What is going on?

In broad terms, the glycolytic oscillations are quite well understood. They are often explained as a result of the enzyme kinetics of phosphofructokinase (PFK), which catalyses the ATP-dependent phosphorylation of fructose-6-phosphate (F6P). This enzyme is inhibited by one of its substrates ATP, and activated by AMP, which is indirectly a product. This gives the autocatalysis needed for the oscillations. In fact, an minimal oscillating glycolysis has been made from the enzymes fructose 1,6-bisphosphatase and PFK.

But the oscillations are a property of glycolysis as a whole, and not a property of a specific enzyme in the pathway. This is evident since things are more complicated in the living cell than in a reconstituted enzyme system. Some obvious features are not accounted for in the above describtion. The oscillations in the NAD+/NADH ratio observed in the laboratory are not included in the description. This goes for most metabolites, and therefore it is a poor description, if one wants to know how glycolytic oscillations interact with the rest of the cell. A detailed investigation of the mechanisms generating glycolytic oscillations is found here.

The fact that we se macroscopic oscillations in the cell suspension shows that the individual yeast cells synchronize their oscillations with each other. You can even mix two suspensions out of phase, and watch how they synchronize. Again, to explain this, a more detailed description is needed.