For much of the later Roman Empire (usually beginning during the Severn dynasty in the third century), the economy was failing. This was largely due to the increasing spending in military and debasement of currency due to the lack of silver content in the coins. This lasted for most of the third century and only began to change under Diocletian. He introduced a number of economic reforms including the Edict on Maximum Prices (albeit this was fairly unsuccessful). Including some reform on the coinage itself (the introduction of gold instead of silver, etc) the economy gradually got much better through Diocletian and Constantine's reign.

So to answer your question, the economy certainly did fluctuate, and there were many depressions, although they did not occur as frequently or quickly as would happen today because of travel time, etc.