Math Physiology: COMPUTER LAB III

The code There are a few new things in this code.

1) First note how the potassium current ik, the sodium current ina, the leakage current il and the applied current ia are defined BEFORE the differential equations. Models of these types often have large and complicated expressions for the currents so this type of coding is convenient.

2) Next, after the differential equations are declared, the are a few auxilliary or "aux" variables defined. These are variables which which you may want to plot once xppaut is running. Here the capitalized versions of the currents have been defined: IK, INA, IL, IA.

3) At the end of the code a new numerical parameter "bound" has been defined. xppaut terminates any numerical run if the absolute value of the dependent variables exceed this value. For the HH model, the current values can get large (in their dimensional forms) so this has been increased to eliminate premature termination of numerical runs. If you ever run into this problem while coding, you can change the value of "bound" intercatively in the "nUmerics" menu of xppaut.

A First Run: In the simulations we will be performing we get to choose the applied current ia(t) and measure the voltage response v(t). The applied current in this case is the sum of two "pulse" functions defined using the Heaviside function "heav". A single "pulse" in the code has the formula:

i(amp,t0,dt)=amp*(heav(t-t0)-heav(t-t0-dt))

which has an amplitude of "amp" only for the times between t0 and t0+dt. Otherwise, it is zero. Given,

ia=i(ia1,t1,dt1)+i(ia2,t2,dt2)

we get to choose parameter values (ia1,t1,dt1) for the the first "pulse" and (ia2,t2,dt2) for the second "pulse".

Finally, execute the code:

xppaut HH.ode &

Experiment 1: Use the default initial conditions (Initial conditions = New by the key strokes "I", "N" then return several times). You should get:

Note that voltage increases (depolarizes), then decreases (hyperpolarizes), goes below the rest potential and then has a long recovery period before going back to rest. This shape is extremely common in most neurons and is called an "action potential". The stimulus in that case comes from synaptic input from other cells (will address this later).

Next view the applied current which caused this action potential:

1) plot the auxilliary variable IA versus time ("Viewaxes", change "V" to "IA", "OK", then "W","F" - for "Window" "Fit"). Note, the sole spike (depolarizing applied current).

2) Now create a plot showing the currents INA, IK and IA:

a) Erase the figure "E"

b) Create a plot of INA using the Viewaxes ("V") menu setting YMIN=-2000 and YMAX=2000, manually.

c) Add a plot of IK using the "Graphics Stuff" menu. Under this menu, select the "Add Curve" menu and type in "IK" (The keystorke for hetting there is "G" then "A"...as always the capital letters are equivalent keystrokes corresponding to the selection of a menu item). Change the color number if you like.

d) Add a plot of IA as in part c).

You should have something that looks like:

Note that soon after the applied current pulse, the sodium current activates. This causes the depolarization (voltage increase). The potassium current (orange above) is slightly delayed, i.e. activates after the sodium current. The potassium current is positive and opposite sign to the negative sodium current. One is an outward current and one is an inward current.

Experiment 2: Here we will demonstrate the notion of "excitability". This is a mathematically imprecise concept having to do with the response V(t) to the stimulus Ia(t). Roughly speaking, a system is "excitable" if it exhibits a threshold response with respect to the stimulus. Through simulations we will show that

(I) You get an action potential with a large V(t) excursion only if the amplitude "ia1" of the applied pulse is big enough.

(II) That increasing the amplitude of the stimulus further does not appreciably change the shape of the action potential.

To do this:

1) Reset the diagram: "Graphic Stuff", "Remove All". If you didn't do this IA,INA and IK would always be superimposed on the same plot....in some instances a good thing but here it will just complicate the plot.

2) Reset the axes for V: "Viewaxes", "2D", V versus T, YMIN=-10, YMAX=100.

3) Select the "Par/Var?" slider on the lower left of the xppaut window. Click on it and set the values as follows:

4) Use the thin black slider to adjust the pulse amplitude ia1 for consecutive runs with the same initial conditions ("I", "N") for the values ia1=0,1000,2000,3000,4000,5000. You should have something that looks like:

Somewhere between a pulse amplitude between ia1=1000 and ia1=2000 there is a significant "qualitative" difference in the resulting response V(t). This illustrates point (I). Also, for ia1 >3000 the action potentials all look similar. This illustrates point (II) This is "excitability"....an inexact concept which we will explore mathematically later.

Experiment 3: You want to add TTX??? That dreaded neurotoxin that shuts down all the sodium channels. To eliminate the sodium channels just set the conductance gna=0 in the "Parameter" menu. Repeat Experiment 2 with gna=0. Is the behavior the same? Is there an action potential? When you are done, set gna=120 again!!!

Experiment 4: Refractoriness is another inexact concept. Roughly speaking, the "refractory period" is the length of time one must wait before the same (superthreshold) stimulus will cause the "same" action potential response.

1) Erase ("E") everything and set up the axes for a V(t) run.

2) As in Experiment 2, set up the two remaining sliders for the parameters ia2 (second pulse amplitude) and t2 (second pulse starting time). Use the respective ranges (0,5000) and (0,40). Set ia1=2000,ia2=2000 and t2=10.

3) Under the "Graphic Stuff" menu, select "Add curve" and add a curve of IAX. This is different than IA. IAX=IA/40, i.e., a scaled version of IA so it can be plotted along with V(t).

4) Now do a run with the standard initial conditions.

you should get something like:

Notice how the second pulse does not cause a second action potential even though it has the same amplitude. If you wait long enough (t2 somewhere around 30), the "cell" has recovered and can fire again. Try changing t2 to see the effect it has on the voltage response. Here, for example is one with t2=16.4:

Here, you see the refractory period extends well into the recovery period of the cell!!

Experiment 5: Change the parameters to create a plot of the applied current needed for a voltage clamp experiment where both the sodium and leakage currents have been blocked (Say using "pharmacological blockers" such as TTX). Consider changing cm (the cell capacitance) and certain conductances.