Ermentrout-Kopell规范模型

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The Ermentrout-Kopell canonical model is better known as the "theta model" and is a simple one-dimensional model for the spiking of a neuron. It is closely related to the quadratic integrate and fire neuron. The model takes the following form:

Ermentrout-Kopell规范模型被称为“θ模型”，是一个尖峰神经元neuron的简单一维模型。它与二次积分和放电神经元quadratic integrate and fire neuron密切相关。模型的形式如下:

[math]\displaystyle{ \label{theta} \frac{d\theta}{dt} = 1-\cos\theta + (1+\cos\theta) I(t) (1) }[/math]

文件:Snic.gif

Saddle node on limit cycle.

where [math]\displaystyle{ I(t) }[/math] are the inputs to the model. The variable [math]\displaystyle{ \theta }[/math] lies on the unit circle and ranges between 0 and [math]\displaystyle{ 2\pi\ . }[/math] When [math]\displaystyle{ \theta=\pi }[/math] the neuron "spikes", that is, it produces an action potential.

其中[math]\displaystyle{ I(t) }[/math]即模型的输入。变量[math]\displaystyle{ \theta }[/math]位于单位圆上，取值范围在0到[math]\displaystyle{ 2\pi\ }[/math]之间。当[math]\displaystyle{ \theta=\pi }[/math]时，神经元“点火”会产生动作电位action potential。

图1:极限环上的鞍节点。

派生Derivation

The theta model is the normal form for the saddle-node on a limit cycle bifurcation (SNIC). (Caution! Do not confuse this with a saddle-node of limit cycles in which a pair of limit cycles collide and annihilate.) Figure 1 shows a schematic of the bifurcation as a parameter varies through the critical value of [math]\displaystyle{ I=0. }[/math] When [math]\displaystyle{ I\lt 0 }[/math] there is a pair of equilibria. One of the equilibria is a saddle point with a one-dimensional unstable manifold. The two branches of the unstable manifold form an invariant circle with the stable equilibrium point. (See saddle-node bifurcation where this is called a saddle-node homoclinic bifurcation.) In neurophysiological terms, the stable manifold of the saddle point forms a true threshold for the neuron. In Figure 1 , the stable manifold is shown in green. Any initial conditions to the left of the manifold will be attracted to the stable equilibrium (in blue), while initial data to the right of the manifold will make a large excursion around the circle before returning to the rest state.

θ模型是极限环分岔(SNIC)鞍节点的标准形式。(注意!不要将此与极限环的鞍节点混淆，在鞍节点中，一对极限环会碰撞和湮灭。)图1显示一分岔示意图，参数随临界值[math]\displaystyle{ I=0. }[/math]变化，当[math]\displaystyle{ I\lt 0 }[/math]时，存在一对平衡。其中一个平衡是具有一维不稳定流形unstable manifold的鞍点。不稳定流形的两个分支构成一个具有稳定平衡点的不变圆。(参见鞍节分岔saddle-node bifurcation，这被称为鞍节同宿分岔。)在神经生理学术语中，鞍点的稳定流形stable manifold为神经元形成了一个真正的阈值threshold。在图1中，稳定歧管以绿色显示。流形左边的任何初始条件都将被吸引到稳定平衡(蓝色部分)，而流形右边的初始数据将在返回到静止状态之前绕圆进行较大的偏移。

Near the transition, the local dynamics is like a saddle-node bifurcation and has the form:

在过渡附近，局部动力学类似于鞍节点分岔，其形式为:

[math]\displaystyle{ \label{sn} \frac{dx}{dt} = x^2 + I. }[/math]

For [math]\displaystyle{ I\lt 0 }[/math] (resp [math]\displaystyle{ I\gt 0 }[/math]) there are two (resp no) equilibria. In the case where [math]\displaystyle{ I\gt 0 }[/math] solutions to this differential equation "blow up" in finite time

如[math]\displaystyle{ I\lt 0 }[/math](resp [math]\displaystyle{ I\gt 0 }[/math]) ，存在两个(resp no)均衡。当[math]\displaystyle{ I\gt 0 }[/math]的情况下，这个微分方程的解在有限时间内“爆炸”

[math]\displaystyle{ T_{blow} =1/2\, \left( -2\,\arctan \left( {\frac { {\it x(0)}}{\sqrt {I}} } \right) +\pi \right) {\frac {1}{\sqrt {I}}}. }[/math]

Here [math]\displaystyle{ x(0) }[/math] is the initial condition. In particular, suppose we reset [math]\displaystyle{ x(t) }[/math] to [math]\displaystyle{ -\infty }[/math] when it blows up to [math]\displaystyle{ +\infty }[/math] Then the total transit time is

这里[math]\displaystyle{ x(0) }[/math]是初始条件。特别地，假设我们将爆炸时间[math]\displaystyle{ x(t) }[/math]重置为[math]\displaystyle{ -\infty }[/math]，那么总穿越时间为

[math]\displaystyle{ T_{per} = \frac{\pi}{\sqrt{I}}. }[/math]

Thus the frequency (the reciprocal of the period) goes to zero as the parameter approaches criticality from the right. This observation led Rinzel & Ermentrout to remark that this bifurcation corresponded to Hodgkin's Class I excitable membranes while the more familiar Andronov-Hopf bifurcation corresponded to Class II excitability. The latter is best exemplified by the classical Hodgkin-Huxley model for the squid axon. Neural models undergoing a SNIC bifurcation include the Connor-Stevens model for crab leg axons, the Wang-Buzsaki model for inhibitory interneurons, the Hindmarsh-Rose model, and the Morris-Lecar model under some circumstances.

因此，频率(周期的倒数)从右边接近临界状态时趋于零。这一观察结果使Rinzel & Ermentrout注意到，这种分岔对应于霍奇金I类兴奋性膜Class I excitable membranes，而更熟悉的Andronov-Hopf分岔对应于II类兴奋性膜Class II excitability。乌贼轴突的经典霍奇金-赫胥黎模型 Andronov-Hopf bifurcation是后者的最佳例证。经历SNIC分支的神经模型包括蟹腿轴突的Connor-Stevens模型、抑制性中间神经元的Wang-Buzsaki模型、Hindmarsh-Rose模型和某些情况下的Morris-Lecar模型。

The quadratic integrate and fire model is essentially equation \eqref{sn} with a finite value for the blow up and a finite reset. It is closely related to the Izhikevich neuron, which has an additional linear variable modeling the dynamics of a recovery variable.

二次积分和火灾模型本质上是方程(2)，爆炸有限值，重置有限值。它与伊兹克维奇神经元lzhikevich neuron密切相关，它有一个额外的线性变量来建模一个恢复变量的动力学。

To derive the theta model (1)from the saddle-node(2), we make a simple change of variables, [math]\displaystyle{ x=\tan(\theta/2) }[/math] from which it is simple calculus to obtain the theta model. We note that as [math]\displaystyle{ \theta }[/math] approaches [math]\displaystyle{ \pi }[/math] from the left, [math]\displaystyle{ x }[/math] goes to [math]\displaystyle{ +\infty\ . }[/math] The theta model collapses the real line to the circle. The SNIC is a global bifurcation, so that to rigorously prove the equivalence of the SNIC and the theta model requires quite a bit more work than is shown in this formal derivation. The interested reader should consult the paper by Ermentrout and Kopell (1986).

为了从鞍节点(2)推导出模型(1)，我们做了一个简单的变量变换[math]\displaystyle{ x=\tan(\theta/2) }[/math]，由此得到θ模型是简单的微积分。我们注意到[math]\displaystyle{ x=\tan(\theta/2) }[/math]，当从[math]\displaystyle{ \pi }[/math]左边接近时，[math]\displaystyle{ x }[/math]会变成[math]\displaystyle{ +\infty\ . }[/math]，θ模型会把到圆的实线折叠起来。SNIC是一个全局分支global bifurcation，因此，要严格地证明SNIC和模型的等价性，需要比在这个形式推导中显示的更多的工作。有兴趣的读者可以参考Ermentrout和Kopell(1986)的论文。

The advantage of the theta model over the quadratic integrate and fire model is that there is no reset to deal with and the resulting dynamics are smooth and stay bounded. However, as the Izhikevich neuron demonstrates, it is sometimes useful to have the freedom to reset the dynamics anywhere.

与二次积分和火灾模型相比，θ模型的优点是不需要处理重置，产生的动力学是平滑的，保持有界的。然而，正如伊兹克维奇神经元所显示的那样，有时可以在任何地方自由重置动态是有用的。

有噪声的θ模型Noisy theta models

To obtain the noisy theta model, we start with the original quadratic model with additive white noise:

为了得到有噪声的θ模型，我们从原始的加性白噪声的二次模型开始:

[math]\displaystyle{ dx = (x^2+I(t))dt + \sigma dW }[/math]

and make the change of variables, [math]\displaystyle{ x=\tan(\theta/2) \ , }[/math] where we are careful to account for the fact that we must use Ito Calculus. The resulting noisy theta model takes the form:

进行变量的替换，[math]\displaystyle{ x=\tan(\theta/2) \ , }[/math]我们要小心地考虑到必须使用伊藤微积分。得到的噪声θ模型的形式如下:

[math]\displaystyle{ d\theta = (1-\cos\theta + [1+\cos\theta](I(t)-\frac{\sigma^2}{2}\sin\theta))dt + \sigma(1+\cos\theta)dW. }[/math]

Note that the sine term in the equation comes from the Ito change of variables. For small noise, that is, [math]\displaystyle{ \sigma \ll 1 \ , }[/math] this term can be neglected and one gets the equation analyzed in Gutkin and Ermentrout.

注意方程中的sin项来自于变量的伊藤变换。对于小噪声，也就是说，[math]\displaystyle{ \sigma \ll 1 \ , }[/math]这一项可以忽略，人们可以在Gutkin和Ermentrout中分析方程。

θ模型的相位重置曲线The phase resetting curve for the theta model

In the oscillatory regime, the phase resetting curve (PRC) can be computed. Izhikevich computed the PRC for finite size stimuli by adding an instantaneous pulse of size [math]\displaystyle{ a }[/math] to the quadratic version of the model. From this, he obtained a map from the old phase to the new phase:

在振荡状态下，可以计算相位重置曲线phase resetting curve(PRC)。Izhikevich通过在模型的二次元中加入一个尺寸为[math]\displaystyle{ a }[/math]瞬时脉冲来计算有限尺寸刺激的PRC。由此，他得到了一张从旧相位到新相位的地图:

[math]\displaystyle{ \theta \to 2 \arctan (\tan \frac{\theta}{2} + a) }[/math]

Note that the PRC is set in phase coordinates rather than in time coordinates.

注意PRC是在相位坐标而不是时间坐标中设置的。

The adjoint or infinitesimal PRC is very easy to compute using the quadratic version of the model (Ermentrout, 1996) . For any scalar oscillator model, [math]\displaystyle{ du/dt=f(u) \ , }[/math] the adjoint is [math]\displaystyle{ u_a(t)=1/du/dt \ . }[/math] Since the "periodic" solution to the quadratic model is

使用二次版的模型，伴随或无穷小PRC是非常容易计算的(Ermentrout, 1996)。对于任何标量振子模型，[math]\displaystyle{ du/dt=f(u) \ , }[/math]。伴随应为[math]\displaystyle{ u_a(t)=1/du/dt \ . }[/math]由于二次模型的周期解为

[math]\displaystyle{ u(t) = -\sqrt{I}\cot(\sqrt{I}t) }[/math]

the PRC is

PRC是

[math]\displaystyle{ PRC(t) =\frac{1}{du/dt} = \frac{1}{2\sqrt{I}}(1-\cos(2\sqrt{I}t)). }[/math]

This is non-negative and has been suggested as the signature of neurons undergoing a SNIC bifurcation.

这是非阴性的，并被认为是神经元经历SNIC分叉的标志。

与其他模型的联系Relation to Other Models

The canonical model described here is closely related to other phase models arising in applications. For example, the classical description of forced oscillations in a damped pendulum is given by the differential equation

本文所描述的正则模型与应用中出现的其他相模型phase models密切相关。例如，阻尼摆的强迫振动的经典描述是由微分方程给出的

[math]\displaystyle{ \label{pendulum} \mu\ddot\theta+f \dot\theta+\ddot H(t)\cos\theta+\ddot V(t)\sin\theta=\omega }[/math]

where [math]\displaystyle{ \theta }[/math] is the angle between the down direction and the radius through the center of mass, [math]\displaystyle{ \mu }[/math] is the mass, [math]\displaystyle{ f }[/math] is the coefficient of friction (damping), [math]\displaystyle{ \ddot H }[/math] and [math]\displaystyle{ \ddot V }[/math] are the horizontal and vertical accelerations of the support point, and [math]\displaystyle{ \omega(t) }[/math] is the torque applied to the support point (see Chester (1975)). This model has been applied to describe mechanical systems (eg., pendulums), micro-electromechanical systems (Hoppensteadt-Izhikevich (2001)), rotating electrical machinery (Stoker (1951)), power systems (Salam (1984)), electronic circuits, such as phase-locked loops (Viterbi (1966)) and parametric amplifiers (Horowitz-Hill (1980)), quantum mechanical devices (Feynman (1963)), and neurons (see VCON).

式中[math]\displaystyle{ \theta }[/math]为向下方向与圆心半径之间的夹角，[math]\displaystyle{ \mu }[/math]为质量，[math]\displaystyle{ f }[/math]为摩擦系数(阻尼)，[math]\displaystyle{ \ddot H }[/math]和[math]\displaystyle{ \ddot V }[/math]分为支撑点的水平和垂直加速度，[math]\displaystyle{ \omega(t) }[/math]为施加在支撑点上的扭矩(见Chester(1975))。这个模型已被用于描述机械系统。、旋转电机(Stoker(1951))、电力系统(Salam(1984))、电子电路，如锁相环(Viterbi(1966))和参数放大器(Horowitz-Hill(1980))、量子力学器件(Feynman(1963))和神经元(参见VCON)。

The model（1） is equivalent to（3） when [math]\displaystyle{ \mu\to 0 }[/math] and [math]\displaystyle{ H(t)\ , }[/math] [math]\displaystyle{ V(t)\ , }[/math] and [math]\displaystyle{ \omega }[/math] are chosen appropriately.

当[math]\displaystyle{ \mu\to 0 }[/math]和[math]\displaystyle{ H(t)\ , }[/math][math]\displaystyle{ V(t)\ , }[/math]和[math]\displaystyle{ \omega }[/math]选择适当时，模型(1)与(3)等价。

Although \eqref{theta} may have been referred to as being the theta-equation, this causes confusion when working with theta rhythms in the brain, and so is not preferred. Hoppensteadt and Izhikevich (1997) suggested to call it the Ermentrout-Kopell canonical model.

虽然(1)可能被称为θ方程，但这在处理大脑中的θ节律时会引起混乱，因此不可取。hoppenstead和Izhikevich(1997)建议将其称为Ermentrout-Kopell规范模型。

参考文献References

W. Chester, The forced oscillations of a simple pendulum, J. Inst. Maths. Appl. 15 (1975) 298-306.

G.B. Ermentrout and N. Kopell, Parabolic bursting in an excitable system coupled with a slow oscillation, SIAM-J.-Appl.-Math, 46 (1986), 233-253.

B. Ermentrout, Type I membranes, phase resetting curves, and synchrony. Neural-Comput. 8, (1996) 979-1001

F.C. Hoppensteadt, E.M. Izhikevich, Synchronization of MEMS resonators and mechanical neurocomputing, IEEE Trans. Circs. Systems, 48 (2001) 133-138.

F. C. Hoppendsteadt and E.M. Izhikevich (1997) Weakly Connected Neural Networks. Springer-Verlag, NY.

J. Stoker, Nonlinear Vibrations, Interscience, 1951.

F.M. Salam, J.E. Marsden, P.P. Varaiya, Arnold diffusion in the swing equations of a power system, IEEE Trans. Circuits and Systems, 31 (1984) 673-688.

A.J. Viterbi, Principles of Coherent Communication, McGraw-Hill, New York, 1966.

P. Horowitz, W. Hill, The Art of Electronics, Camb. U. Press, 1989.

R.P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, Addison-Wesley, Menlo Park, 1965.

Internal references

Yuri A. Kuznetsov (2006) Andronov-Hopf bifurcation. Scholarpedia, 1(10):1858.

John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.

Valentino Braitenberg (2007) Brain. Scholarpedia, 2(11):2918.

James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.

Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.

Harold Lecar (2007) Morris-Lecar model. Scholarpedia, 2(10):1333.

James Murdock (2006) Normal forms. Scholarpedia, 1(10):1902.

Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.

Yuri A. Kuznetsov (2006) Saddle-node bifurcation. Scholarpedia, 1(10):1859.

Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

Frank Hoppensteadt (2006) Voltage-controlled oscillations in neurons. Scholarpedia, 1(11):1599.