Two kinetic models for the auroral topside ionosphere are compared. The collisionless plasma distributed along an auroral magnetic field line behaves like a non-Ohmic conducting medium with highly non-linear characteristic curves relating the parallel current density to the potential difference between the cold ionosphere and the hot plasmasheet region. The (zero-electric current) potential difference, required to balance the current carried by the precipitating plasmasheet particles and the current transported by the outflowing ionospheric particles, depends on the ratio nps.e/nth.e and Tps.e/Tth.e of the plasmasheet and ionospheric electron densities and temperatures. When in the E-region the magnetic field lines are interconnected by a high conductivity plasma the resulting field-aligned currents driven by the magnetospheric potential distribution are limited by the integrated Pedersen conductivity of the ionospheric layers. These currents are not related to the parallel electric field intensity as they would be in Ohmic materials. The parallel electric field intensity is necessarily determined by the local quasi-neutrality of the plasma.