Final answer:
The process that occurs when a train of action potentials stops firing involves an initial influx of sodium ions for depolarization, followed by an efflux of potassium ions for repolarization, leading to a refractory period and the restoration of the resting potential.
Step-by-step explanation:
When a train of action potentials stops firing down an axon, it's because the process of depolarization and repolarization has completed. Initially, sodium ion channels open, allowing sodium ions to influx and depolarize the membrane, which starts the action potential. Following this, potassium ion channels open to allow potassium ions to exit the cell, leading to repolarization. During the subsequent refractory period, the neuron cannot fire another action potential. Meanwhile, the sodium-potassium pump works to restore the resting potential. Propagation of the action potential is either continuous in unmyelinated axons or saltatory in myelinated axons, where the action potential 'jumps' from node to node at the Nodes of Ranvier. The process ends at the terminal buttons, where the action potential can stimulate neurotransmitter release.