What is the mechanism responsible for the periodicity of hiccups?
Check out Van Der Pol (chaotic) oscillator. It can be used to model heart rhythms. If a stimulus is applied to the oscillator, it will return to the curved envelope discussed in the article.

However, if "pushed just right", the oscillator's current state will be pushed towards the very center of the loop, and it will stop oscillating. This is the mechanism by which a sudden blow to the chest can stop the heart.

Reading up on hiccups, their "almost regular" interval sounds similar to a chaotic oscillator. The abrupt stop may represent pushing the oscillator towards the middle of the limit cycle, from which it cannot recover, effectively stopping hiccups