Identification of Ca2+ oscillations with the 0-1 test for periodic and chaotic time-series: One- and two-parameter bifurcations

This paper examines the oscillatory responses (periodic and chaotic) of a biosystem store model for bursting and complex Ca²⁺ oscillations in which three compartments have been taken into consideration: the cytosol, endoplasmic reticulum (ER) and mitochondria. The oscillatory model is used to examine the reliability of the 0-1 test for chaos in the bifurcation analysis of continuous signals obtained when the frequencies of oscillatory responses vary significantly with a relatively small changes of the bifurcation parameters. The illustrative examples in both the one- and two-parameter cases are designed to show that for a periodic time-series the test's reliability may be questioned when a periodic series is classified as a chaotic one - the 'false-positive' case. To prevent the incorrect result an additional computational work is needed to examine the frequency spectrum of the periodic time-series. The illustrative examples utilize an autonomous dynamical model of cytosolic calcium oscillations with three dynamical variables and sixteen parameters. The dynamical model is such that the frequency of oscillations may change by the factor of about 200, when a certain dynamical system's parameter changes from its minimum to maximum values, making selection of the parameters in the 0-1 test extremely difficult. The extra computational work improves the test's reliability and eliminates the 'false-positive' outcomes of the test. The paper is focused on the computational aspects of the 0-1 test for periodic and chaotic oscillations rather than on the properties of the store model for bursting and complex Ca²⁺ oscillations.