Stabilization to the standing wave in a simple caricature of the nerve equation
Stabilization to the standing wave in a simple caricature of the nerve equation
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神经方程简单漫画中驻波的稳定性
DOI：
10.1002/cpa.3160390403
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发表时间：
1986
影响因子：
3
通讯作者：
V. Moll
中科院分区：
文献类型：
--
作者：
H. McKean;V. Moll
1952年，Hodgkin-Hwdey描述了鱿鱼视神经中神经冲动的传导；见Hodgkm[9]。要模拟的生理事实是，如果神经被刺激到低于阈值水平，那么信号就会衰减，并且不会传递任何信息，而如果它被刺激到阈值以上，那么信号就会经历快速变化，变成或多或少固定形状的脉冲序列，这些脉冲沿着神经行进，几乎没有失真。方程的形式是其中f是u及其过去的非线性函数。数学问题是对可能的波解Ti(x-kt)进行分类，并证明每个具有适当初始大小和形状的解u(x，t)，以适当的速度k跟踪，稳定到其中之一的平移：lirntT，u(x+kt，t)=U(x-m)。Rinzel[16]、Cohen[L]和Hadeler[7]回顾了数学模型及其几个简化。第一个简化是由Fitzhugh[6]提出的，他注意到(1)与范德波尔方程之间的相似性。Nagumo等人[15]使用这个方程来模拟隧道二极管网络的神经传递。方程是a2u+u(1-u)(u-u)+u，Au在轴2
The conduction of the nervous impulse in the optical nerve of a squid was described by Hodgkin-Hwdey in 1952; see Hodgkm [9]. The physiological fact to be modeled is that if the nerve is stimulated below a threshold level then the signal damps out and no information is transmitted, while if it is stimulated above threshold then the signal undergoes a fast change into a train of pulses of more or less fixed shape which travel down the nerve with little distortion. The form of the equation is in which f is a non-linear function of u and its past. The mathematical problem is to classify the possible wave solutions Ti (x-kt) and to prove that every solution u (x, t) of a proper initial size and shape, tracked at the proper speed k, stabilizes to a translate of one of them: lirntT, u (x+ kt, t)= U (x-m). Rinzel [16], Cohen [l], and Hadeler [7] review the mathematical model and its several simplifications. The first simplification was proposed by Fitzhugh [6] who noticed the similarity between (1) and van der Pol's equation. Nagumo et al.[15] used this equation to model nerve transmission by a tunnel diode network. The equation is a2u+ u (1-u)(u-u)+ u, au at ax2