Analysis of degenerate hopf bifurcations for a nonlinear model of calcium metabolism

Abstract

RESULTS of the in viva studies related to calcium metabolism have been generally analysed within the framework of linear mathematical models (e.g. classical compartmental analysis [ 11). This approach assumes the maintenance of the extracellular calcium concentration at a nearly constant value under physiological conditions (plasma calcium homeostasis), and the adequacy of a sum of exponential terms (the analytical solution of a first order linear differential system) to fit kinetic data such as the time-course of tracer concentration. Nonlinear models which exhibit a self-oscillatory behaviour have been proposed recently. A model of the hormonal control of calcium homeostasis in chicken includes the consideration of a growth function and takes into account complex interrelationships among the calciumregulating hormones and between these hormones and specific metabolic functions [2]. Another model, constructed in our laboratory, is a self-oscillating model for calcium metabolism in the rat; it includes some nonlinear processes involved in bone calcium metabolism and adequately reproduces the kinetic behaviour of radiocalcium in plasma, including circadian variations [3]. In contrast to linear systems for which the simple dynamic behaviour patterns are predictable and their analytical solutions are commonly applied in calcium metabolism field, it is wellestablished that nonlinear systems have potentially numerous and complex spatio-temporal expressions (excitability, periodic oscillations, chaos, etc.). Therefore, the relevance of the proposed nonlinear models and their significance with regard to the control of calcium metabolism require a concomitant knowledge of the set of the dynamic patterns that, in theory, such nonlinear structures are able to produce. The variety of the dynamic behaviours exhibited by the self-oscillating model we have established in rat, or rather its nonlinear sub-structure [4, 51, motivated this theoretical work. Our model of calcium metabolism has been constructed in the general context of compartmental formalism (Fig. 1). Briefly, this model was based on the explicit consideration the nonlinear properties of the phase transition and nucleation mechanisms associated with bone calcification and their formulation through a single second-order autocatalytic process [5]. In addition to its ability to generate periodic temporal variations fitting to experimental data (Fig. 2), the model correctly approximates the growth-curve of calcium mass as a function of age, and proposes a dynamic organization for bone calcium metabolism which distinguishes