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.