6
$\begingroup$

I'm a high school student who's currently in AP Biology, so my background in biology is quite limited. I'm interested in mathematically modelling action potentials.

Doing a quick google search, I discovered the Fitzhugh-Nagumo model, a simplified case modelling the voltage of the cell during an action potential as a second order differential equation $ C \frac{d^2V}{dt^2} + \epsilon(V^2-1) + \frac{V}{L} = 0$. I want to solve this equation numerically, but I can't find the numerical values of $\epsilon$, $L$, or $C$ online. What are some reasonable parameters for this model for a human neuron?

$\endgroup$
2
  • 2
    $\begingroup$ A high school student would wants to solve a differential equation numerically. Woah. I definitely did not have your level in math and programming when I was in high school. +1. $\endgroup$
    – Remi.b
    Apr 28, 2015 at 0:12
  • 2
    $\begingroup$ I didn't know about the Fitzhugh-Nagumo model. I quickly scrolled down this paper and this scholarpedia article. Maybe you can indicate where your differential equation come from and link to some info about the meaning of the parameters $\epsilon$, $L$ and $C$. $\endgroup$
    – Remi.b
    Apr 28, 2015 at 0:13

1 Answer 1

5
$\begingroup$

This is actually not the Fitzhugh-Nagumo model of the a neuron, however it is highly related to it. I believe the the Equation you have there, is capable of oscilations for any positive values of $\epsilon$, $C$, $L$. I generated this graph with the parameter values all set to 1. Note how it is very similar to a sine wave.

I believe you where trying to write Van der Pol's equation here: $$C\frac{d^2V}{dt^2} +\epsilon (V^2-1) \frac{dV}{dt}+ \frac{V}{L}$$ Note the subtle difference of multiplying by the first derivative. Here is again a image with all parameters set to 1

enter image description here

The Fitzhugh-Nagumo equation can be found on wikipedia here and on peer reviewed scholarpedia here. For reference it is a two state model described by the equation:

$$ \frac{dV}{dt}=V-V^3/3-W+I$$ $$ \tau \frac{dW}{dt}= V + a -bW$$

A common value for the parameters are $a=.7$,$b=.8$,$\tau=12.5$ $I$ can be anything really, I used $I=.5$ to generate the graph below. Also note that the model wont spike if I below a certain value. The Fitzhugh Equation is is slightly more general, but does simplify to the Van der pol under the assumption $a=b=0$. Here is the graph of it with the constants I described. enter image description here

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .