查看“︁菲茨休-南云方程”︁的源代码

[[File:Fitzhugh Nagumo Model Time Domain Graph.png|thumb|right|当刺激电流强度I=0.5时，膜电位对于时间的函数。]]
[[File:Fitzhugh Nagumo Phase Space Graph.png|thumb|right|蓝线为菲茨休-南云模型在相空间中的轨迹，粉线为三次{{le|零斜率線|nullcline}}，黄线为线性零斜率線。这里的刺激电流强度被设为0.5。]]
'''菲茨休-南云方程'''（Fitzhugh-Nagumo equation）是一个[[非线性偏微分方程]]，最早由[[理查德·菲茨休]]（Richard FitzHugh）于1961年提出<ref>{{cite journal |last1=FitzHugh |first1=Richard |title=Impulses and Physiological States in Theoretical Models of Nerve Membrane |journal=Biophysical Journal |date=1961-07 |volume=1 |issue=6 |pages=445–466 |doi=10.1016/S0006-3495(61)86902-6 |url=https://doi.org/10.1016/S0006-3495(61)86902-6}}</ref>，描述了在高于阈值的常电流刺激下神经元动作电位的周期性振荡<ref>{{cite book |last1=Griffiths |first1=Graham |title=Traveling Wave Analysis of Partial Differential Equations : Numerical and Analytical Methods with Matlab and Maple. |publisher=Elsevier Science |location=Burlington |isbn=9780123846532 |page=147-172}}</ref>。当时菲茨休将其称为“朋霍费尔-[[范德波尔振荡器|范德波尔模型]]（Bonhoeffer-van der Pol model）”。次年，南云仁一等人也提出了一个与该方程等效的电路<ref>{{cite journal |last1=Nagumo |first1=J. |last2=Arimoto |first2=S. |last3=Yoshizawa |first3=S. |title=An Active Pulse Transmission Line Simulating Nerve Axon |journal=Proceedings of the IRE |date=1962-10 |volume=50 |issue=10 |pages=2061–2070 |doi=10.1109/JRPROC.1962.288235}}</ref>。该方程为{{le|霍奇金-赫胥黎模型|Hodgkin-Huxley model}}的二维情形<ref name="scholarpedia">{{cite journal |last1=Izhikevich |first1=Eugene |last2=FitzHugh |first2=Richard |title=FitzHugh-Nagumo model |journal=Scholarpedia |date=2006 |volume=1 |issue=9 |pages=1349 |doi=10.4249/scholarpedia.1349}}</ref>；后者因揭示了[[枪乌贼巨大轴突]]中[[动作电位]]的产生和传导机制而分享了1963年的[[诺贝尔生理学或医学奖]]。

==方程==

用于描述枪乌贼巨大轴突中动作电位的菲茨休-南云方程如下<ref name="scholarpedia"/>：

:<math>\dot{V} = V-V^3/3 - W + I</math> 
:<math>\dot{W} = 0.08(V+0.7 - 0.8W)</math> 

其中，<math>V</math>为[[膜电位]]，<math>W</math>为回复变量，<math>I</math>为[[刺激 (生理学)|刺激]]电流的强度。该方程的一般形式可写作：

:<math>\dot{V} = f(V)  - W +  I</math> 
:<math>\dot{W} = a(bV - cW)</math> 

其中<math>f(V)</math>为三次多项式；a，b，c为常数。

==行波解==
[[File:Fitzhugh-Nagumo pde Maple animation.gif|thumb|250px|菲茨休-南云方程行波解的动画]]
菲茨休 - 南云方程的解析解如下：

<math>\frac{\partial u}{\partial t}=D\frac{\partial^2 u}{\partial^2 x^2}-u(1-u)(a-u)</math><ref>{{cite book |last1=Griffiths |first1=Graham |title=Traveling Wave Analysis of Partial Differential Equations : Numerical and Analytical Methods with Matlab and Maple. |publisher=Elsevier Science |location=Burlington |isbn=9780123846532 |page=166}}</ref>

利用[[Maple]]软件包TWSolution可得以下行波解<ref>{{cite book |last1=Griffiths |first1=Graham |title=Traveling Wave Analysis of Partial Differential Equations : Numerical and Analytical Methods with Matlab and Maple. |publisher=Elsevier Science |location=Burlington |isbn=9780123846532 |page=436}}</ref>{{#tag:ref|行波解可通过使用[[Tanh函数展开法]]得到的方程组来实现<ref>{{cite journal |last1=Wazwaz |first1=Abdul-Majid |title=The tanh method for traveling wave solutions of nonlinear equations |journal=Applied Mathematics and Computation |date=2004-07 |volume=154 |issue=3 |pages=713–723 |doi=10.1016/S0096-3003(03)00745-8}}</ref>：
<br/>
u(x, t) = 1/2+(1/2)*tanh(_C1+(1/4)*sqrt(2)*x-(1/4)*t)
<br/>
u(x, t) = 1/2+(1/2)*tanh(_C1-(1/4)*sqrt(2)*x-(1/4)*t)
<br/>
u(x, t) = 1/2-(1/2)*tanh(_C1-(1/4)*sqrt(2)*x+(1/4)*t)
<br/>
u(x, t) = 1/2-(1/2)*tanh(_C1+(1/4)*sqrt(2)*x+(1/4)*t)|group="注"}}：

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== 相关条目 ==
* {{le|霍奇金-赫胥黎模型|Hodgkin-Huxley model}}
* {{le|生物神经元模型|Biological neuron model}}
* [[计算神经科学]]
* [[反应-扩散系统]]

== 注释 ==
{{Reflist|group="注"}}

==参考文献==
{{reflist|2}}

== 拓展阅读 ==
* 谷超豪 《[[孤立子]]理论中的[[达布变换]]及其几何应用》 上海科学技术出版社
* 阎振亚著 《复杂非线性波的构造性理论及其应用》 科学出版社 2007年
* 李志斌编著 《非线性数学物理方程的行波解》 科学出版社
* 王东明著 《消去法及其应用》 科学出版社 2002
* 何青 王丽芬编著 《[[Maple]] 教程》 科学出版社 2010 ISBN 9787030177445
* Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
* Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
* Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
* Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
* Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
* Dongming Wang, Elimination Practice,Imperial College Press 2004
* David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
* George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759

{{非线性偏微分方程}}

[[category:非线性偏微分方程]]
[[category:孤立子]]