In electrodialysis desalination processes, the operating current density is limited by concentration polarization. In contrast to other membrane processes such as ultrafiltration, in electrodialysis, current transport above the limiting current is possible. In this work, the origin of the overlimiting current at cation-exchange polymers is investigated. We show that, under certain experimental conditions, electroconvection is the origin of the overlimiting conductance. The theory concerning electroconvection predicts a shortening of the plateau length of membranes with increased conductive or geometrical heterogeneity. We investigate the influence of these two parameters and show that the creation of line undulations on the membrane surface normal to the flow direction, having distances in the range of approximately 50-200% of the boundary-layer thickness, lead to an earlier onset of the overlimiting current. The plateau length of the undulated membranes is reduced by up to 60% compared to that of a flat membrane. These results verify the existence of electroconvection as a mechanism destabilizing the laminar boundary layer at the liquid-membrane interface and causing ionic transport above the limiting current density.