Preface to the special issue nonlinear economic dynamics: A tribute to Tonu Puu

Pituitary cells are electrically excitable. A core set of intracellular ion channels cooperate in providing the basic conditions for the generation of electro-excitable patterns. Although much data have been accumulated on electrophysiological recordings of corticotrophs, there is an unclear triggering relationship between the secretory function of the cells and the electrical activity pattern. Fletcher et al. established a conductance-based 6-dimensional mathematical model to capture the detailed features of three spontaneous firing activities observed in electrophysiological experiments. This paper employs eigenvalue analysis and curve fitting to simplify the original model to a 4-dimensional one. The simplified model maintains many aspects of the firing activity of the original model. Then, considering the development of system variables on multiple time scales, the geometric singular perturbation theory and fast/slow analysis are utilized to observe the system's intrinsic transients and the long-term evolution of each attractor from a multi-geometric perspective. In particular, singular periodic orbits of the corticotropic model are constructed to explore the dynamic mechanism of cells firing. The dynamic mechanism of mixed-mode oscillations in this model is also revealed. The analysis helps us gain insight into the corticotrophs model from a low-dimensional subsystem and provides a new perspective on the origin of cellular spontaneous firing activity.