Mixed-mode oscillation genealogy in a compartmental model of bone mineral metabolism

Abstract

A well-supported self-oscillating eight-compartment model has been proposed by Staub et al. to account for thein vivo rat calcium metabolism (Staub et al.,Am. J. Physiol.254, R134–139, 1988). The nonlinear nucleus of this model is a three-compartment subunit which represents the dynamic autocatalytic processes of phase transition at the interface between bone and extracellular fluids. The organization of the temporal mixed-mode oscillations which successively appear as the calcium input is varied is analyzed. On one side of the bifurcation diagram, the generation of periodic trajectories with a single large amplitude oscillation is governed by homoclinic tangencies to small amplitude limit cycles and follows the universal sequence (U-sequence) given for the periodic solutions of unimodal transformations of the unit interval into itself. On the other side, the progressive appearance and interweaving of trajectories with multiple large amplitude oscillations per period is linked to homoclinic tangencies to large amplitude unstable cycles. The bifurcation sequence responsible for the temporal pattern generation has been analyzed by modeling the first return map of the differential system associated with the compartmental subunit. We establish that this genealogy does not follow the usual Farey treelike organization and that a comprehensive view of the resulting fractal bifurcation structure can be obtained from the unfolding of singular points of bimodal maps. These theoretical features can be compared with those reported in experiments on dissolution processes, and the extent to which the knowledge of the subunit bifurcation structure provides new conceptual insights in the field of bone and calcium metabolism is discussed.