While we can only help so much with Asterisk issues (since we did not make the product), in general terms here's how registration works:
1) Express Talk will send a register request to the PBX, including a suggested expiry time for the register
2) the PBX will accept the register (after authorization) and either acknowledge the suggested expiry time OR give the softphone a different expiry time.
3) Express Talk will always "honour" whatever the expiry time is, and will only re-register at the specified time.
4) The default registration expiry time in most cases is 1 hour (3600 seconds).
5) Now if you are having problems with asterisk and getting disconnected within an hour, it is expected behaviour that Express Talk would not try to re-register again until the hour is up. If asterisk has a setting to force the registration expiry times, then shorten it to something like 300 secs or 600 secs.
6) If you can't get Express Talk to register with Asterisk, that would point to a networking/firewalling/routing issue, we have no problems using it with our Axon software.