Mckean model

• Dr. Arnaud Tonnelier, INRIA, Montbonnot, France

Figure 1: The two piecewise linear functions proposed by McKean.

The McKean model, named after H. P. McKean, is a two-dimensional model of spiking neurons \[ \begin{matrix} \dot{v} & = & f(v) - w + I \\ \dot{w} & = & b (v - c w) \end{matrix} \] where \(f\) is a piecewise linear function of "cubic" shape, \(v\) is the membrane potential, \(w\) is a recovery variable, \(I\) is an input current and \(b\ ,\) \(c\) are constant parameters. Two piecewise linear models were proposed by McKean (see Figure 1) \[ \begin{matrix} f(v) =\left \{ \begin{array}{ll} - v, & \textrm{for} \ v < a/2\\ v - a, & \textrm{for} \ a/2 < v < (a+1)/2 \\ -v - 1, & \textrm{for} \ v > (a+1)/2 \end{array} \right. \end{matrix} \] and \[ f(v) = -v + h(v-a) \] where \(0<a<1\) is a threshold and \(h\) the Heaviside function. The McKean model is a piecewise linear caricature of the FitzHugh-Nagumo model. The interest lies in the fact that the model preserves the essential features of neuronal behavior while allowing explicit calculations. A third piecewise linear model of FitzHugh-Nagumo type referred as the "Pushchino" model has been introduced as a model for the ventricular action potential.

History

The McKean model has been originally introduced in the study of nerve conduction. Propagating pulses in the nerve are well approximated by the Nagumo's equation \[ \begin{matrix} \dot{v} & = & v_{xx} + f(v) - w \\ \dot{w} & = & b v \qquad \qquad \end{matrix} \] where \(f\) is a cubic polynomial \(f(v) = v(1-v)(v-a)\ .\) Since the Nagumo model is itself a caricature of a more detailed model, namely the the Hodgkin-Huxley model, it is natural to simplify still further. The crucial point is to preserve the shape of the function \(f\) but its exact expression is not relevant. Taking a piecewise non-linearity allowed McKean to compute analytically the wave solutions and to give some mathematical indications to the picture numerically obtained. Rinzel and Keller (1973) found solitary pulse solutions and periodic wave solutions for the McKean caricature of nerve conduction. Wang (1988) further showed the existence and stability of multiple impulse solutions.

Basically, the piecewise linear approach, also known as sector bound method, allows a tractable analysis by calculating the solutions in the linear regimes using standard techniques and connecting the solutions at the boundaries.

Geometrical analysis, multistability and binary neuron

Figure 2: Monostable behavior with one globally attractive fixed point.

Figure 3: Bistability between a fixed point and a stable limit cycle.

Figure 4: Multistability between two stable fixed points and a limit cycle.

Since the McKean system is planar, its dynamics has a nice geometrical interpretation in the phase plane. Intersections of the v-nullcline (obtained from \(\dot{v}=0\)) and w-nullcline (obtained from \(\dot{w}=0\)) define the fixed points. In Figure 2-Figure 4 nullclines are shown in thin dashed lines. As the parameters vary, different phase portraits are obtained. Existence of a limit cycle in the phase plane is related to the existence of oscillations (tonic spiking). The stable limit cycle is in red and unstable limit cycles are in green. Note that the unstable limit cycle is a sliding solution. Appearance by pairs of periodic solutions for the McKean nonlinearity \(f(v)=-v+h(v-a)\) indicates a saddle node bifurcation of limit cycles leading to stable and unstable limit cycles in the phase plane (see Figure 3).

Another way for the onset of oscillations is the non smooth Andronov-Hopf bifurcation obtained when the fixed point lies on the middle branch of the \(v\)-nullcline. In both cases the frequency at the onset of oscillations is non-zero corresponded to type II excitability (also called class II excitability). This excitability is also exemplified by the FitzHugh-Nagumo model and many other properties of the FitzHugh-Nagumo model are reproduced by the McKean model (excitation block, post-inhibitory rebound, traveling pulse , \(\ldots\)).

Figure 5: The phase plane of the McKean relaxation oscillator and the related binary variable. .

The McKean model presents interesting relation to binary neuron models. In the limit of fast relaxation \(b \rightarrow 0\) the membrane potential \(v\) spends most of time on the two attracting branchs of the nullcline \(\dot{v}=0\ .\) It is convenient to introduce the binary variable \[ \begin{matrix} S = \left \{ \begin{array}{ll} 1, & v > (1+a)/2 \\ 0, & v < a/2 \end{array} \right. \end{matrix} \] to describe the behavior of the system. In Figure 5 the fixed point is unstable and trajectories tend towards a stable limit cycle. The neuron is described using the binary variable \(S\) that oscillates between the \(S=1\) and \(S=-1\) states. A tractable mathematical analysis with explicit expressions for the period of oscillations and the phase response curve can be obtained.

Analytical treatment

The McKean model with the discontinuous nonlinearity has an equivalent spike-response formulation : \[\tag{1} \begin{matrix} v(t) & = & \sum \limits_{t^f \in \mathcal{F}} \eta(t-t^f) - \sum \limits_{t^r \in R} \eta(t-t^r) + \int \limits_0^{\infty} \epsilon(s) I(t-s) ds \end{matrix} \]

where \(\mathcal{F}\) and \(\mathcal{R}\) are the so-called firing and resetting set defined as the time of threshold crossing of the membrane potential from below and from upper respectively. The two kernels \(\eta\) and \(\epsilon\) are pulse shaped functions and can be explicitly calculated. For \(c = 0\) and \(b>1/4\ ,\) \(\eta(t)= 1/r \ e^{-t/2} \sin(r t)\) and \(\epsilon(t) = e^{-t/2} ( \cos(r t) - 1/(2r) \sin(r t) )\) for \(t \geq 0\) and \(0\) otherwise, where \(r=\sqrt{b-1/4}\ .\)

Figure 6: Time variation of the membrane potential for different initial stimulations.

Figure 7: Oscillatory regime.

Constant input and fixed points

Let \(\alpha=c/(1+c)\) and assume that \(\alpha < a\ ,\) i.e. without external current (\(I = 0\)) the McKean model has the point \((0,0)\) as unique fixed point. The two critical currents \(I_{th,1} = a/\alpha-1\) and \(I_{th,2} = a/\alpha\) determine the fixed points of the system. For \(I<I_{th,1}\) the McKean neuron has a stable subthreshold fixed point, for \(I_{th,1}<I<I_{th,2}\) two stable fixed points and for \(I>I_{th,2}\) the subthreshold fixed point disappears and the McKean model presents a stable superthreshold fixed point \(v_2=\alpha(1+I)\ .\)

Brief pulse and oscillations

Let us assume that the neuron receives a brief current pulse \(I(t) = q_0 \delta(t)\) (where \(\delta\) is the Dirac function). In the excitable regime the neuron emits spike(s) (see Figure 6). If an infinite number of spikes is emitted (see Figure 7) the neuron oscillates and the periodic solution has the following analytical expression ( obtained from (1)) : \[ \begin{matrix} v_{\infty}(t) & =& \bar{v}_{\infty} + \eta_{\infty}(t+\tau) - \eta_{\infty}(t-\tau) \end{matrix} \] where \(\bar{v}_{\infty}=2c\tau/((1+c)T)\) is the mean value of \(v_{\infty}\ ,\) \(T\) is the period, \(\tau\) is a parameter and \(\eta_{\infty}\) is a combination of trigonometric functions. Existence of an oscillatory regime is related to the existence of \((\tau,T)\) that are obtained as roots of transcendental equations. Roots appear by pairs indicating the existence of two periodic solutions. A necessary condition for the existence of \((\tau,T)\) and thus for oscillations is \(a < 1/\pi\ .\)

References

• McKean H. P. (1970) Nagumo's equation. Advances in Mathematics 4, 209-223.

• Tonnelier A. (2002) The McKean's caricature of the FitzHugh-Nagumo model. The space-clamped system, SIAM J. Appl. Math. 63, 2, 459-484.

Internal references

• Yuri A. Kuznetsov (2006) Andronov-Hopf bifurcation. Scholarpedia, 1(10):1858.
• James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
• Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
• Eugene M. Izhikevich and Richard FitzHugh (2006) FitzHugh-Nagumo model. Scholarpedia, 1(9):1349.
• Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
• Carmen C. Canavier (2006) Phase response curve. Scholarpedia, 1(12):1332.
• Gregoire Nicolis and Anne De Wit (2007) Reaction-diffusion systems. Scholarpedia, 2(9):1475.
• Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

Further reading

• Abbott L. F. (1990) A network of oscillators. Journal of Physics A: Mathematical and General 23, 3835-3859.

• Coombes S. (2001) Phase locking in networks of synaptically-coupled McKean relaxation oscillators. Physica D, 160, 173-188

• Rinzel J. and Keller J. B. (1973) Traveling wave solutions of a nerve conduction equation. Biophys. J. 13:1313.

• Wang W.P (1988) Multiple impulse solutions to McKean's caricature of the nerve equation. 1. Existence. Communications on pure and applied mathematics 41: 71.

• Wang W.P (1988) Multiple impulse solutions to McKean's caricature of the nerve equation. 2. Stability. Communications on pure and applied mathematics 41: 997.

See also

FitzHugh-Nagumo model, Relaxation Oscillator, Periodic Orbit, Reaction-Diffusion Systems, Traveling Wave

Sponsored by: Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia

Reviewed by: Anonymous

Accepted on: 2007-04-26 17:12:31 GMT