This paper discusses mechanisms for the passage of an 'overlimiting' efficient steady electric current through a cation-exchange membrane. In particular, the possibility is studied for the above effect being due to a nonlinear amplification by the depletion-diffusion layer of localized electric current or salt concentration disturbances, produced at the membrane solution interface. The study is based on the solution of Nernst-Planck-Poisson equations for the ionic transport, accounting for the formation of a space charge near the membrane solution interface in the course of concentration polarization. Respective experimental results are reported, concerning the effect of elimination of gravitational convection in the depleted diffusion layer, along with those concerning the effect of voltage modulation upon the limiting current Cu** plus ** plus reduction on a copper cathode from aqueous CuSO//4 solution.