Bifurcation and Geometric Singular Perturbation Analysis of a Multi-timescale Pituitary Model

Abstract

This paper thoroughly discusses the electric activities generated by ion communication between cells and their surrounding environment. Specifically, it focuses on the transients of firing activities of a four-dimensional pituitary model that evolves on three disparate timescales. To examine the impact of inward rectifying [Formula: see text] current and calcium concentration on the firing activities, a bifurcation analysis is conducted, categorizing three primary behaviors: resting, tonic spiking, and bursting. Each behavior is validated through their respective time courses, with the pituitary cells showing higher secretion rates of hormones and neurotransmitters during bursting than spiking. The geometric singular perturbed theory is applied to reveal hidden geometric features and the transient mechanisms associated with bursting, particularly mixed-mode oscillations (MMOs). Singular orbit construction performed in two-timescale separation with different viewpoints offers clarity on the underlying dynamic mechanisms. Canard-induced MMOs are observed in the context of 1 fast/3 slow and 2 fast/2 slow separations, facilitated by the presence of folded saddle-node and folded node, respectively. Additionally, the fast–slow analysis of the 3 fast/1 slow subsystem, which treats calcium concentration [Formula: see text] as a parameter, in conjunction with the singular orbit constructions, effectively illustrates the system’s complex dynamics. Furthermore, the information obtained in 1 fast/3 slow and 3 fast/1 slow discussions is interplayed in the context of three-timescale separation. The singular orbits identified within three-timescale framework offer a supplemental perspective to the delicate firing patterns observed in two-timescale analysis, enriching the overall understanding of the transient and long-term firing behaviors of the pituitary cells. This study presents valuable insights into the firing features in pituitary cells from the perspectives of dynamic systems. The singular perturbation analysis provides useful viewpoints for accessing firing patterns in multi-timescale systems.