Expired membership subscriptions retain access

We were under the impression that a Member who purchases a subscription will have a membership level only until that level expiry date passes.

A year ago we began selling year long subscriptions via a slightly customized version of the PayPal Express Gateway. Now these subscriptions are coming up for renewal. For users who have not cancelled or turned off the auto-renewal, PayPal will send us an IPN notifying us that they have received payment at the end of the year when the subscription comes due. With this IPN, when it hits our Gateway, we then drop the subscription and create a new one.

However, for users who have cancelled or turned off the auto-renewal via PayPal, we will not get a subscription payment IPN from PayPal. When we log in with a user that has the subscription with an expiry date in the past, that user still has access. Is there a configuration setting we missed while setting up these subscriptions? Does membership require that we manually drop these subscriptions? How does the user ultimately lose Membership Level access?

We are logging all IPNs from PayPal in our customized version of the PayPal Express Gateway, so we have a history to work with.

Thanks in advance.