Reading action potentials 4 der

• The initial form of the equation is $$\frac{dn}{dt} = \alpha_{n} (1 - n) - \beta_{n} n. \quad\text{(Equation 10)}$$
• Let us focus on the right hand side. If we multiply the first rate constant through, we get $$\alpha_{n} *1 - \alpha_{n} n - \beta_{n} n. \quad\text{(Equation 10.1)}$$
• Let's group together the factors multiplying the gate probability n: $$\alpha_{n} - (\alpha_{n} + \beta_{n}) n. \quad\text{(Equation 10.2)}$$
• Let's factor out the term multiplying the gate probability n: $$(\alpha_{n} + \beta_{n}) (\frac{\alpha_{n}}{(\alpha_{n} + \beta_{n})} - n). \quad\text{(Equation 10.3)}$$
• We can re-write this by turning the multiplying factor into a factor that divides the right hand side, if we recall that $x = 1/(1/x)$: $$\frac{ (\frac{\alpha_{n}}{(\alpha_{n} + \beta_{n})} - n)}{\frac{1}{(\alpha_{n} + \beta_{n})}}. \quad\text{(Equation 10.4)}$$
• We can simplify the right hand side by defining $$\frac{\alpha_{n}}{(\alpha_{n} + \beta_{n})} = n_{\infty} \quad\text{(Equation 10.5)}$$ and by defining $$\frac{1}{(\alpha_{n} + \beta_{n})} = \tau_{n} \quad\text{(Equation 10.6)}$$ which allows us to re-write Equation 10.4 as $$\frac{(n_{\infty} - n)}{\tau_{n}}. \quad\text{(Equation 10.7)}$$ and this allows us to re-write the original equation as $$\frac{dn}{dt} = \frac{(n_{\infty} - n)}{\tau_{n}}. \quad\text{(Equation 11)}$$