### 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.