Suspected crimes are reported to the police every day. It is sort of what they are there for in the first place. Just because the police and the prosecutor decide not to prosecute the case doesn't mean that a falsified complaint was made. It means there wasn't enough evidence to try you for the case or they simply didn't feel it was important enough to prosecute you for.

Let's say you got a copy of the complaint. You don't get to choose to prosecute the neighbor. The person that would choose to prosecute is the same prosecutor's office that already chose not to file such a charge.