RESUMEN

With the use of perturbation expansions and asymptotic matching, bursting in the three-dimensional Bonhoeffer–van der Pol equations is shown to be the result of the interaction of two oscillatory modes, one of small amplitude and the other of large amplitude. The large oscillations are similar to the relaxation oscillations found in the two-dimensional van der Pol system, but the small oscillations have not been fully understood before. This analysis also explains the transition in the two-dimensional system from stable equilibrium to relaxation oscillations: the intermediate stage between the two are the small oscillations.

RESUMEN

Various physiological systems display bursting electrical activity (BEA). There exist numerous three-variable models to describe this behavior. However, higher-dimensional models with two slow processes have recently been used to explain qualitative features of the BEA of some experimentally observed systems [T. Chay and D. Cook, Math. Biosci., 90 (1988), pp. 139–153; P. Smolen and J. Keizer, J. Memb. Biol., 127 (1992), pp. 9–19; R. Bertram et al., Biophys. J., 79 (2000), pp. 2880–2892; R. Bertram et al., Biophys. J., 68 (1995), pp. 2323–2332; J. Keizer and P. Smolen, Proc. Nat. Acad. Sci. USA, 88 (1991), pp. 3897–3901]. In this paper we present a model with two slow and two fast variables. For some parameter values the system has stable equilibria, while for other values there exist bursting solutions. Singular perturbation methods are used to define a one-dimensional return map, wherein fixed points correspond to singular bursting solutions. We analytically demonstrate that bursting solutions may exist even with a combination of activating and inactivating slow processes. We also demonstrate that for different parameters, bursting solutions may coexist with stable equilibria. Hence small variations in the initial conditions may drastically affect the dynamics.