菲茨休-南雲方程

菲茨休-南雲方程（Fitzhugh-Nagumo equation）是一個非線性偏微分方程，最早由理察·菲茨休（Richard FitzHugh）於1961年提出^[1]，描述了在高於閾值的常電流刺激下神經元動作電位的周期性振盪^[2]。當時菲茨休將其稱為「朋霍費爾-范德波爾模型（Bonhoeffer-van der Pol model）」。次年，南雲仁一等人也提出了一個與該方程等效的電路^[3]。該方程為霍奇金-赫胥黎模型（英語：Hodgkin-Huxley model）的二維情形^[4]；後者因揭示了槍烏賊巨大軸突中動作電位的產生和傳導機制而分享了1963年的諾貝爾生理學或醫學獎。

方程

用於描述槍烏賊巨大軸突中動作電位的菲茨休-南雲方程如下^[4]：

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

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

其中， $V$ 為膜電位， $W$ 為回復變量， $I$ 為刺激電流的強度。該方程的一般形式可寫作：

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

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

其中 $f(V)$ 為三次多項式；a，b，c為常數。

菲茨休 - 南雲方程的解析解如下：

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

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

^ 行波解可通過使用Tanh函數展開法得到的方程組來實現^[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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