methods of hyperbolic sine, hyperbolic tangent and others at first. We observed the breather in the small amplitude state.
5. We have given the numerical simulation of the neuron discharge condition under external stimulus.
非线性动力学的突飞猛进为相关科学的发展创造了契机,在生物和医学领域体现得尤为突出,因为该领域的研究对象和系统一般都具有很强的非线性性质。非线性动力学方法的引入使得人们对过去实验观察和记录到的大量杂乱无规则的生物医学信号现在有了全新的理解和认识,它们不能再像过去那样只被看作是随机过程,而完全可能是由确定性机制所产生的混沌运动。现在,围绕着神经元放电模式、心律变化、脑功能等方面的复杂运动形式引起了科研工作者的极大兴趣和广泛注意。
神经系统是人体生理机能的重要调节系统,是通过神经元感受外界刺激,并以神经放电的方式对外界刺激信息进行编码、传递和解码。不同的放电模式反映了不同的外界刺激,相应的产生不同的生理效应。因此,深入研究外刺激与神经放电方式之间的量化关系,对更好的利用电刺激治疗疾病有重要的理论和实际意义。神经电生理活动具有复杂的非线性动力学行为。
本文以Hodgkin-Huxley(HH)模型、FitzHugh-Nagumo(FHN)模型以及神经传播型方程为研究对象,利用稳定性、定性等理论对模型本身进行理论分析;由于HH模型采用了高维微分方程组的数学表述形式,具有多变量强耦合非线性的特点,给分析工作带来了极大的困难,本文导出HH模型的一种实用合理的简化模型,并对它们进行分析;利用非线性动力学理论和方法,研究矩形脉冲和指数衰减波形脉冲刺激下神经元的放电情况,取得了以下一些主要新结果:
1.导出了HH模型的一种实用合理的简化模型,首次给出它们的孤波解,以及孤波出现的条件。
2.利用常微分方程定性理论对新的简化的HH模型、FHN模型以及神经传播型方程进行定性分析,首次给出了不同情况下,HH简化方程以及FHN方程可以用来描述神经元放电机制的数学证明。
3.构造了Lyapunov函数,对FHN模型以及非线性神经传播型方程进行了稳定性分析,首次证明了一组神经传播过程的稳定性定理,给出了其稳定性存在的充分条件。
4.利用多种方法,首次求出神经元的传播是以扭结孤波的形式传播。并得出波幅很小时出现呼吸子现象,随波幅的增大而趋于稳定状态。
5.首次给出了神经元在外部刺激作用下放电情况的数值仿真。
The research of neuron discharge activity, a some-what new subject intersecting nonlinear science and nerve science, raises many scholars concerns in the fields of mathematical science and life science. It is also a marginal and mutual scientific research field.
The rapid development of nonlinear dynamics creates a chance for the development of related sciences, which is shown especially obviously in the biological and medical science fields, because the research objects and system in the very fields generally have strong nonlinear character. The introduction of the method of nonlinear dynamics makes people have a fully new understanding of large quantities of messy and irregular biological and medical signals observed and recorded in the past researches. They are no longer regarded as random processes, but on the contrary, they may surely be chaos movement caused by qualitative mechanisms. Now, the complex movement forms, surrounding neuron discharge patterns, heart ratio variation and brain functions, arouse great interests and wide attention among scientific workers.
The nerve system is an important modulating system in the human physiological enginery. The external stimulus are taken by the neurons of the nerve system, and coded, propagated and decoded by means of various patterns of neuron discharge. Different patterns of discharge reflect the difference of the external stimulus information and induce diverse physiological effects. Therefore, a deep research into the quantitive relationship between external stimulus and nerve discharge pattern has important theological and practical meanings for the better usage of electric stimulus to cure diseases. The electrophysiological function of nerve system contains very complex nonlinear dynamical behaviors.
This paper, taking Hodgkin-Huxley(HH) model, FitzHugh-Nagumo(FHN) model and nerve propagation equation as the research objects, analyzes theologically the models themselves with the theories such as stability and quality. Many variables and strong coupling nonlinear, due to HH models adopting such math expression as high dimensional differential coefficient equation groups, bring a huge difficulty for analytic work. This paper educes the practical rational simplified model of HH model, makes an analysis of them, studies the neuron discharge conditions under stimulus of rectangle impulse and exponent attenuation wave mode impulse, using nonlinear dynamics theory and method, and gets the following results:
1. We have given a practical rational simplified model of HH model, their solitary wave solutions and the solutions exited conditions.
2. We give a new specific analysis of the simplified HH model, FHN model and nerve propagation equation, by using the qualitative theory as a series of differential equations. We give the mathematical proof which simplified HH equations; FHN equations may describe neuron discharge mechanism under different condition.
3. We have constructed a Lyapunov function and given a stability analysis of FHN model and nerve propagation equation. We proved series stability theorems and given the full conditions for stability existence.
4. We get the solitary wave is of the kink mode waves and its solution of nerve wave propagation equation by using the expansion methods of hyperbolic sine, hyperbolic tangent and others at first. We observed the breather in the small amplitude state.
5. We have given the numerical simulation of the neuron discharge condition under external stimulus.
引文
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