  Understanding the dynamics of magnetic vortices has emerged as an important
challenge regarding the recent development of spin-torque vortex oscillators.
Either micromagnetic simulations or the analytical Thiele equation approach are
typically used to study such systems theoretically. This work focuses on the
precise description of the restoring forces exerted on the vortex when it is
displaced from equilibrium. In particular, the stiffness parameters related to
a modification of the magnetic potential energy terms are investigated. A
method is proposed to extract exchange, magnetostatic and Zeeman stiffness
expressions from micromagnetic simulations. These expressions are then compared
to state-of-the-art analytical derivations. Furthermore, it is shown that the
stiffness parameters depend not only on the vortex core position but also on
the injected current density. This phenomenon is not predicted by commonly used
analytical ans\"atze. We show that these findings result from a deformation of
the theoretical magnetic texture caused by the current induced Amp\`ere-Oersted
field.
