You can have clients loose connectivity and APs loose connectivity.
What is telling you that the APs are loosing connectivity? Can you see that in the dashboard? Anything interesting in the event log? If you hold your mouse over the "red bar" does it say what causes the outage?
If this is only started happening in the last week - what changed a week ago?
I have heard of an LLDP bug affecting Cisco Enterprise APs, but not Meraki APs. Do you have LLDP enabled on the 3850 ports (can you do a "show lldp ne" and get a result)? If so, try an experiment and disable LLDP on AP ports.