In guiding-center simulations of a tokamak plasma edge, bifurcations of both the radial electric field and the electrode current are observed for a threshold value of the electrode bias voltage, in qualitative agreement with experiments. The simulations are neoclassical and Monte Carlo based, also containing the perpendicular viscosity effect. Above the voltage threshold, an inward moving, solitary radial electric field structure is generated with a concomitant drop in the electrode current. The position, width, and height of the soliton depend on the magnitude and radial variation of the conductivity (and, thus, on local plasma parameters), finite orbit effects, and also on the perpendicular viscosity and neutral density. The obtained Er structures are narrower and higher than the ones measured in the edge polarization experiments in the Jülich tokamak TEXTOR-94 [S. Jachmich et al., Plasma Phys. Controlled Fusion 40, 1105 (1998)].