Math Physiology: COMPUTER LAB V
Math Physiology: COMPUTER LAB V
The program "xtc"
Much like xppaut, the program xtc (also written by Bard Ermentrout) requires a file "file.xtc" which defines the problem to be solved, parameter values, numerical parameters and the like. xtc gets its name since x=space, t=time and c=continuum. It uses the method of lines to solve systems of reaction diffusion equations on finite spatial intervals 0 < x < L. It is currently no longer supported by Dr. Ermentrout so the version we have is old. Some documentation can be downloaded (in latex format): xtc_doc.tex
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 &
You should get a window that looks like:
There are few menus. Like xppaut, you can select a menu by typing the capital letter associated with, i.e., "I" for "Init. cond" and "N" for "Numerics". Roughly, each menu is for the following:
Init. data Set initial conditions for the run.
3D graphs Choose the type of 3D graphics for displaying v(x,t), i.e., contour, color, surface etc.
2D Graphs Choose the type of 2D graphics for a "slice" of the solution v(x,t) at a particular x or t value.
Numerics Adjust numerical tolerances, parameters, run durations, number of spatial grids, etc.
File save runs, read parameter sets, write parameter sets,...
Redraw refresh the diplay
More Time continue the run using a specified amount of more time
Erase erase the displays
Go execute a run
Window fit run to window, rescale graphics, etc.
Parameter change model parameters
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.
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:
Do more than one pulse ever occupy the same axon at the same time?