A simplified model for the membrane potential

Limited exponential growth: Applications of limited exponential growth models

A simplified model for the membrane potential

When we discussed applications of unlimited growth, the discharge of a capacitor and the link with the membrane potential via the differential \[g_k\cdot V+C_m\cdot \frac{\dd V}{\dd t} = 0\] was discussed (where \(V\) is the membrane potential with respect to the resting membrane potential, \(C_m\) the capacity of the membrane, and \(g_k\) the conductivity of an ion channel through which leakage current). Here, we will discuss two improvements of this model:

The trajectory of the membrane potential at a constant stimulus
A model for the conductivity of an ion channel

1. The trajectory of the membrane potential at a constant stimulus Suppose that the membrane potential is equal to the resting membrane potential, i.e., \(V=0\), and that at a certain moment, say at time \(t=0\), a stimulus is given, which is a constant current intensity \(I\) through the membrane. Then we can write the following initial value problem \[g_k\cdot V+C_m\cdot \frac{\dd V}{\dd t} = I, \quad V(0)=0\] In other words: \[ \frac{\dd V}{\dd t}=\frac{I}{C_m}-\frac{g_k}{C_m}\cdot V, \quad V(0)=0\] This is a differential equation of limited exponential growth and the solution is \[V(t)= \frac{I}{g_k}\cdot \left(1-e^{-\frac{g}{C_m}\cdot t}\right)\]

2. A model for the conductivity of an ion channel In a simple description of the behaviour of an ion channel, we use two states of the channel, namely "closed" (C) and "open" (O), and a model can be written as a so-called Markov chain: \[\text{C }\mathop \rightleftharpoons \limits_{{\beta}}^{{\alpha}} \text{ O}\] By not looking at a single ion channel, but at the fractions of the ion channels that are open and close per surface, the following system of differential equations can be written \[\begin{aligned}\frac{\dd C}{\dd t}&=-\alpha\, C+\beta\, O\\ \\ \frac{\dd O}{\dd t}&=\alpha\,C-\beta\, O\end{aligned}\] with \[O+C=1\] Uncoupling of the two equations leads to \[\frac{\dd O}{\dd t}=\alpha\,(1-O)-\beta\, O\] which can be written as \[\frac{\dd O}{\dd t}=\alpha-(\alpha+\beta)\cdot O\] This is again a differential equation of limited exponential growth.

In reality, the mathematical modelling of electrical properties of cell membranes and ion channel is more complex. Hodgkin and Huxley have postulated in their model for action potential generation in nerve cells, that the conductivity of the potassium channel can be modelled as \[g_{\mathrm{K}}=\overline{g}_{\mathrm{K}}\cdot n^4\] where \(n(t)\) is a function which satisfies the following differential equation \[\frac{\dd n}{\dd t}=\alpha_n\cdot(1-n)-\beta_n\cdot n\] with \(a_n\) and \(\beta_n\) constants that depend on the membrane potential.

The differential equation for \(n\) can be rewritten as \[\frac{\dd n}{\dd t}=\alpha_n-(\alpha_n+\beta_n)\cdot n\] so you can easily recognize that this is a differential equation of limited exponential growth with general solution \[n(t)=\frac{\alpha_n}{\alpha_n+\beta_n}+c\cdot e^{-(\alpha_n+\beta_n)\cdot t}\] for some constant \(c\). When the initial value \(n(0)=n_0\) is given, then the solution can be rewritten as \[n(t)=\frac{\alpha_n}{\alpha_n+\beta_n}\cdot \left(1-e^{-(\alpha_n+\beta_n)\cdot t}\right)+n_0\cdot e^{-(\alpha_n+\beta_n)\cdot t}\]