Article No. bulm.1999.0095 Available online at http://www.idealibrary.com on Bulletin of Mathematical Biology (1999) 61, 507–530 Bursting, Chaos and Birhythmicity Originating from Self-modulation of the Inositol 1,4,5-trisphosphate Signal in a Model for Intracellular Ca 2+ Oscillations GÉrald Houart, GeneviÈve Dupont and Albert Goldbeter Unit´ e de Chronobiologie Th´ eorique, Facult´ e des Sciences, Universit´ e Libre de Bruxelles, Campus Plaine, C.P. 231, B-1050 Brussels, Belgium E-mail: agoldbet@ulb.ac.be We investigate the various types of complex Ca 2+ oscillations which can arise in a model based on the mechanism of Ca 2+ -induced Ca 2+ release (CICR), that takes into account the Ca 2+ -stimulated degradation of inositol 1,4,5-trisphosphate (InsP 3 ) by a 3-kinase. This model was previously proposed in the course of an investigation of plausible mechanisms capable of generating complex Ca 2+ oscillations (Borghans et al., 1997). Besides simple periodic behavior, this model for cytosolic Ca 2+ os- cillations in nonexcitable cells shows complex oscillatory phenomena like bursting or chaos. We show that the model also admits a coexistence between two stable regimes of sustained oscillations (birhythmicity). The occurrence of these various modes of oscillatory behavior is analysed by means of bifurcation diagrams. Com- plex oscillations are characterized by means of Poincar´ e sections, power spectra and Lyapounov exponents. The results point to the role of self-modulation of the InsP 3 signal by 3-kinase as a possible source for complex temporal patterns in Ca 2+ signaling. c  1999 Society for Mathematical Biology 1. I NTRODUCTION A large variety of cell types display Ca 2+ oscillations after stimulation by an extracellular agonist such as a hormone or a neurotransmitter (Berridge, 1993, 1997; Berridge and Dupont, 1994; Thomas et al., 1996). In most cell types, these oscillations take the form of repetitive sharp spikes in the level of cytosolic Ca 2+ . Typically, the cytosolic Ca 2+ concentration varies from less than 0.1 μM up to 1 μM with a periodicity that ranges from a few seconds to 30 minutes. In nearly all cell types the frequency of Ca 2+ oscillations is seen to increase with the level of stimulation, a phenomenon that strongly suggests that the external signal is encoded in terms of the temporal pattern of Ca 2+ oscillations. Although the pathway between the successive Ca 2+ spikes and the cellular response remains poorly understood in most cell types, there is increasing evidence pointing to a physiological role for these 0092-8240/99/030507 + 24 $30.00/0 c  1999 Society for Mathematical Biology