When we say 'communication between UE and Network', we normally think about only signaling message (RRC or NAS message). When I say 'communication' in this case, it means 'control command exchange' between UE and network, not the data traffic.
In UMTS case, it is true that only RRC and NAS message functions as communication between UE and Network, but in LTE case there are several communication path at MAC layer. It implies that there are special MAC structure that carries special control information. These special MAC structure carrying the control information is called 'MAC CE', which means 'MAC Control Element'.
This special MAC structure is implemented as a special bit string in LCID field of MAC Header (Refer to LTE MAC page for the details of MAC header).
There are several MAC CE in downlink MAC and also several MAC CE in uplink MAC. Following table from 36.213 shows the LCID types of MAC header. The parts marked in red rectangle is LCID representing various MAC CE. These are relatively small set of MAC CE defined in 36.321 Rel 8 - 6.1.3 MAC Control Elements. You will see this gets longer as new release comes out (Refer to LTE-Advanced MAC for the snapshot of the evolution )
MAC CE for LTE Advanced
With the introduction of LTE Advanced, MAC CE has been extended a little bit further to meet the operations in LTE Advanced. Refer to MAC in LTE Advanced page for the details.