Friday, February 24, 2023

A Simple Mathematical Function Representing the Intracellular Action Potential

In Problem 14 of Chapter 7 in Intermediate Physics for Medicine and Biology, Russ Hobbie and I chose a strange-looking function to represent the intracellular potential along a nerve axon, vi(x). It’s zero everywhere except in the range −a < x < a, where it’s

Why this function? Well, it has several nice properties, which I’ll leave for you to explore in this new homework problem.
Section 7.4

Problem 14 ¼. For the intracellular potential, vi(x), given in Problem 14
(a) show that vi(x) is an even function,
(b) evaluate vi(x) at x = 0,
(c) show that vi(x) and dvi(x)/dx are continuous at x = 0, a/2 and a, and
(d) plot vi(x), dvi(x)/dx, and d2vi(x)/dx2 as functions of x, over the range −2a < x < 2a.
This representation of vi(x) has a shape like that of an action potential. Other functions also have a similar shape, such as a Gaussian. But our function is nice because it’s non-zero over only a finite region (−a < x < a) and it’s represented by a simple, low-order polynomial rather than a special function. An even simpler function for vi(x) would be triangular waveform, like that shown in Figure 7.4 of IPMB. However, that function has a discontinuous derivative and therefore its second derivative is infinite at discrete points (delta functions), making it tricky (but not too tricky) to deal with when calculating the extracellular potential (Eq. 7.21). Our function in Problem 14 ¼ has a discontinuous but finite second derivative.

The main disadvantage of the function in Problem 14 ¼ is that the depolarization phase of the “action potential” has the same shape as the repolarization phase. In a real nerve, the upstroke is usually briefer than the downstroke. The next new homework problem asks you to design a new function vi(x) that does not suffer from this limitation.
Section 7.4

Problem 14 ½. Design a piecewise continuous mathematical function for the intracellular potential along a nerve axon, vi(x), having the following properties. 
(a) vi(x) is zero outside the region −a < x < 2a
(b) vi(x) and its derivative dvi(x)/dx are continuous. 
(c) vi(x) is maximum and equal to one at x = 0. 
(d) vi(x) can be represented by a polynomial bi + ci x + di x2, where i refers to four regions: 
        i = 1,    −a < x < −a/2 
        i = 2, −a/2 < x < 0 
        i = 3,      0 < x < a
        i = 4,      a < x < 2a.
Finally, here’s another function that I’m particularly fond of.
Section 7.4

Problem 14 ¾. Consider a function that is zero everywhere except in the region −a < x < 2a, where it is

(a) Plot vi(x) versus x over the region −a < x < 2a,
(b) Show that vi(x) and its derivative are each continuous. 
(c) Calculate the maximum value of vi(x).
Simple functions like those described in this post rarely capture the full behavior of biological phenomena. Instead, they are “toy models” that build insight. They are valuable tools when describing biological phenomena mathematically.

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