Mathematical Analysis of Complex Cellular Activity

Dynamical systems theory provides a number of powerful tools for analyzing biological models, providing much more information than can be obtained from numerical simulation alone. In this chapter, we demonstrate how geometric singular perturbation analysis can be used to understand the dynamics of bursting in endocrine pituitary cells. This analysis technique, often called “fast/slow analysis,” takes advantage of the different time scales of the system of ordinary differential equations and formally separates it into fast and slow subsystems. A standard fast/slow analysis, with a single slow variable, is used to understand bursting in pituitary gonadotrophs. The bursting produced by pituitary lactotrophs, somatotrophs, and corticotrophs is more exotic, and requires a fast/slow analysis with two slow variables. It makes use of concepts such as canards, folded singularities, and mixed-mode oscillations. Although applied here to pituitary cells, the approach can and has been used to study mixed-mode oscillations in other systems, including neurons, intracellular calcium dynamics, and chemical systems. The electrical bursting pattern produced in pituitary cells differs fundamentally from bursting oscillations in neurons, and an understanding of the dynamics requires very different tools from those employed previously in the investigation of neuronal bursting. The chapter thus serves both as a case study for the application of recently-developed tools in geometric singular perturbation theory to an application in biology and a tutorial on how to use the tools.

Oscillations and travelling waves in the concentration of free cytosolic calcium are complex dynamical phenomena that play vital roles in cellular function, controlling such processes as contraction, secretion and differentiation. Although, nowadays, these oscillations and waves may be observed experimentally with relative ease, we still lack a rigorous understanding of, firstly, the mechanisms underlying these waves and oscillations in different cell types, and, secondly, the mathematical structures that underlie these complex dynamics. Thus, the study of calcium waves and oscillations is one area in which modellers have, over the years, played a major role. Here, we review our current understanding of the nonlinear dynamics of calcium waves and oscillations, restricting our attention almost wholly to deterministic models.