Why Don’t Ions Neutralize in an Applied Current Solution?

[ak33m98]

If a current is being applied to a solution with mobile ions in it, why don't the cations lose electrons and the anions gain electrons creating neutral atoms like they do in electrolysis, what causes these ions to stay charged and form a charged lager when they could just gain/lose charge and become neutral?

 

[DrDu]

The potential not being high enough for the redox reaction to happen?