Lab 8

INTRODUCTION

The Pancreas has several functions in the digestion of food. One of the most well known functions is that of insulin secretion. If the Pancreas is no longer able to properly regulate insulin production, diabetes may be the result.

The pancreas has many groupings of cells called the islets of Langerhan. Within these islets are beta cells. These cells are excitable with their transmembrane potentials playing a large role in release of insulin into the islet's capillaries.

From http://www3.umdnj.edu/histsweb/lab21/lab21pancreas.html
Red arrows indicate beta cells.

In 1980, Atwater et al. published a paper describing the biology of the electrical behavior of the beta cell. Chay and Keizer followed three years later with a simplified model of the excitation. Rinzel, three years after that, simplified the model further to a three variable model with two fast and one slow variable.

This is the model we study today. We will write down a fuller version of the model and reduce it to three variables. We will study the fast two variable system in the phase plane. We will construct the bifurcation diagram and encounter a homoclinic bifurcation. This is a global bifurcation which eludes local analysis.

An interesting electrical behavior of the beta cell in response to glucose is bursting. Bursting typically has a rise in transmembrane potential until some threshold is reached and rapid large excursions in potential (firing) occur. Firing abates with a slower variable reaching a some other threshold. Examples of many types of bursting in many different systems are found in Keener and Sneyd.

We model the transmembrane potential and changes thereof with the Hodgkin and Huxley formalism. In the beta cell, not only is there sodium and potassium currents (along with a background leak current) but there is also calcium movement across the membrane. Furthermore, there is believed to be an independent potassium channel which has a calcium controlled conductance rather than voltage dependent conductance. The balance of all of these transmembrane currents yields the equation including the capacitive current

CmdV/dt = -gim3h(V-Vi) -gkn4(V-Vk) -gkCaCa/(1+Ca)(V-Vk) -gL(V-VL)

where the first term is a mixed inward current due to both sodium and calcium; the second term is the potassium current; the third term is the calcium gated potassium current, and the fourth is the leak current. We have the same gating variables, m, h, and n as in the Hodgkin-Huxley model with their dynamic equations

dy/dt = (yinf-y)/Ty

for y = m, h, and n. Now in the beta cell model we are also concerned about the movement of calcium so that we get the equation for intercellular calcium as influx due to calcium current minus efflux due to buffering and uptake into the calcium storage areas.
dCa/dt = r(-m3h(V-VCa)-kCaCa)

To simplify things (and because it doesn't matter a great deal to the dynamics) we set m and h to their infinity values which reduces the system to three variables V, n and Ca. We note that r is typically a very small value making the Ca dynamics slow compared to the other two variables. We can thus think about Ca as a parameter in the two variable system V, n.

Notice the bursting in the full three variable model. chay.m

Look at the fast variable system in the phase plane. chay.pps

ASSIGNMENT
1) Create the Bifurcation Diagram for the Reduced System:
  1. Look at the fast system model in Pplane for many values of fixed Ca (including negative values).
  2. Sketch a possible bifurcation diagram in the V vs. Ca plane.
  3. Use Matlab to generate the steady state curve including stability (consider Ca as a function of V).
  4. Denote and label the bifurcations.
  5. Plot the amplitude (both max and min) of the stable periodic solutions emanating from the Hopf bifurcation.
  6. Explain why the ending of the periodic solution is considered a homoclinic bifurcation.

2) Show representative phase planes and time plots for interesting regions of Ca.

3) Relate the mathematical analysis to the physical situation of the full three variable system with dynamic Ca.

References (from Gerda De Vries's web page):

I. Atwater, C.M. Dawson, A. Scott, G. Eddlestone, and E. Rojas, The nature of oscillatory behaviour in electrical activity from pancreatic beta-cell, in Biochemistry and Biophysics of the Pancreatic Beta-cell, Hormone and Metabolic Research Supplement Series 10, W.J. Malaisse and I.B. Taljedal, eds., Georg Thiem Verlag, Stuttgart, 1980, pp. 100-107. First descriptive model.

T.R. Chay and J. Keizer, Minimal model for membrane oscillations in the pancreatic beta-cell, Biophys. J., 42 (1983), pp. 181-190. First mathematical model.

T.R. Chay and J. Keizer, Theory of the effect of extracellular potassium on oscillations in the pancreatic beta-cell, Biophys. J., 48 (1985), pp. 815-827. Simplification of the first mathematical model.

J. Rinzel, Bursting oscillations in an excitable membrane model, in Ordinary and Partial Differential Equations, Lecture Notes in Mathematics 1151, B.D. Sleeman and R.J. Jarvis, eds., Springer, New York, 1985, pp. 304-316. Fast-slow analysis.