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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:
:<math>\label{theta}
\frac{d\theta}{dt} = 1-\cos\theta + (1+\cos\theta) I(t)
</math>
[[Image:Snic.gif|thumb|400px|right|Saddle node on limit cycle.|链接=Special:FilePath/Snic.gif]]
where <math> I(t) </math> are the inputs to the model. The variable <math> \theta </math> lies on the unit circle and ranges between 0 and <math>2\pi\ .</math> When <math> \theta=\pi </math> the neuron "spikes", that is, it produces an [[action potential]].
==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.) <figref>Snic.gif</figref> shows a schematic of the bifurcation as a parameter varies through the critical value of <math> I=0. </math> When <math> I<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 <figref>Snic.gif</figref>, 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.
Near the transition, the local dynamics is like a saddle-node bifurcation and has the form:
:<math>\label{sn}
\frac{dx}{dt} = x^2 + I.
</math>
For <math> I< 0 </math> (resp <math> I>0 </math>) there are two (resp no) equilibria. In the case where <math> I>0 </math> solutions to this differential equation "blow up" in finite time
:<math>
T_{blow} =1/2\, \left( -2\,\arctan \left( {\frac { {\it x(0)}}{\sqrt {I}} } \right)
+\pi \right) {\frac {1}{\sqrt {I}}}.
</math>
Here <math> x(0)</math> is the initial condition. In particular, suppose we reset <math> x(t) </math> to <math> -\infty </math> when it blows up to <math> +\infty </math> Then the total transit time is
:<math>
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.
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.
To derive the theta model \eqref{theta} from the saddle-node \eqref{sn}, we make a simple change of variables, <math> x=\tan(\theta/2) </math> from which it is simple calculus to obtain the theta model. We note that as <math>\theta</math> approaches <math>\pi</math> from the left, <math>x</math> goes to <math>+\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).
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.
==Noisy theta models==
To obtain the noisy theta model, we start with the original quadratic model with additive white noise:
:<math>
dx = (x^2+I(t))dt + \sigma dW
</math>
and make the change of variables, <math> 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:
:<math>
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> \sigma \ll 1 \ ,</math> this term can be neglected and one gets the equation analyzed in Gutkin and Ermentrout.
==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> a </math> to the quadratic version of the model. From this, he obtained a map from the old phase to the new phase:
:<math>
\theta \to 2 \arctan (\tan \frac{\theta}{2} + a)
</math>
Note that the PRC is set in phase coordinates rather than in time coordinates.
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> du/dt=f(u) \ ,</math> the adjoint is <math> u_a(t)=1/du/dt \ .</math> Since the "periodic" solution to the quadratic model is
:<math>
u(t) = -\sqrt{I}\cot(\sqrt{I}t)
</math>
the PRC is
:<math>
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.
==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
:<math>\label{pendulum}
\mu\ddot\theta+f \dot\theta+\ddot H(t)\cos\theta+\ddot
V(t)\sin\theta=\omega
</math>
where <math>\theta</math> is the angle between the down direction
and the radius through the center of mass, <math>\mu</math> is the
mass,
<math>f</math> is the coefficient of friction (damping), <math>\ddot H</math>
and
<math>\ddot V</math> are the horizontal and vertical accelerations of the
support point, and <math>\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]]).
The model \eqref{theta} is equivalent to \eqref{pendulum} when <math>\mu\to 0</math> and <math>H(t)\ ,</math> <math>V(t)\ ,</math> and <math>\omega</math> are chosen appropriately.
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''.
==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.
<!-- Authors, please check this list and remove any references that are irrelevant. This list is generated automatically to reflect the links from your article to other accepted articles in Scholarpedia. -->
<b>Internal references</b>
*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.
== See Also==
[[Integrate-and-fire neuron]],
[[Neural excitability]],
[[Quadratic integrate-and-fire neuron]],
[[Saddle-node on invariant circle bifurcation]],
[[Voltage-controlled oscillations in neurons]]
[[Category:Computational Neuroscience]]
[[Category:Models of Neurons]]
[[Category:Eponymous]]
此词条由神经动力学读书会词条梳理志愿者（Glh20100487）翻译审校，未经专家审核，带来阅读不便，请见谅
此词条由神经动力学读书会词条梳理志愿者（Glh20100487）翻译审校，未经专家审核，带来阅读不便，请见谅
= Ermentrout-Kopell canonical model =
=Ermentrout-Kopell canonical model=
'''Post-publication activity'''
'''Post-publication activity'''
Curator: Bard Ermentrout
Curator: Bard Ermentrout
* Dr. Bard Ermentrout, Dept of Mathematics, Univ Pittsburgh, Pittsburgh PA
*Dr. Bard Ermentrout, Dept of Mathematics, Univ Pittsburgh, Pittsburgh PA
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:
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:
{| class="wikitable"
{| class="wikitable"
|
|
== Contents ==
== Contents==
[hide]
[hide]
* 1 Derivation
*1 Derivation
* 2 Noisy theta models
*2 Noisy theta models
* 3 The phase resetting curve for the theta model
*3 The phase resetting curve for the theta model
* 4 Relation to Other Models
*4 Relation to Other Models
* 5 References
*5 References
* 6 See Also
*6 See Also
|}
|}
== Derivation ==
== 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 When 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.
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 When 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.
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。
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。
== Noisy theta models ==
==Noisy theta models==
To obtain the noisy theta model, we start with the original quadratic model with additive white noise:
To obtain the noisy theta model, we start with the original quadratic model with additive white noise:
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，这一项可以忽略，人们可以在Gutkin和Ermentrout中分析方程。
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，这一项可以忽略，人们可以在Gutkin和Ermentrout中分析方程。
== The phase resetting curve for the theta model ==
==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 to the quadratic version of the model. From this, he obtained a map from the old phase to the new phase:
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 to the quadratic version of the model. From this, he obtained a map from the old phase to the new phase:
这是非阴性的，并被认为是神经元经历SNIC分叉的标志。
这是非阴性的，并被认为是神经元经历SNIC分叉的标志。
== Relation to Other Models ==
==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
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
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。
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。
== References ==
==References==
* W. Chester, The forced oscillations of a simple pendulum, J. Inst. Maths. Appl. '''15''' (1975) 298-306.
*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.
*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
*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. 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.
*F. C. Hoppendsteadt and E.M. Izhikevich (1997) Weakly Connected Neural Networks. Springer-Verlag, NY.
* J. Stoker, Nonlinear Vibrations, Interscience, 1951.
*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.
*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.
*A.J. Viterbi, Principles of Coherent Communication, McGraw-Hill, New York, 1966.
* P. Horowitz, W. Hill, The Art of Electronics, Camb. U. Press, 1989.
*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.
*R.P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, Addison-Wesley, Menlo Park, 1965.
'''Internal references'''
'''Internal references'''
* Yuri A. Kuznetsov (2006) Andronov-Hopf bifurcation. Scholarpedia, 1(10):1858.
*Yuri A. Kuznetsov (2006) Andronov-Hopf bifurcation. Scholarpedia, 1(10):1858.
* John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
*John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
* Valentino Braitenberg (2007) Brain. Scholarpedia, 2(11):2918.
*Valentino Braitenberg (2007) Brain. Scholarpedia, 2(11):2918.
* James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
*James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
* Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
*Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
* Harold Lecar (2007) Morris-Lecar model. Scholarpedia, 2(10):1333.
*Harold Lecar (2007) Morris-Lecar model. Scholarpedia, 2(10):1333.
* James Murdock (2006) Normal forms. Scholarpedia, 1(10):1902.
*James Murdock (2006) Normal forms. Scholarpedia, 1(10):1902.
* Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
*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.
*Yuri A. Kuznetsov (2006) Saddle-node bifurcation. Scholarpedia, 1(10):1859.
* Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
*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.
*Frank Hoppensteadt (2006) Voltage-controlled oscillations in neurons. Scholarpedia, 1(11):1599.
== See Also ==
==See Also==
Integrate-and-fire neuron, Neural excitability, Quadratic integrate-and-fire neuron, Saddle-node on invariant circle bifurcation, Voltage-controlled oscillations in neurons
Integrate-and-fire neuron, Neural excitability, Quadratic integrate-and-fire neuron, Saddle-node on invariant circle bifurcation, Voltage-controlled oscillations in neurons
{| class="wikitable"
{| class="wikitable"