本論文給出了細胞內外離子濃度與電位勢的動態模擬，進而產生出動作電位。動作電位的產生來自於細胞膜兩側的電化學電勢差。我們討論的離子種類包含鈉、鉀及氯離子。這像模擬涉及泊松─能斯特─普朗克方程及霍奇金-赫克斯利模型。前者給出了描述離子擴散和電泳行為的標準模型。後者給出了離子通道機制與電路的轉換關係。我們想要結合並比較兩者的結果，然後試著證實後者可由前者得出。此研究中用到的空間離散方法包含有限體積法及擬譜法。在將偏微分方程以半離散形式轉成系統常微分方程後，透過MATLAB裡的ode15s求解器處理對時間的積分。

This thesis presents a dynamic simulation of intracellular and extracellular ionic concentrations and electric potential, then create an action potential, which is generated by a difference of the electrochemical potential between two sides of a cell membrane. Ion species including Sodium, Potassium and Chlorine. This simulation would involve Poisson-Nernst-Planck (PNP) system and Hodgkin–Huxley (HH) model. The former gives a standard model for describing behaviors of ionic diffusion and electrophoresis. The latter gives a transformation between mechanism of ion channels and a circuit. We want to combine and compare the results of these two models, then try to verify that the PNP equations can reduce to the HH model. In this study, methodologies are based on finite volume method and pseudospectral method for space discretization. After changing the semi-discrete scheme to a system of ODE by method of lines(MOL), we use ode15s solver on MATLAB to handle for time integration.

口試委員審定書iii
誌謝iii
Acknowledgements iv
摘要iv
Abstract vi
List of Figures ix
List of Tables ix
1 Introduction 1
2 Mathematical Models 4
2.1 PNP equations: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 HH model: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Nondimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Dimensionless equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.6 Interface conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Mapping 11
4 Numerical Method I :Finite Volume Method 13
4.1 Cartesian coordinate: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Numerucal Scheme: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3 Polar coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.4 Numerical scheme: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5 Numerical Method II: Pseudospectral Method 21
5.1 Derivative matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.2 Poisson equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.2.1 Cartesian coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.2.2 Polar coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.3 Nernst-Planck equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.3.1 Cartesian coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.3.2 Polar coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.3.3 Treatment for the pole conditions . . . . . . . . . . . . . . . . . . . . 27
6 Numerical Results and Discussions 29
6.1 Comparisons of action potentials . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.2 Comparisons between PNP and HH . . . . . . . . . . . . . . . . . . . . . . . 31
6.2.1 At resting state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.2.2 At depolarization stage . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.3 Reduce PNP to HH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.3.1 Cartesian coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.3.2 Polar coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
7 Conclusion 39
References 40
A Notation Table 40
B Initial 42

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