From simple to complex oscillatory behaviour: analysis of bursting in a multiply regulated biochemical system.

We analyze the transition from simple to complex oscillatory behaviour in a three-variable biochemical system that consists of the coupling in series of two autocatalytic enzyme reactions. Complex periodic behaviour occurs in the form of bursting in which clusters of spikes are separated by phases of relative quiescence. The generation of such temporal patterns is investigated by a series of complementary approaches. The dynamics of the system is first cast into two different time-scales, and one of the variables is taken as a slowly-varying parameter influencing the behaviour of the two remaining variables. This analysis shows how complex oscillations develop from simple periodic behaviour and accounts for the existence of various modes of bursting as well as for the dependence of the number of spikes per period on key parameters of the model. We further reduce the number of variables by analyzing bursting by means of one-dimensional return maps obtained from the time evolution of the three-dimensional system. The analysis of a related piecewise linear map allows for a detailed understanding of the complex sequence leading from a bursting pattern with p spikes to a pattern with p + 1 spikes per period. We show that this transition possesses properties of self-similarity associated with the occurrence of more and more complex patterns of bursting. In addition to bursting, period-doubling bifurcations leading to chaos are observed, as in the differential system, when the piecewise-linear map becomes nonlinear.