CBS 613: COMPUTER LAB IV (Dynamics of Single Cells)

In the first part of this lab we will use the program "xppaut" to examine bistability in a phenomenological model of "bursting". The model mimics the glucose stimulated electrical activity of the insulin secreting pancreatic B-cell. These same type of oscillations are observed in several types of neurons and the ionic mechanisms are all very similar.

xppaut Overview and Intro:

The program "xppaut" is specialized to solve and analyze the dynamics of ordinary differential equations and maps. The program was written by Prof. B Ermentrout (U. Pitt, Math Dept) and, unlike matlab, it is free. It comes with a full web based tutorial and PDF documentation. "xppaut" is pronounced "X-P-P-AUT" which is a shortened acronym for "X window based program for Phase Planes and AUTo". The later subprogram "AUTO" creates something called bifurcation diagrams which we will see later.

To use xppaut, the user must first create a file with a *.ode suffix which defines the problem. If the code has the name "file.ode", one executes the following UNIX command in a terminal window:

xppaut file.ode &

The (phenomenological) model

We are going to use xppaut to explain the bursting oscillations in the model:

There is a long story behind this model. We need only know a few things. For example, the functions f anf g 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 potassium channels
c	is	proportional to intracellular calcium concentration
f(u)	represents	the voltage-gated calcium current
g(u)	is	hard to explain
h(u)	represents	the calcium influx from channels

and t is time. In 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. To change this value and re-run the simulation do the following:

(3) Select "Erase" (or type "e")

(4) Select "Parameters" (or type "p")

(5) Type "alpha" then hit the return key at the top by the word "Parameter:"

(6) Backspace and enter the new beta value

(7) Type return twice

(8) Select "Initialconds" (or type "i")

(9) Select "Go" (or type "g")

Steps (3)-(9) can be repeated for several alpha values. Repeat these steps for the following increasing alpha values (increasing glucose)

alpha = -1.25,-0.95,-0.35,-0.255,-0.20,0

Question How does the cell's qualitative electrical activity change as glucose concentration increases?

The classifying of stable (and unstable) behavior of solutions as a function of a parameter(s) in mathematics is called "bifurcation theory". We just examined the bifurcations of the electrical activity as a function of glucose concentration.

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.

(10) Reset alpha=-1.5

(11) Select "Initialconds"

(12) make u=-1.21, w=-1.147,c=1.147 at the top

(13) hit the return key

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

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

Question How do the bifurcating behaviors in applied current "ia" compare to those in glucose "alpha"?

Viewing other variables

Before we do anything else, we should reset the parameter values to those initially. They should be:

Now, do the following:

(15) Select "Initialconds"

(16) Select "Go"

(17) Select "Viewaxes" (keystroke "v")

(18) Select "2D" (keystroke "2")

(19) Enter: C for Y-axis, Ymin=0, Ymax=2, then click "OK"

You should have the following

This shows the calcium concentration c(t) as a function of time. Note there are no large amplitude "spikes" like in the active phase of the voltage burst. Yet the calcium oscillation has the same period of the bursting of the voltage. We will try to explain this in a moment.

The model has three time dependent variables. Like 2-D phase planes, we can render a three-dimensional phase space portrait of the oscillation by plotting (u(t),w(t),c(t)) in space for every time t. The following commands will do this:

(20) Select "Viewaxes" (keystroke "v")

(21) Select "3D" (keystroke "3")

(22) Change "X-axis" to "U", then click "OK"

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

(24) Select "Fit" (keystroke "f")

You should have a nice 3-D rendering of the oscillation

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 &

(28) keystroke: "v", then "2", then change all parameters to match the following:

For all of these runs c=1.4. The solution depends on the initial conditions (state). The figure is u versus w (i.e., a phase plane). To chose an initial condition do:

(29) keystroke: "i" then "m", then click the mouse button on a (u,w) pair you want to start at.

Repeat step (29) for several different initial conditions. You should get something like:

Question: Given the morning lecture, what is this figure illustrating? What are the stable states?

(30) Keep track (say on a sheet of paper) how many stable states (and type) there are ranging c from -16 to 6 (paying careful attention to the range 0.9-2.0). You may have to "Erase" (keystroke "e") and "Window/Fit" (keystroke "w" then "f") to see everything.

Question Imagining that c decreases at low voltages but increases at larger voltages, can you explain how bursting in the full model works?

Want more??????

Hodgkin-Huxley Model and Excitability:

Click here for the last part of the lab.