In 1963, Hodgkin and Huxley were awarded the Nobel Prize for their pioneering work on characterizing the transmission of electric impulses along nerves. The classical (1952) Hodgkin-Huxley (HH) model for the action potential of a space-clamped squid giant axon remains the subject of mathematical investigation to this day, due to its impressively rich dynamics. I will focus on certain solutions of the HH equations classified as mixed-mode oscillations, consisting of long periods of low-amplitude oscillations abruptly interrupted by large, rapid spikes. I will introduce some of the theory needed to understand how these solutions arise, including recently developed theory on canards in systems of ordinary differential equations with two slow variables.