A chaotic bursting-spiking transition in a pancreatic beta-cells system: observation of an interior glucose-induced crisis

Nonlinear systems are commonly able to display abrupt qualitative changes (or transitions) in the dynamics. A particular type of these transitions occurs when the size of a chaotic attractor suddenly changes. In this article, we present such a transition through the observation of a chaotic interior crisis in the Deng bursting-spiking model for the glucose-induced electrical activity of pancreatic $β $-cells. To this chaos-chaos transition corresponds precisely the change between the bursting and spiking dynamics, which are central and key dynamical regimes that the Deng model is able to perform. We provide a description of the crisis mechanism at the bursting-spiking transition point in terms of time series variations and based on certain amplitudes of invariant intervals associated with return maps. Using symbolic dynamics, we are able to accurately compute the points of a curve representing the transition between the bursting and spiking regimes in a biophysical meaningfully parameter space. The analysis of the chaotic interior crisis is complemented by means of topological invariants with the computation of the topological entropy and the maximum Lyapunov exponent. Considering very recent developments in the literature, we construct analytical solutions triggering the bursting-spiking transition in the Deng model. This study provides an illustration of how an integrated approach, involving numerical evidences and theoretical reasoning within the theory of dynamical systems, can directly enhance our understanding of biophysically motivated models.