A model of the nerve impulse using two first-order differential equations

Abstract

The Hodgkin–Huxley model¹ of the nerve impulse consists of four coupled nonlinear differential equations, six functions and seven constants. Because of the complexity of these equations and the necessity for numerical solution, it is difficult to use them in simulations of interactions in small neural networks. Thus, it would be useful to have a second-order differential equation which predicted correctly properties such as the frequency–current relationship. Fitzhugh² introduced a second-order model of the nerve impulse, but his equations predict an action potential duration which is similar to the inter-spike interval³ and they do not give a reasonable frequency–current relationship. To develop a second-order model having few parameters but which does not have these disadvantages, we have generalized the second-order Fitzhugh equations², and based the form of the functions in the new equations on voltage-clamp data obtained from a snail neurone. We report here an unexpected property of the resulting equations—the x and y null clines in the phase plane lie close together when the phase point is on the recovery side of the phase plane. The resulting slow movement along the phase path gives a long inter-spike interval, a property not shown clearly by previous models^2,4. The model also predicts the linearity of the frequency–current relationship, and may be useful for studying detailed interactions in networks containing small numbers of neurones.