On a Model Stationary Nonlinear Wave in an Active Medium

A Lienard-type model equation describing stationary nonlinear wave motion in an active medium is investigated. This model equation has been derived from the evolution equation governing pulse transmission in a nerve fibre. Two models (the classical third-order FitzHugh-Nagumo equation and the model second-order equation) are compared and analysed. The possibility of calculating a single pulse in the whole `time' range, without any convergence problems, offers a certain advantage to a Lienard-type model equation. In physical terms, this equation is able to describe both a threshold effect and the possibility of amplification or attenuation. A qualitative analysis of the phase portrait of this model is presented. It is proved that limit cycles in the phase plane cannot exist under the conditions arising from the physics of the nerve fibre. The threshold problem is analysed in detail and an algorithm is presented to find the threshold distinguishing the processes of amplification and attenuation. The results obtained in this work permit a full map of the solutions to be given for a general Lienard-type equation. The model equation under consideration describes a single pulse, but the full map of the solutions also contains periodic solutions corresponding to the well-known Van der Pol equation.