As the final product of stellar evolution, neutron stars are exotic bjects both in terms of their internal structure than their electromagnetic activity. We will study the electrodynamics of their magnetosphere using a theoretical and numerical techniques for plasma simulations of highly magnetized relativistic electron/positron pairs bathed in an intense photon field. We will decipher their enigmatic properties by including general relativity effects and of quantum electrodynamics on the dynamics of particles and electromagnetic waves. We will deduce the signatures from it associated observational data in the form of multi-wavelength pulsed radiation from existing and future terrestrial and space telescopes accessible to the various partners of the project.