Dynamical systems theory in physiology.
Dynamical systems theory in physiology.
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DOI：
10.1085/jgp.201110668
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发表时间：
2011-07
期刊：
影响因子：
--
通讯作者：
Sherman A
中科院分区：
文献类型：
--
作者：
Sherman A
当系统受到与摆速度成比例的“摩擦”作用时，解就会以不断增加的振幅振荡。在这种情况下，我们可以说解0是不稳定的。(One可能想知道负摩擦力从何而来;我们将在下面回到这个问题。）在这些稳定和不稳定参数区域之间的边界处，B= 0，ε也将为0，并且解将是纯正弦振荡。在一个非线性系统中，类似的事情也会发生，这被称为霍普夫分叉（HB）。像XPPAUT这样的程序通过将系统线性化并计算线性化系统的常数（称为特征值）来检测这一点，从中可以获得ε和ε。当参数改变符号时，整个系统的振荡产生，最初是正弦的，但由于非线性效应，当参数远离分叉时会发生扭曲，并且可以变得高度不对称，就像神经元或神经元细胞的动作电位一样。摆没有固定的振幅;振幅取决于摆的初始位置和速度。图1中B= 0时的实线和虚线分别具有较小和较大的初速度。也可以通过简单地敲击摆锤（未示出）来改变振幅。在非线性系统中，如果振荡受到扰动，比如说，在单电池的情况下，受到短暂注入电流的扰动，那么振荡将返回到其原始幅度或转向其他行为。因此，非线性振荡也可以被表征为稳定或不稳定，并且通常被称为“极限环”，因为当时间到达无穷大时，系统在极限中被它们吸引或排斥。相比之下，线性振荡是中性稳定的;就像平面上的大理石一样，如果移动它，它不会恢复到原来的位置。
friction” that injects energy into the system in proportion to the speed of the pendulum, then the solutions oscillate with ever increasing amplitude. In this case, we would say that the solution 0 is unstable.(One may wonder where negative friction comes from; we will return to that below.) At the border between these stable and unstable parameter regimes, b= 0, would also be 0, and the solution would be a pure sinusoidal oscillation. In a nonlinear system, something similar can happen, and it is called a Hopf bifurcation (HB). Programs like XPPAUT detect this by linearizing the system and calculating constants of the linearized system, called eigenvalues, from which and can be obtained. When changes sign, an oscillation of the full system is born that is initially sinusoidal but distorts because of nonlinear effects as the parameter is moved away from the bifurcation and can become highly asymmetrical, like a neuronal or-cell action potential.There is one critical difference between the oscillations of the nonlinear system and those of the pendulum. A pendulum has no fixed amplitude; the amplitude depends on the initial position and velocity of the pendulum. The solid and dashed traces in Fig. 1 for b= 0 have smaller and larger initial velocity, respectively. The amplitude can also be changed by briefly tapping the bob (not depicted). In the nonlinear system, if the oscillation is perturbed, say, by a brief injected current in the case of the cell, then the oscillation will either return to its original amplitude or run away to some other behavior. Thus, nonlinear oscillations can also be characterized as stable or unstable and are usually referred to as “limit cycles” because the system is either attracted to or repelled by them in the limit as time goes to infinity. The linear oscillation, in contrast, is neutrally stable; like a marble on a flat surface, it does not revert to its old position if it is moved.
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DOI：
10.1085/jgp.201110611
发表时间：
2011-07
期刊：
The Journal of general physiology
影响因子：
--
作者：
Cha CY;Nakamura Y;Himeno Y;Wang J;Fujimoto S;Inagaki N;Earm YE;Noma A
通讯作者：
Noma A
影响因子：
1.2
作者：
Tabak, Joel;Toporikova, Natalia;Bertram, Richard
通讯作者：
Bertram, Richard
DOI：
10.1152/ajpendo.00194.2002
发表时间：
2003-07-01
影响因子：
5.1
作者：
Fridlyand, LE;Tamarina, N;Philipson, LH
通讯作者：
Philipson, LH
影响因子：
2
作者：
CHAY, TR
通讯作者：
CHAY, TR
影响因子：
2.1
作者：
Goel, Pranay;Sherman, Arthur
通讯作者：
Sherman, Arthur