具位阻效應之普瓦松-能斯特-普朗克模型漸近解的研究

Asymptotic Analysis of Steric Poisson-Nernst-Planck Model for Multi-Species Ions

張偉楨(Wei-Chan Chang)

指導教授 ： 林太家

國立臺灣大學/理學院/數學研究所/碩士(2014年)

https://doi.org/10.6342/NTU.2014.00961

摘要

普瓦松-能斯特-普朗克模型模擬離子流經狹宰的離子通道中的現象。不同於基本的普瓦松-能斯特-普朗克模型，具位阻效應之普瓦松-能斯特-普朗克模型更準確地描述離子的運動行為因為具位阻效應之普瓦松-能斯特-普朗克模型加入了空間效應項以強調離子在狹宰的通道中會互相競爭去占據一定的空間。我們的目標就是去分析具位阻效應之普瓦松-能斯特-普朗克模型的漸近行為亦或是漸近穩定性。在文中我們利用線性代數中有關特徵值的技巧簡化分析上會遇到的困難。更重要的是在本文中的方法比之前文獻中運用變數變換的方法更為簡單且可以延伸到更多的狀況。另一方面，我們寫了一些程式去模擬我們遇到的各種情況。雖然我們得到了很多數值結果，然而所有的結果皆支持分析上的結論。在未來，我們將更進一步的運用具位阻效應之普瓦松-能斯特-普朗克模型去修正用來描述神經行為的霍奇金 - 赫胥黎模型。

關鍵字

具位阻效應之普瓦松-能斯特-普朗克模型 ； 漸近穩定 ； 特徵值 ； 半正定理論 ； 電導 ； 霍奇金 - 赫胥黎模型

並列摘要

Poisson-Nernst-Planck system models ionic flow in ion channels. Differ from basic Poisson-Nernst-Planck, steric Poisson-Nernst-Planck accurately describes the behaviour of ions for size effect terms are added into steric Poisson-Nernst-Planck in order to make a description of the fact that ions complete to each other to hold exact spaces in ion channel. The goal in this paper is to analyse the asymptotic behaviour we may say asymptotic stability of steric Poisson-Nernst-Planck. In the text, We make use of the skill for eigenvalues including positive semi-definite pencil in linear algebra to simplify the difficulties in the analysis. More important thing is that the way in this paper is easier and extends more cases than one which is changing variable in former literature. On the other hand, we also write some programs to simulate various situations we encounter. Though we get lots of numerical results; however, all of them support the conclusions in the analysis. Furthermore, in the future, the further goal is to take advantage of steric Poisson-Nernst-Planck systems to modify Hodgkin-Huxley equations which describe the behaviour of neuron.

並列關鍵字

Steric Poisson-Nernst-Planck ； Asymptotic stability ； Eigenvalues ； Positive semi-definite pencil ； Conductance ； Hodgkin-Huxley equations

參考文獻

[1] Richard E. Plant. The geometry of the hodgkin-huxley model. 1976.

[2] B. Hassard. Bifurcation of periodic solutions of the hodgkin-huxley model for the squid giant axon. 1977.

[3] J. Rinzel. Numerical calculation of stable and unstable periodic solutions to the hodgkin-huxley equations. 1979.

[6] M. N. Rasband. Ion channels and excitable cells. Nature Education, 2010.

[8] A. L. Hodgkin and A. F. Huxley. A quantitative description of ion currents and its applications to conduction and excitation in nerve membranes. 1952.