The ability to form gigaseals is essential for patch-clamp electrophysiology; however, ion channels located in the seal can produce measureable currents. To explore the expected properties of channels in the seal (i.e., rim channels), we created a mathematical model. The seal was a two-dimensional cable filled with saline and bounded on one side by membrane (with resistance and capacitance) and on the other side by glass (nonconductive and noncapacitive). We included ion depletion/accumulation around the channels. The channels were ohmic with a conductance that increased with the concentration of permeant ions. The aqueous seal thickness was set nominally to 1 nm. Imaging with fluorescent dyes in the pipette showed that the hydrophilic dye Alexa 488 is impermeant, but lipophilic FM1-43 labels the seal. The model showed that to obtain high-resistance seals, the conductivity of the seal media has to be <10% that of the bath. Stimulus voltages decreased with distance down the seal. In agreement with results in the literature, channels in the seal can produce currents similar to those in the pipette-spanning dome. The transition times of currents are slower due to membrane capacitance. If channel densities are uniform, patch currents are dominated by channels in the dome.