Fast-slow analysis of a stochastic mechanism for electrical bursting

Abstract

Electrical bursting oscillations in neurons and endocrine cells are activity patterns that facilitate the secretion of neurotransmitters and hormones and have been the focus of study for several decades. Mathematical modeling has been an extremely useful tool in this effort, and the use of fast-slow analysis has made it possible to understand bursting from a dynamic perspective and to make testable predictions about changes in system parameters or the cellular environment. It is typically the case that the electrical impulses that occur during the active phase of a burst are due to stable limit cycles in the fast subsystem of equations or, in the case of so-called “pseudo-plateau bursting,” canards that are induced by a folded node singularity. In this article, we show an entirely different mechanism for bursting that relies on stochastic opening and closing of a key ion channel. We demonstrate, using fast-slow analysis, how the short-lived stochastic channel openings can yield a much longer response in which single action potentials are converted into bursts of action potentials. Without this stochastic element, the system is incapable of bursting. This mechanism can describe stochastic bursting in pituitary corticotrophs, which are small cells that exhibit a great deal of noise as well as other pituitary cells, such as lactotrophs and somatotrophs that exhibit noisy bursts of electrical activity.

Analysis on the symmetric fast-slow behaviors in a van der Pol-Duffing-Jerk oscillator

“…In order to investigate the important question, many analytical methods have been proposed, such as experimental analysis [9,10], geometric singular perturbation method [11,12], numerical simulation method [13,14] and fast-slow decomposition analysis [15,16]. Based on these methods, many bursting patterns have been obtained, such as 'Homoclinic/Homoclinic' bursting [17], 'fold/fold' bursting [18], 'subHopf/LPC' bursting [19], 'fold/fold limit cycle' bursting [20] and 'Hopf/Hopf' bursting [21].…”