Electrical conduction in silica-based glasses under a low electric field is dominated by high mobility ions such as sodium, and there is a transition from ionic transport to electronic transport as the electric field exceeds 108 V/m at low temperatures. Electrical conduction under a high electric field was investigated in thin low-alkali boroaluminosilicate glass samples, showing nonlinear conduction with the current density scaling approximately with E1/2, where E is the electric field. In addition, thermally stimulated depolarization current (TSDC) characterization was carried out on room-temperature electrically poled glass samples, and an anomalous discharging current flowing in the same direction as the charging current was observed. High electric field conduction and TSDC results led to the conclusion that Poole-Frenkel based electronic transport occurs in the mobile-cation-depleted region adjacent to the anode, and accounts for the observed anomalous current.