Taking the derivative of an Action Potential to find the inflection point

Hello,

MATLAB newbie here, please go easy on me. I need to determine the slope of a membrane depolarization before the marked inflection point in this voltage vs. time graph:

The purpose is to compare the pre-inflection point slopes of several action potentials. The recording was taken at 20kHz. Since this is discrete data, would it be possible to take a derivative of the data so that I can observe when the slope changes? I'd rather not do this arbitrarily.

Thanks

Answer 1

[Stefan]

Thanks for the response! I've gotten MATLAB to define dy. Both y and t are (301x1 double), however dy comes out as (300x1 double).

This means that I can't plot (dy, t), and I'm also losing a data point somewhere. I suppose by definition, taking diff(y) of n will have n-1 time points. I'll try removing a point from t to make it (300x1 double) as well.

[Matt Tearle]

Yes, that's expected because it's a finite difference approximation -- the derivative at t is approximated using y(t) and y(t+dt). This means there's no derivative for the last t value. (Or you can shift t by one and think of it as a backward difference, so there's no derivative for the first point.) The simplest solution is just plot(t(1:end-1),dy). But if you really want all 301 derivatives, you'll have to use a different finite diff approximation for the last point... but you could just use a backward difference there, in which case the derivative at the 301 point is the same as the 300th:

dy = zeros(size(y));

dy(1:end-1) = diff(y)*Fs;

dy(end) = dy(end-1);

plot(t,dy)

[Stefan]

Wohoo!

http://imgur.com/0BWsK

Answer 2

[Matt Tearle]

Approximate derivative of a discrete signal y (with discrete independent variable t) can be obtained using

dy = diff(y)./diff(t)

If you want more accuracy, you'd need to find a higher-order finite difference formula.

In your case, the sampling is, presumably, uniform (20kHz), so you can just do

Fs = 2e4;
dy = diff(y)*Fs;

Another approach would be to perform some kind of interpolation or local least-squares fit, then use the interpolant to determine the derivative. This works better if the signal is noisy.

Answer 3

[andy]

hi, @stevan, can you give your script about action potential ?

[Stefan]

Absolutely!

Attached is the function that I ended up with. It takes the sample rate in kHz (I use 50), the voltage trace, and an onset time (when the postsynaptic potential begins, I use 0.6).

I'm also attaching a sample spike that works with this code, marked 'V'. I commented it fairly well on my first pass through, but feel free to ask questions if you get stuck.

To run the script, load V into your workspace, and input this command:

V_thresh_fun3(50,V,.6);

All the best,

Stefan