We describe a planar MEMS silicon structure to record ion channel currents in biological cells. The conventional method of performing an electrophysiological experiment, `patch-clamping', employs a glass micropipette. The micropipette tip is a source of thermal noise because of its inherent, tapered, conical structure, giving rise to a large pipette resistance. This pipette resistance, when coupled with the biological cell capacitance, limits the available bandwidth of single ion channel recording. In this work, we propose a current transport model to characterize the series resistance and capacitance of a planar pipette fabricated on a silicon BioChip. Our model provides a deeper insight into how currents injected into a micropore are quantitatively partitioned into the individual ion transports, and goes beyond just describing the solute and solvent kinetics inside pores of microscale dimensions. The device topology and fabrication sequence of the planar patch-clamp setup are also discussed. The theoretical predictions by the model are in close agreement with the experimental results.