Math Physiology: COMPUTER LAB IV

A phenomenological model of the pancreatic B-cell

We are going to use xppaut to explain the oscillations in the following "phenomenological" model of bursting electrical activity in the insulin secreting pancreatic B-cell:

There is a long story behind this model. We need only know a few things. For example, the functions f(u) and g(u) are "cubic-like" and h(u) is linear in u. In the model:

u is transmembrane voltage
w is proportional to the fraction of open K+ channels
c is proportional to intracellular calcium concentration
f(u) represents the voltage-gated Ca-current
g(u) is activation variable for V-gated K+ channel
h(u) represents the calcium influx from channels

and t is time. In experimental labs, the voltage u(t) and the calcium concentration c(t) can be measured. Critical to what we will examine is that the "epsilon" parameter is small. That means that calcium changes slowly. We call u and w "fast" variables and c a "slow" variable.

Bursting solutions:

Download the following code:

Burster.ode Bursting Model

and then execute the code in a terminal window:

xppaut Burster.ode &

Close all but the black window and then

(1) Select "Initialconds"

(2) Select "Go"

Typing "i" then "g" is the equivalent keystroke. You should get something like:

This bursting occurs in the pancreatic B-cell if glucose is at intermediate values. In the model there is a parameter "alpha" which increases with glucose concentration.

(3) Set up a parameter slider for "alpha" for a range of values between -1.5 < alpha < 0. Then vary alpha to see how the increase of glucose "alpha" affects the behavior.

Elevated calcium levels have been correlated to an increase of the insulin secretion rate in these cells. The simulations in (3) support this if one observes the calcium concentration c(t). To see this:

(4) Under "Graphics" menu, add a plot of c(t) in a different color and repeat your simulations varying alpha. Note that on average, c(t) is higher when alpha is higher.

In neurons of course, glucose has little to do with the electrical activity. The stimulus in neurons is primarily a current pulse (either from a synapse or experimentally applied). At zero glucose levels, pancreatic B-cells also exhibit different electrical responses depending on the magnitude of an experimentally applied current "ia". A parameter in the model "ia" models the level of such a current.

(5) Reset alpha=-1.5

(6) Set up another parameter slider for ia in the range 0< ia < 5.

(7) Under the "Initial Conditions" , "New" menu set u=-1.21, w=-1.147,c=1.147 and then run the simulation

You should have level voltage. Now let's apply a current "experimentally".

(8) Do several runs varying the parameter "ia" from 0 to 5 (in steps of 0.5, say)

3 Dimensional "Limit Cycle"

Bursting oscillations have interesting geometries. To see a nice example set the parameter values to:

then, rerun the simulation and:

(9) Select "Graphic Stuff", "Deleted last" (10) Select "Viewaxes" (keystroke "v")

(11) Select "3D" (keystroke "3")

(12) Set the X,Y,Z axes to u,w,c, respectively

(13) Select "Window/zoom" (keystroke "w")

(14) Select "Fit" (keystroke "f")

You should have a nice 3-D rendering of the oscillation, something like the following one rotated in space.

Fast-Slow Dynamics in Bursting

Before we begin, quit xppaut:

(25) Select "File" (keystroke f)

(26) Select "Quit" (keystroke q)

(27) Select "Yes"

Now download the following code:

BursterFS.ode Fast Subsystem of Bursting Model

This code is the first two equations of the previous model. Remember that calcium was "slow". Since it is "slow" one might wonder what the model would do if calcium were held constant (this is not possible experimentally that I know of). The argument here is that over short durations of time t (roughly t=1, say) calcium remains nearly constant so the first two equations should help us explain the dyanmics in the full three variable model. So, in this code, "c" is a parameter.

Now, execute the code:

xppaut BursterFS.ode &

(15) Under "View Axes" "2D" set (X,Y)=(u,w) with the ranges -4< u < 4 and -4 < w <6

(16) Set up a slider so "c" has a range of -3.5 < c < 6 with an initial value of c=1.4.

(17) For a variety of different c values create phase portraits with nullclines ("Nulcline", then "New"). For c=1.4 it show look something like

In the figure above, I have added the stable and unstable manifolds of the center equilibria (which is a saddle). To do this

(18) Select "Sing pts"

(19) Select "Mouse"

(20) click the mouse cursor near the equilibria

(21) Answer "yes" to eigenvalues and invariant sets. You'll need to hit the "Esc" key for each of the Four invariant sets.

As you vary c making new phase portraits you'll see that a Hopf bifurcation occurs at low values, a saddle node bifurcation at c=1 and a homoclinic bifurcation in the range 1< c < 2 (to see the latter better, you should adjust the Slider range for c and the (X,Y)-axes of the plot. Ask if you're not sure what these bifurcations are.

AUTO Here we'll use the "aut" part of "xppaut" to make a bifurcation diagram for the (FS) of the model.

(22) Set c=-4, redraw the nullclines, select an initial condition near the equilibria, run the simulation and then "Initial Cond", "Last" several times. By the end of such a simulation the (u,w) values should be very close to the equilibria and AUTO needs this to be the case to properly initialize.

(23) "File" then "Auto" to get the AUTO window up

(24) Under the AUTO "Parameter" menu make "c" the Main parameter

(25) Under "Axes" set the range of x to (-4,6) and y to be in (-3,4)

(26) Under the "Numerics" menu, set them to be:

(27) Then "Run" and "Steady States"

(28) Using Computer Lab II as a guide, "Grab" the Hopf point and continue the periodic orbits from it. You should get something like

I'll guide you verbally from here

Chay_Cook88.ode Chay Cook 1988 Model of Bursting