The dynamics underlying pseudo-plateau bursting in a pituitary cell model

Bursting is a common pattern of electrical activity in excitable cells such as neurons and many endocrine cells. Bursting oscillations are characterized by the alternation between periods of fast spiking (the active phase) and quiescent periods (the silent phase), and accompanied by slow variations in one or more slowly changing variables, such as the intracellular calcium concentration. Bursts are often more efficient than periodic spiking in evoking the release of neurotransmitter or hormone [1-3].The endocrine cells of the anterior pituitary gland display bursting patterns with small spikes arising from a depolarized voltage [2-5]. Similar patterns have been observed in single pancreatic β-cells isolated from islets [6-8]. Figure 1(a) shows a representative example from a GH4 pituitary cell. Several mathematical models have been developed for this bursting type [5,8-10]. Prior analysis showed that the dynamic mechanism for this type of bursting, called pseudo-plateau bursting, is significantly different from that of square-wave bursting (also called plateau bursting) which is common in neurons [11-13]. Yet this analysis did not determine the possible number of spikes that occur during the active phase of the burst. The goal of this paper is to understand the dynamics underlying pseudo-plateau bursting, with a focus on the origin of the spikes that occur during the active phase of the oscillation.Minimal models for pseudo-plateau bursting can be written aswhere V is the membrane potential, n is the fraction of activated delayed rectifier K+ channels, and c is the cytosolic free Ca2+ concentration. The velocity functions are nonlinear, and ε1 and ε2 are parameters that may be small.The variables V, n, and c vary on different time scales (for details, see Section 2). By taking advantage of time-scale separation, the system can be divided into fast and slow subsystems. In the standard fast/slow analysis one considers ε2 ≈ 0, so that V and n form the fast subsystem an