Threshold, excitability and isochrones in the Bonhoeffer-van der Pol system

Some new insight is obtained for the structure of the Bonhoeffer-van der Pol system. The problems of excitability and threshold are discussed for all three types of the system classified according to the existing attractors: a focus only, a limit cycle only and a limit cycle together with a focus. These problems can be treated by the T-repellers and the T-attractors of the system which are mutually reciprocal under time inversion. The threshold depends on the structure of the T-repeller (unstable part of integral manifold). This structure is then used to understand the behavior and the properties of the two different types of isochrones: Winfree isochrones (W-isochrones) and regular isochrones. Winfree's description of a W-isochrone is extended to excitable systems. Both W-isochrones and regular isochrones are calculated for the Bonhoeffer-van der Pol system in its limit cycle and excitable regimes. The important role of the T-repeller as an asymptotic limit for both types of isochrones is manifested.