Most synthetic channels are Ohmic in conductance, that is, the current passed (both individually and as an ensemble) is proportional to the potential across the membrane.  Some rare channels, however, show current-voltage characteristics that is non-linear.  Specifically, two different types of non-Ohmic conductance are known: 
# a rectifying behaviour, where current passes depends on the sign of the applied potential, and
# a exponential potential dependence, where the current passed scales exponentially with the applied potential.
The former requires asymmetry with respect to the mid-plane of the lipid bilayer, and is realized often by introducing an overall molecular dipole.  The latter, demonstrated in natural channels such as alamethicin, is rarely encountered in synthetic ion channels.  They may be related to lipid ion channels, but to date their mechanism remains elusive.