[PERFORM] How can I find the source of postgresql per-connection memory leaks?

I'm using postgresql 9.5.4 on amazon RDS with ~1300 persistent connections from rails 4.2 with "prepared_statements: false". Over the course of hours and days, the "Freeable Memory" RDS stat continues to go down indefinitely but jumps back up to a relatively small working set every time we reconnect (restart our servers). If we let it go too long, it goes all the way to zero and the database instance really does start to go into swap and eventually fail. Subtracting the freeable memory over days from the peaks when we restart we see that there are 10's of MB per connection on average.