CBS 613: COMPUTER LAB V (Wave Phenomena in Excitable Media)

v = transmembrane voltage (potential)

w = fraction of potassium channels "open"

f(v) = net sodium current

i₁ = applied intra-axon current at left side

i₂ = applied intra-axon current at right side

I = experimentally applied transmembrane current (transverse)

The code: Download the following "xtc" codes for the model described above:

FHN_Wavefront_1.xtc

FHN_Wavefront_2.xtc

FHN_Wavefront_3.xtc

FHN_Wavefront_4.xtc

FHN_Wavefront_5.xtc

Experiment 1

In the first experiment, i₁ = i₂=0 but a pulse current is applied of amplitude ia=1 is applied on the spatial interval x0 < x < x0+dx (here x0=0.5,dx=0.05) for the duration interval t0 < t < t0+ddt (here t0=1, ddt=0.7). Initially v(x,0)=w(x,0)=0, so the "axon" is at rest. The model simulates a brief superthreshold transmembrane current being applied (by an electrode) at x=x0. The source code is in the file "FHN_Wavefront_2.xtc". To execute it, type:

xtc FHN_Wavefront_2.xtc &

For the purpose of this lab, I have preset everything in the source files so that the runs work out nicely. To execute this first experiment, type the following:

1) "G" or click "Go". Wait for the run to complete.

2) "W" or click "Window", then "Fit 3D".

You should have something that looks like:

The horixontal axis is the spatial coordinate x. The vertical axis is time t with t=0 at the top and the duration of the run (here 19.6) at the bottom. Black indicates a larger value of v(x,t) on this grid. White represents a lower value of v(x,t). Note that 2 waves were initiated by the applied pulse!! Color is better....type:

3) "3D graphs"

4) "0.Color"

5) "OK"

6) "Window"

7) "Fit 3D"

You now should have:

Or, to visualize the results as a surface, select:

p 8) "3D graphs"

9) "3.SurfCol"

10) "OK"

11) "Window"

12) "Fit 3D"

to get v(x,t) as a surface plot:

Now, repeat steps 3)-7) above to prepare for another run (xtc will dynamically display the solution as it is being computed only in the "0.Color" or "2.Grey" graphical modes). Now select:

13) "Init. data"

14) "Last"

15) "Go"

This will restart the run at t=0 but using initial values for v(x,0) and w(x,0) from the last run. As the code is written, an applied pulse near x=x0=0.5 will occur at time t0=1. What you should notice is that the two previous waves continue to the boundary of the domain (x=0 and x=L=1) AND that the applied pulse does NOT initiate new pulses. This is an example of refractoriness in the spatially dependent model. As the pulses travel, one can think of the waves leaving a refractory wake behind it.

Now, exit xtc by selecting "File", then "Quit".

Experiment 2

Here current only i₂ is nonzero. Moreover, it is a time dependent pulse of duration approximately "ddt" and amplitude "ia" (in the parameter set under "Parameters"). This simulation mimics a short duration axial current originating from the cell's soma located at x=1 (and perhaps more closely related to how travelling action potentials are initiated in cells "in situ"). Execute the UNIX command:

xtc FHN_Wavefront_3.xtc &

and then "Go". The code runs slower because the spatial grid is finer and the run is for a longer time interval. You should get something like:

The "slope" of the pulse is approximately constant indicating a constant ("unique") wavespeed. The theory for existence of pulse solutions for this model (on the infinite spatial domain) predicts a unique wavespeed.

Question: Does the magnitude or duration of the applied current affect the wavespeed? Exit xtc.

Experiment 3

Is like the previous experiment except that the axial current is not a pulse but constant, i.e. the soma is supplying a constant superthreshold current (axially) to the axon.

xtc FHN_Wavefront_1.xtc &

Execute this code as is. You should get something like:

Question: Do more than one pulse ever occupy the same axon at the same time??

Experiment 4

In this experiment there is no axial current applied at the ends. There are now two transmembrane currents that you can apply at independent locations, at different times for different durations. Instead of just I(x,t), there is an I₁(x,t) and I₂(x,t). The parameters for I₁(x,t) are

x01 , ddx1 , t01 , ddt1 , ia1

The respective parameters for I₂(x,t) all end in "2" instead of "1". Execute:

xtc FHN_Wavefront_4.xtc &

Before you start, click on the "Numerics" menu, select "Euler" and change

GRID =50

NOUT=100

Now "xtc" will use a courser space grid and a different time integration technique (Euler's method). The results will be less accurate but the calculations much quicker.

Initially the parameters for I₁(x,t) are set to initiate a travelling pulse action potential (Verify this).

Project Choose the parameters defining I₂(x,t) to stop the action potential half way down the axon. What should the sign of ia2 be? Here's an example:

Experiment 5

Something completely different. Here the diffusivity is zero. The model can then be thought of a line of GRID=100 cells acting independently. The applied current is:

I(x,t) = ia sin( (mx+b)*t )

(You may want to change to "Euler" like before). Execute:

xtc FHN_Wavefront_5.xtc &

and do a run. From the resulting (rather cool looking) figure, what can you tell about the frequency response of the cell?