Math 1370 Homework 2
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Homework 2 Math 1370 26 points
As you can guess from this exercise, it will be on the HH equations. You will find three codes in the Files/Codes folder: hh.ode, hh.py, HHpul.m which are the HH equations in XPP, Python, and MatLab. You are welcome to code them in anything else that you want, but you must use the parameters as given in these codes. I have tested all three of them (Python3, MatLab R2018B, xppaut8) and they seem to work. MatLab and Python will produce graphics windows. XPP will show the voltage on the screen and you can either take a screen shot or you can export the graph in Postscript and convert to pdf. If you want to install and run XPP, then you can read the documentation on installation that I uploaded.
There are several parameters that are in the codes: I0 or I 0 which is a bias current; iapp, an applied current, ton, the start of the applied current, toff, the end of the applied current. (The latter three are hard-coded in Python). In MatLab, you can set tmax as the duration of the simulation. In XPP, at the bottom, you can set the total inside the program or in the code (total=500, eg).
Run the codes as follows:
HHPul(-65,0 .0,10,500,100,200) inside the MatLab prompt
xpp hh .ode (Windows; on a Mac, /Applications/xppaut hh .ode from the terminal) python3 hh .py (or whatever version you have)
XPP is interactive so that you can do stuff once inside. (I have also uploaded a short tutorial)
So, with that lengthy preface, here are your tasks:
1. (6 points) Run the model for 500 msec with a stimulus of strength (iapp)
10 starting at 100 and ending at 200. (I think these are the defaults, except maybe XPP). Answer the following: (i) How many action potentials were elicited (ii) What is the peak voltage of the first action potential? (iii) Subsequent action potentials? (iv) Why do you think that the first is larger? (v) what is the minimum voltage that is reached? (vi) What is the time difference between the 3rd and 4th AP? between the 4th and 5th? Note they are pretty much the same. Produce a figure that has the voltage and 3 variables during a single action potential and during
repetitive spiking. (You and draw the m,n,h curves on the same plot)
2. (4 points) Now we want to establish the rheobase. This is the minimum current needed to elicit an action potential. Change toff to 500. Then vary Iapp (the size of the applied current) to find the minimum current needed to elicit an action potential. (You should try to get this within 0.2 mV.) What is the minimum current required to elicit sustained oscillations until the end of the simulation?
3. (5 points) An interesting phenomena in the HH equations and the squid axon is called anodal break excitation. Set ton to 100 and toff to 125.
Set iapp to -2 and run the simulation. You should see the membrane potential drop to below rest but then when the stimulus turns off the potential goes above the resting potential before returning to rest. What do you think is going on? (Hint, look at the h variable) Now make iapp the applied current, -3 and rerun the simulation. You should see a nice action potential. This is anodal break excitation and also called a rebound spike. Produce a plot that shows the voltage during this stimulus.
4. (5 points) Set the bias current, I0 = 8 and set the initial conditions as follows: (V, m, n, h) = ( −60, 0.09, 0.4, 0.4) (In MatLab, you can just set the initial voltage and the rest of them will be taken care of, but in Python & XPP, you should hand set them.) Set the current input stimulus to ton=100, toff=105, iapp=1 and run the simulation for tmax=500. You should see some small oscillations for the voltage but it returns to rest. The rest state is stable. Now, increase iapp to 2 and run again. What happens? You should see a sustained oscillation. This shows that the HH model is bistable; it has two different stable states, an equilibrium point and a sustained oscillation. Produce two plots: one showing the relaxing back to 0 and the other showing the transition to oscillation.
5. (6 points) John Rinzel noticed the bistability in the 1970’s and made a prediction that you could stop the repetitive firing by giving an appropri- ately timed stimulus. This was tested and confirmed in a paper with Rita Guttman & Steve Lewis in 1980. You can repeat their experiment. This time set I0 = 6.3 and V (0) = −25. This should result in an oscillation. Set iapp=1 or 2 and try to time a 2 msec pulse of current to stop the oscillation. Hint: look at the voltage trace and time your pulse to when the AP is headed for an upstroke at around -65 mV. Show a plot with this simulation. By setting the bias current to a low value, I’ve made the oscillation quite sensitive. At the high value of 8, you will not be able to kill the oscillation. There is an apocryphal story about this in regards to the cardiac action potential and the dangers of self-experimentation.
2023-02-01