菲茨休-南云方程

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^[1]，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^[2]。当时菲茨休将其称为“朋霍费尔-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^[3]。该方程为霍奇金-赫胥黎模型（英语：Hodgkin-Huxley model）的二维情形^[4]；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^[4]：

{\dot {V}}=V-V^{3}/3-W+I

{\dot {W}}=0.08(V+0.7-0.8W)

其中， $V$ 为膜电位， $W$ 为回复变量， $I$ 为刺激电流的强度。该方程的一般形式可写作：

{\dot {V}}=f(V)-W+I

{\dot {W}}=a(bV-cW)

其中 $f(V)$ 为三次多项式；a，b，c为常数。

菲茨休 - 南云方程的解析解如下：

${\frac {\partial u}{\partial t}}=D{\frac {\partial ^{2}u}{\partial ^{2}x^{2}}}-u(1-u)(a-u)$ ^[5]

利用Maple软件包TWSolution可得以下行波解^[6]^{[注 1]}：

^ 行波解可通过使用Tanh函数展开法得到的方程组来实现^[7]：
u(x, t) = 1/2+(1/2)*tanh(_C1+(1/4)*sqrt(2)*x-(1/4)*t)
u(x, t) = 1/2+(1/2)*tanh(_C1-(1/4)*sqrt(2)*x-(1/4)*t)
u(x, t) = 1/2-(1/2)*tanh(_C1-(1/4)*sqrt(2)*x+(1/4)*t)
u(x, t) = 1/2-(1/2)*tanh(_C1+(1/4)*sqrt(2)*x+(1/4)*t)

^ FitzHugh, Richard. Impulses and Physiological States in Theoretical Models of Nerve Membrane. Biophysical Journal. 1961-07, 1 (6): 445–466. doi:10.1016/S0006-3495(61)86902-6.
^ Griffiths, Graham. Traveling Wave Analysis of Partial Differential Equations : Numerical and Analytical Methods with Matlab and Maple.. Burlington: Elsevier Science. : 147-172. ISBN 9780123846532.
^ Nagumo, J.; Arimoto, S.; Yoshizawa, S. An Active Pulse Transmission Line Simulating Nerve Axon. Proceedings of the IRE. 1962-10, 50 (10): 2061–2070. doi:10.1109/JRPROC.1962.288235.
^ ^4.0 ^4.1 Izhikevich, Eugene; FitzHugh, Richard. FitzHugh-Nagumo model. Scholarpedia. 2006, 1 (9): 1349. doi:10.4249/scholarpedia.1349.
^ Griffiths, Graham. Traveling Wave Analysis of Partial Differential Equations : Numerical and Analytical Methods with Matlab and Maple.. Burlington: Elsevier Science. : 166. ISBN 9780123846532.
^ Griffiths, Graham. Traveling Wave Analysis of Partial Differential Equations : Numerical and Analytical Methods with Matlab and Maple.. Burlington: Elsevier Science. : 436. ISBN 9780123846532.
^ Wazwaz, Abdul-Majid. The tanh method for traveling wave solutions of nonlinear equations. Applied Mathematics and Computation. 2004-07, 154 (3): 713–723. doi:10.1016/S0096-3003(03)00745-8.

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