Formula for this curve?

  • A
  • Thread starter TULC
  • Start date
  • #1
4
0
Summary:
Can someone help me figure out a formula for an action potential of a neuron? (attached)
I want to find a formula for an action potential (illustrated with the curve in the attachment). I would like to use the formula to graph this in Desmos graphics calculator. I don't have much of a math background, but a sine function comes to mind...I would like to get the precise shape though. See the source of the image here (top figure).
 

Attachments

  • Action Potential with values.png
    Action Potential with values.png
    8.2 KB · Views: 26

Answers and Replies

  • #2
FactChecker
Science Advisor
Gold Member
6,374
2,518
It doesn't look like any standard, closed-form, equation to me. What accuracy are you looking for? There are interpolation methods that can give a reasonable approximation, but they are messier than you might want.
 
  • #3
35,505
11,959
The difference of two exponentials can look roughly like that. Example: 6*exp(-x^2*36)-exp(-(x-0.4)^2)
To model the tail better a Crystal Ball function can be used.
sin(x)/x could be interesting, too.

But generally you won't get a really good approximations just with random functions unless you add an absurd amount of parameters.
 
  • #4
FactChecker
Science Advisor
Gold Member
6,374
2,518
Just eyeballing it, it might be possible to approximate it with the sum of a normal distribution and a piecewise continuous function like
\begin{equation*}
g(x)=\begin{cases}
0 \quad &\text{if} \, x \lt a_1 \\
a_2-(x-a_3)/a_4 \quad &\text{otherwise} \\
\end{cases}
\end{equation*}
Where the parameters ##a_1##, ##a_2##, ##a_3##, and ##a_4## are TBD.
 
  • #5
4
0
It doesn't look like any standard, closed-form, equation to me. What accuracy are you looking for? There are interpolation methods that can give a reasonable approximation, but they are messier than you might want.
Thank you. I would like to be able to at least convey the shape of the function to someone who is visually impaired (Desmos can sonify the functions). However, the higher the precision, the better. The values on the y-axis are of special interest. It would be good to capture the change in voltage from -70 mv to +40 mv (vertical)...
 
  • #6
4
0
Thank you all...I will check these out.
 
  • #7
4
0
Just eyeballing it, it might be possible to approximate it with the sum of a normal distribution and a piecewise continuous function like
\begin{equation*}
g(x)=\begin{cases}
0 \quad &\text{if} \, x \lt a_1 \\
a_2-(x-a_3)/a_4 \quad &\text{otherwise} \\
\end{cases}
\end{equation*}
Where the parameters ##a_1##, ##a_2##, ##a_3##, and ##a_4## are TBD.
Interesting...I'll explore this.
 
  • #8
pbuk
Science Advisor
Gold Member
2,352
1,088
Desmos is not really the ideal tool for this (and I have no idea if anything more suitable is available on the web) but I think your best chance may be modelling each part of the curve separately something like this:
https://www.desmos.com/calculator/v9hyi8qj0o
 
  • #9
DaveE
Science Advisor
Gold Member
1,590
1,227
This looks similar to a 2nd order band-pass filter step response, as shown below. The parameter Q will adjust how oscillatory the curve is. The parameter ωo is just a scaling factor for time.

LCR_pulse2.jpg
 
  • #10
DaveE
Science Advisor
Gold Member
1,590
1,227
The longer negative tail in your picture is a more nonlinear response. Probably best done in some piecewise approach. For example, the previous function where ωo is larger when the signal goes negative.
 

Related Threads on Formula for this curve?

  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
5
Views
2K
Replies
4
Views
2K
Replies
1
Views
617
Replies
4
Views
2K
  • Last Post
Replies
5
Views
1K
Replies
4
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
5
Views
2K
Top