For an isolated membrane, the resting (zero current) potential is stable if the slope conductance is positive, and is unstable if the slope conductance is negative. Recent work suggests that the properties of many preparations are influenced by the presence of an extracellular space that is not in good diffusive contact with the bulk extracellular fluid. Ionic current flow across the membrane changes the ion concentrations in this space. These concentration changes affect the stability of the membrane potential. Even if the slope conductance is negative, the presence of the extracellular space can confer stability on the resting potential. Conversely, even if the slope conductance is positive, the extracellular space can produce instability of the resting potential. Evaluation of the relevant parameters for cardiac Purkinje fibres, from published experimental data, suggests that concentration changes in the extracellular space may play a significant role in determining when an action potential is initiated.