Are cycles possible in circuits of neurons?
There are tons of cycles in the brain. The study of how the brain uses these cycles are used for computation is called Theoretical Neuroscience. Browsing the tag on this website is one way to become introduced to those ideas.

Alternatively, these loops are framed in different ways in "How to Build a Brain" by Chris Eliasmith. Loops involving a population node connecting to itself are considered to be memories. Longer loops, such as the basal-ganglia$\rightarrow$thalamus$\rightarrow$cortex loop could be considered analogous to the "main event loop" of a program, but with a few caveats.