The problem of obtaining Helmholtz equivalents for nonlinear resistive one-ports is considered. Two fundamentally different classes of equivalent are described, one local and the other global. For each, necessary and sufficient conditions are derived for the existence and uniqueness of either the Thévenin equivalent or the Norton equivalent or both. These concepts are illustrated (i) by proving that a cell whose channels and pumps are monotone in the membrane potential will, in the absence of net state changes in these ionophores, possess a unique stable resting potential and (ii) by demonstrating that it is in principle impossible to assign unique equivalent circuits to such ionophores.