经典Hodgkin-Huxley (H-H)离子传导模型是离子跨膜流动的基础模型,也是神经网络研究的基础.Chua在最近的研究工作中指出经典H-H模型中关于离子通道电导的定义是不合理的,证明了钾、钠离子通道应是阶数分别为一、二的两个具有时间记忆效应的忆阻元件.Chua把忆阻引入H-H模型,对此新模型的研究结果能合理解释神经元研究中出现的一些奇异现象.考虑到分数阶微积分是描述记忆特征的恰当数学工具,因此,本项目首先对H-H模型中的钾、钠离子通道建立分数阶忆阻模型,然后建立具有分数阶忆阻的H-H模型.其次,推广整数阶系统的分岔理论至分数阶系统,利用相应的分岔理论研究具有分数阶忆阻的H-H模型的动力学,分析模型的动力学现象与分数阶微积分、忆阻之间的因果关系,阐述模型的各类动力学特征的实际生理意义.最后,针对现有的分数阶系统的数值算法,结合本项目,优化相应的数值算法.

The classical Hodgkin-Huxley (H-H) model is the basic model for the transmembrane flow of ions and is also the base of the neural network research. It is pointed out in Chua's recent studies that the definition of the ion-channel conductance in H-H model is unreasonable, and it is also proved that the potassium ion-channel、sodium ion-channel is the first-order, the second-order memristor with time memory effect. By putting the memristors into H-H model, some anomalies and confusions shown in the neural network research are clearly resolved in Chua’s studies. Fractional calculus is an appropriate mathematical tool for describing memory characteristics, so the project will build the fractional-order memristor models for the potassium ion-channel and sodium ion-channel firstly, then to build H-H model with the fractional-order memristor. Secondly, some bifurcation theories for the integer-order system will be generalized in the fractional-order systems. Then the nonlinear dynamics of H-H model with the fractional-order memristor will be studied. The relationship among the nonlinear phenomenon of the new model, fractional calculus and memristor will be analysed, the biological meaning of the nonlinear dynamics shown in the new model will be described. Finally, the existed numerical methods of the fractional-order system will be optimized.