臺灣博碩士論文加值系統

這篇論文主要是在觀察波頌-能斯特-普朗克方程組的解的行為。首先我們用擾動方法做了兩個情形：第一、在低離子濃度的假設下，我們可以推得高門-霍金-凱茲公式，這個式子可以用來解釋神經脈衝的產生；第二、得到離子流動所引起的電流和膜電壓為線性關係的充分條件。另外還以數值方法模擬離子通過不同形狀的通道時，位置和離子濃度以及電流和電壓的關係圖。最後我們用能量變分方法推導出對應不同形狀的離子通道的新波頌-能斯特-普朗克方程組；藉由新的方程，我們可以做出通道形狀為指數函數的近似解。

In this paper, we use the perturbation methods and numerical simulations to observe the behavior of the solution of Poisson-Nernst-Planck equations. First, we do the low-concentration limit case to derive the Goldman-Hodgkin-Katz formula which can be used to explain the occurrence of the nerve impulse. In addition, we obtain a sufficient condition such that current and voltage hold a linear relationship. And for the channel with different wall shapes, we find the numerical solutions. Furthermore, by the energetic variational approach, we derive the modifier Nernst-Planck equation corresponding different geometries of channel. By the modifier equation, we find an approximate solution when the wall shape function of ion channel is an exponential function.

誌謝………………………………………………………………………………..... i
中文摘要…………………………………………………………………………… ii
英文摘要…………………………………………………………………………... iii
1. Introduction …...………………………………………………………… 1
1.1 Ion Channels of Nervous System ………….…………………………… 1
1.2 Poisson-Nernst-Planck Equations …..………………………………….. 3
2. Asymptotic Analysis …………………............……………………………….. 4
2.1 Low-Concentration Case …………………..…………………………… 5
2.2 Channel with a Linear i-v Relation …………….…………………….... 10
3. Numerical Simulation ……………………………………………………….. 15
3.1 Wall Function is a Straight Line ……………………………………… 16
3.2 Wall Function is a Parabola …………………………………………... 17
3.3 Wall Function is a Cosine Function …………………………………... 20
3.4 Wall Function is an Exponential Function ……………………………. 20
4. Energetic Variational Approach ……………………………………………… 21
4.1 Derive the original NP equation by EVA ……………………………... 21
4.2 Derive the modified NP equation …………………………………..…. 23
4.3 Application of the modified NP equation ……………………………... 27
參考文獻…………………………………………………………………….…… 29

[1] W. K. Purves, D. Sadava, G. H. Orians and H. C. Heller, Life: The Science of Biology, W H Freeman & Co, 2004.
[2] J. p. Keener and J. Sneyd, Mathematical Physiology, Springer-Verlag, New York, 1998.
[3] V. Barcilon, D. -P. Chen, R. S. Eisenberg and J. W. Jerome, Qualitative prop- erties of steady-state PNP systems: perturbation and simulation study, SIAM J. APPL. MATH. Vol. 57, No. 3, p. 631-648, June 1997.
[4] Declan A. Doyle, Joao Morais Cabral, Richard A. Pfuetzner, Anling Kuo, Jacqueline M. Gulbis, Steven L. Cohen, Brian T. Chait and Roderick MacKinnon, The Structure of the Potassium Channel: Molecular basis of K+conductionandSelectivity; SCIENCE Vol. 280, p. 69-77, April 1998:
[5] R. Ryham, An Energetic Variational Approach To Mathematical Modeling Of Charged Fluids: Charge Phases, Simulation And Well Posedness, thesis, Penn- sylvania State University, 2006.