The three functions of (1) controlling cabin pressure under normal circumstances, (2) preventing maximum differential pressure being exceeded and (3) preventing negative pressure limit exceedance are NORMALLY handled by two separate mechanical valves (all three can be built into the same valve eg the BAe /Hawker 125)
The negative relief valve is a simple lightly sprung valve and is important because conventional civil aircraft structures are designed to withstand significant positive pressure but will collapse like a Coke can with 10% of that value under negative pressure (possible for example in an emergency descent).
Over pressure protection is crucial (as seen from the photo above) because although the pressure number sounds low (less than 10 Pounds Per Square Inch Differential P.S.I.D.) the load taken by the structure is huge. A door 40 inches wide 100 inches height would have 40,000 pounds pressure trying to push it outwards. The way over pressure (failure of auto/manual pressurisation) is dissipated is through the outflow valves so this detection is normally bolted onto the outflow valves and it overrides any signal given by the pressurisation control system.
Outflow valves (normally two) are just connected electronically (or pneumatically in the case of some Garrett systems) to the pressurisation controller and modulate around a slightly open position under normal control (rarely closed as pax need constant air change).