Geometric Singular Perturbation Analysis of Bursting Oscillations in Pituitary Cells

Abstract

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.

8 Appendix

Computer codes for all models discussed in this chapter are available as freeware from http://www.math.fsu.edu/~bertram/software/pituitary.

1.1 8.1 The Chay-Keizer Model

The Chay-Keizer model for bursting in the pancreatic β-cell (Chay and Keizer (1983)) was one of the first bursting models analyzed using a fast/slow analysis technique (Rinzel and Lee (1985)). A modified form of the model was used in Teka et al. (2011b) to analyze the transition between plateau and pseudo-plateau bursting, and was used for this purpose in Fig. 4. The modified Chay-Keizer model is similar in many ways to the lactotroph model used in most of this article. The differential equations are:

$$\displaystyle\begin{array}{rcl} C_{m}\frac{dV } {dt} & =& -[I_{Ca}(V ) + I_{K}(V,n) + I_{SK}(V,c) + I_{K(ATP)}(V )]{}\end{array}$$

(1.79)

$$\displaystyle\begin{array}{rcl} \frac{dn} {dt} & =& \frac{n_{\infty }(V ) - n} {\tau _{n}} {}\end{array}$$

(1.80)

$$\displaystyle\begin{array}{rcl} \frac{dc} {dt}& =& -f_{c}(\alpha I_{Ca}(V ) + k_{c}c).{}\end{array}$$

(1.81)

where in place of the V -dependent BK K⁺ current there is a K⁺ current whose conductance is regulated by intracellular levels of ATP. If, as assumed here, the ATP concentration is constant, then g _K(ATP) is a constant-conductance K⁺ current given by

$$\displaystyle{ I_{K(ATP)}(V ) = g_{K(ATP)}(V - V _{K})\;\;. }$$

(1.82)

The form of the steady state activation curves $m_{\infty }(V )$ and $n_{\infty }(V )$ are the same as for the lactotroph model (Eqs. 1.5, 1.7). The steady state activation function for the SK current is a third-order Hill function, rather than second-order:

$$\displaystyle{ s_{\infty }(c) = \frac{c^{3}} {c^{3} + K_{d}^{3}}. }$$

(1.83)

Parameter values for this model are given in Table 3.

Table 3 Parameter values for the simplified Chay-Keizer model

1.2 8.2 The Lactotroph Model with an A-Type K⁺ Current

An alternate lactotroph model was developed by Toporikova et al. (2008) and analyzed in Vo et al. (2010) and Vo et al. (2012). In this model, the equation for intracellular Ca²⁺ concentration is removed, as are the SK and BK currents. Instead, an A-type K⁺ current is included that activates instantaneously and inactivates on a slower time scale. The differential equations are:

$$\displaystyle\begin{array}{rcl} C_{m}\frac{dV } {dt} & =& -[I_{Ca}(V ) + I_{K}(V,n) + I_{A}(V,e) + I_{L}(V )]{}\end{array}$$

(1.84)

$$\displaystyle\begin{array}{rcl} \frac{dn} {dt} & =& \frac{n_{\infty }(V ) - n} {\tau _{n}} {}\end{array}$$

(1.85)

$$\displaystyle\begin{array}{rcl} \frac{de} {dt}& =& \frac{e_{\infty }(V ) - e} {\tau _{e}} \;\;.{}\end{array}$$

(1.86)

where the I _Ca and I _K currents are as before, I _L is a constant-conductance leakage current, and I _A is the A-type K⁺ current:

$$\displaystyle\begin{array}{rcl} I_{L}(V )& =& g_{L}(V - V _{K}){}\end{array}$$

(1.87)

$$\displaystyle\begin{array}{rcl} I_{A}(V,e)& =& g_{A}a_{\infty }e(V - V _{K})\;\;.{}\end{array}$$

(1.88)

The activation function for I _A  is

$$\displaystyle{ a_{\infty }(V ) = \left (1 +\exp (\frac{v_{a} - V } {s_{a}} )\right )^{-1} }$$

(1.89)

and the inactivation function is

$$\displaystyle{ e_{\infty }(V ) = \left (1 +\exp (\frac{V - v_{e}} {s_{e}} )\right )^{-1}\;\;. }$$

(1.90)

Parameter values are given in Table 4.

Table 4 Parameter values for the lactotroph model with A-type K⁺ current