Buckling dielectric elastomer actuators are special type of electro-mechanical transducers that exploit electro-elastic instability phenomena to generate large out-of-plane axial-symmetric deformations of circular membranes made of non-conductive rubbery material. In this paper a simplified explicit analytical model and a monolithic finite element model are described for the coupled electro-mechanical analysis and simulation of buckling dielectric elastomer membranes which undergo large electrically induced displacements. Experimental data are also reported which validate the developed models.