Ermentrout-Kopell canonical model

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Bard Ermentrout (2008), Scholarpedia, 3(3):1398. doi:10.4249/scholarpedia.1398 revision #122128 [link to/cite this article]
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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:

\[\tag{1} \frac{d\theta}{dt} = 1-\cos\theta + (1+\cos\theta) I(t) \]

Figure 1: Saddle node on limit cycle.

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

Contents

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 \( I=0. \) When \( I<0 \) 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.

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

\[\tag{2} \frac{dx}{dt} = x^2 + I. \]


For \( I< 0 \) (resp \( I>0 \)) there are two (resp no) equilibria. In the case where \( I>0 \) solutions to this differential equation "blow up" in finite time

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

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

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

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 (2) 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 (1) from the saddle-node (2), we make a simple change of variables, \( x=\tan(\theta/2) \) from which it is simple calculus to obtain the theta model. We note that as \(\theta\) approaches \(\pi\) from the left, \(x\) goes to \(+\infty\ .\) 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:

\[ dx = (x^2+I(t))dt + \sigma dW \]

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

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

Note that the sine term in the equation comes from the Ito change of variables. For small noise, that is, \( \sigma \ll 1 \ ,\) 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 \( a \) to the quadratic version of the model. From this, he obtained a map from the old phase to the new phase:

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

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, \( du/dt=f(u) \ ,\) the adjoint is \( u_a(t)=1/du/dt \ .\) Since the "periodic" solution to the quadratic model is

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

the PRC is

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

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

\[\tag{3} \mu\ddot\theta+f \dot\theta+\ddot H(t)\cos\theta+\ddot V(t)\sin\theta=\omega \]


where \(\theta\) is the angle between the down direction and the radius through the center of mass, \(\mu\) is the mass, \(f\) is the coefficient of friction (damping), \(\ddot H\) and \(\ddot V\) are the horizontal and vertical accelerations of the support point, and \(\omega(t)\) 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 (1) is equivalent to (3) when \(\mu\to 0\) and \(H(t)\ ,\) \(V(t)\ ,\) and \(\omega\) are chosen appropriately.

Although (1) 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.

Internal references

  • John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
  • Valentino Braitenberg (2007) Brain. Scholarpedia, 2(11):2918.
  • Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
  • James Murdock (2006) Normal forms. Scholarpedia, 1(10):1902.
  • Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
  • Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.


See Also

Integrate-and-fire neuron, Neural excitability, Quadratic integrate-and-fire neuron, Saddle-node on invariant circle bifurcation, Voltage-controlled oscillations in neurons

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