Synchronization of Regular and Chaotic oscillations: the Role of Local Divergence and the Slow Passage Effect - a Case Study on calcium oscillations

In this paper, coupling properties of regular and chaotic calcium oscillations are examined. Synchronized calcium signals among coupled cells in tissue, where calcium ions were found to be one of the most important second messengers, have proven indispensable for proper and reliable functioning of living organisms. When modeling such systems, it is of particular interest to determine, which internal system properties guarantee best coupling abilities and herewith physiologically the most efficient signal transduction between cells. We found that local contractive properties of attractors in phase space, quantified by the local divergence, represent one of the crucial system properties that determine synchronization abilities of coupled regular and chaotic oscillators. In particular, parts of attractors with close to zero local divergence largely facilitate synchronization of initially unsynchronized oscillators. For bursting oscillations, this is fully in agreement with previous studies showing that synchronization abilities of bursters are closely related with the slow passage effect. We extended this concept with the help of local divergence and succeeded to apply our theory also to other oscillatory regimes, like regular spiking and complex chaotic oscillations.