MIXED-MODE OSCILLATION GENEALOGY IN A COMPARTMENTAL MODEL OF BONE-MINERAL METABOLISM

A well-supported self-oscillating eight-compartment model has been pro posed by Staub et al. to account for the in vivo rat calcium metabolis m (Staub et al., Am. J. Physiol. 254, R134-139, 1988). The nonlinear n ucleus of this model is a three-compartment subunit which represents t he dynamic autocatalytic processes of phase transition at the interfac e between bone and extracellular fluids. The organization of the tempo ral mixed-mode oscillations which successively appear as the calcium i nput is varied is analyzed. On one side of the bifurcation diagram, th e generation of periodic trajectories with a single large amplitude os cillation is governed by homoclinic tangencies to small amplitude limi t cycles and follows the universal sequence (U-sequence) given for the periodic solutions of unimodal transformations of the unit interval i nto itself. On the other side, the progressive appearance and interwea ving of trajectories with multiple large amplitude oscillations per pe riod is linked to homoclinic tangencies to large amplitude unstable cy cles. The bifurcation sequence responsible for the temporal pattern ge neration has been analyzed by modeling the first return map of the dif ferential system associated with the compartmental subunit. We establi sh that this genealogy does not follow the usual Farey treelike organi zation and that a comprehensive view of the resulting fractal bifurcat ion structure can be obtained from the unfolding of singular points of bimodal maps. These theoretical features can be compared with those r eported in experiments on dissolution processes, and the extent to whi ch the knowledge of the subunit bifurcation structure provides new con ceptual insights in the field of bone and calcium metabolism is discus sed.