霍奇金-赫克斯利模型與泊松─能斯特─普朗克方程之比較

Comparison of Hodgkin Huxley model and Poisson Nernst Planck equations

陳博允(Bo-Yun Chen)

指導教授 ： 林太家

國立臺灣大學/理學院/應用數學科學研究所/碩士(2019年)

https://doi.org/10.6342/NTU201903092

摘要

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。

關鍵字

泊松─能斯特─普朗克方程 ； 霍奇金-赫克斯利模型 ； 有限體積 ； 擬譜法

並列摘要

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.

並列關鍵字

Poisson-Nernst-Planck equations ； Hodgkin–Huxley model ； finite volume method ； pseudospectral method

參考文獻

[1] A. L. Hodgkin and A. F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology, 117(4):500–544, 1952.

[2] Brian J. Kirby. Micro- and Nanoscale Fluid Mechanics, chapter 11. Cambridge University Press, 2013.

[3] Zilong Song, Xiulei Cao, and Huaxiong Huang. Electroneutral models for dynamic poisson-nernst-planck systems. Phys. Rev. E, 97:012411, Jan 2018.

[4] RANDALL J. LEVEQUE. Finite Volume Methods for Hyperbolic Problems, chapter 10.4. Cambridge University Press, 2002.

[5] Li-Lian Wang Jie Shen, Tao Tang. Spectral Methods: Algorithms, Analysis and Applications, chapter 3.3. Springer Science and Business Media, 2011.