@raboynics, not sure I understand what you are saying. For a 1D problem, the current on a boundary will indeed determine the voltage, but for 2D problems, knowing the total current through an electrode is a scalar constraint and can only determine a voltage if the voltage is also a scalar. Thus for a 2D problem, we must assume in addition that the voltage along the boundary in question is constant. This is why specifying total current is insufficient information to alone determine the voltage on the boundary.
I wasn’t able to find any of the codes you referred to above. Can you send them to me or send me a link to them?