Quantitative Models of The Action Potential - Fitzhugh-Nagumo Model

Fitzhugh-Nagumo Model

Because of the complexity of the Hodgkin-Huxley equations, various simplifications have been developed that exhibit qualitatively similar behavior. The Fitzhugh-Nagumo model is a typical example of such a simplified system. Based on the tunnel diode, the FHN model has only two independent variables, but exhibits a similar stability behavior to the full Hodgkin-Huxley equations. The equations are

C \frac{dV}{dt} = I - g(V),
L\frac{dI}{dt} = E - V - RI

where g(V) is a function of the voltage V that has a region of negative slope in the middle, flanked by one maximum and one minimum (Figure FHN). A much-studied simple case of the Fitzhugh-Nagumo model is the Bonhoeffer-van der Pol nerve model, which is described by the equations

C \frac{dV}{dt} = I - \epsilon \left(\frac{V^{3}}{3} - V \right),
L\frac{dI}{dt} = - V

where the coefficient ε is assumed to be small. These equations can be combined into a second-order differential equation

C \frac{d^{2}V}{dt^{2}} + \epsilon \left( V^{2} - 1 \right) \frac{dV}{dt} + \frac{V}{L} = 0.

This van der Pol equation has stimulated much research in the mathematics of nonlinear dynamical systems. Op-amp circuits that realize the FHN and van der Pol models of the action potential have been developed by Keener.

A hybrid of the Hodgkin-Huxley and FitzHugh-Nagumo models was developed by Morris and Lecar in 1981, and applied to the muscle fiber of barnacles. True to the barnacle's physiology, the Morris-Lecar model replaces the voltage-gated sodium current of the Hodgkin-Huxley model with a voltage-dependent calcium current. There is no inactivation (no h variable) and the calcium current equilibrates instantaneously, so that again, there are only two time-dependent variables: the transmembrane voltage V and the potassium gate probability n. The bursting, entrainment and other mathematical properties of this model have been studied in detail.

The simplest models of the action potential are the "flush and fill" models (also called "integrate-and-fire" models), in which the input signal is summed (the "fill" phase) until it reaches a threshold, firing a pulse and resetting the summation to zero (the "flush" phase). All of these models are capable of exhibiting entrainment, which is commonly observed in nervous systems.

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