How does the membrane potential change so drastically during an action potential?

Question

I understand that the Nernst potentials

$$V = \frac{k_\mathrm{B}T}{e} \ln\left(\frac{c_\mathrm{out}}{c_\mathrm{in}}\right)$$

for all ions inside and outside a cell barely change when an action potential travels along the cell, because barely any ions transverse the membrane compared to the amount of ions present. However, the membrane potential has an enormous change of 100 mV.

Can someone provide an intuition for that behaviour?