Chapter 2.3, Problem 40E

The Nobel Prize in Physiology or Medicine in 1963 was shared by A. L. Hodgkin and A. F. Huxley in recognition of their model for the firing of neuronal synapses. As will be discussed in Chapter 12, they proposed that the opening/closing of certain ion channels in the neuron cell was governed by a combination of probabilistic “gating variables,” each satisfying a differential equation that they expressed as

$\frac{du}{dt}=\alpha \left(1-u\right)-\beta u\mathrm{...}(23)$

with positive parameters $\alpha ,\beta$.

a. Use a direction field diagram (Section 1.3) to show that the solutions of equation (23) are “probabilistic” in the sense that if their initial values lie between $0$ and $1$, all subsequent values also lie on $\left[0,1\right]$.

b. Solve (23) and show that all solutions approach the value $\alpha /\left(\alpha +\beta \right)$ exponentially.