FitzHugh-Nagumo Oscillator

By simplifying the classical Hodgkin-Huxley equations [5] for modeling nerve membranes and action potential gener­ation, FitzHugh [7] and Nagumo et al. [8] gave the follow­ing two-variable equation, widely known as the FitzHugh — Nagumo model,

(3)

x — c[y — f (x) + I ]

(3)

y — —(x + by — a)/c

where f(x) is as defined in Eq. (2), I is the injected current, and a, b, and c are system parameters satisfying the condi­tions: 1 > b > 0, c2 > b, and 1 > a > 1 — 2b/3. In neurophys — iological terms, x corresponds to the neuronal membrane potential, and y plays the aggregate role of three variables in the Hodgkin-Huxley equations. Given that the x null­cline is a cubic and the y nullcline is linear, the FitzHugh — Nagumo equation is mathematically similar to the van der Pol equation. Typical relaxation oscillation with two time scales occurs when c > 1. Because of the three parameters and the external input I, the FitzHugh-Nagumo oscillator has additional flexibility. Depending on parameter values, the oscillator can exhibit a stable steady state or a stable periodic orbit. With a perturbation by external stimulation, the steady state can become unstable and be replaced by an oscillation; the steady state is thus referred to as the excitable state.

Morris-Lecar Oscillator

In modeling voltage oscillations in barnacle muscle fibers, Morris and Lecar [9] proposed the following equation,

2

x — —gCam<x (x)(x — 1) — gKy(x — xk) — gL(x — xl ) + I (4)

y — — s[y^ (x) — y]/ty (x) (4)

where

m^ (x) — {1 + tanh[(x — x1)/x2]}/2

y^ (x) — {1 + tanh[(x — x3)/x4]}/2

Ty (x) — 1/cosh[(x — x3)/(2×4)]

and xi x2, x3, x4, gCa, gK, gL, xK, and xL are parameters. Ca stands for calcium, K for potassium, L for leak, and I is the injected current. The parameter є controls relative time scales of x and y. Like Eq. (3), the Morris-Lecar oscillator is closely related to the Hodgkin-Huxley equations, and it is used as a two-variable description of neuronal membrane properties or the envelope of an oscillating burst [10]. The x variable corresponds to the membrane potential, and y corresponds to the state of activation of ionic channels.

The x nullcline of Eq. (4) resembles a cubic and the y nullcline is a sigmoid. When є is chosen to be small, the Morris-Lecar equation produces typical relaxation oscilla­tions. From the mathematical point of view, the sigmoidal y nullcline marks the major difference between the Morris — Lecar oscillator and the FitzHugh-Nagumo oscillator.

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