Potential induced in a spherical cell by an intracellular point source and an extracellular point sink

Summary The potential is calculated for all time, inside and outside a spherical cell for a point source of current inside the cell and a point sink located a finite distance outside the cell. The source and sink are step functions in time. An eigenfunction expansion is obtained, valid for arbitrary =_m a/_i , where _i and _m are the conductivities inside the cell and in the membrane, respectively, a is the cell radius and the membrane thickness. For small , the eigenfunction expansion is expanded in powers of . The time dependence of the potential contains transients with two widely differing time constants =C_m a/_i, where C_m is the membrane surface capacitance, and _m=/. Closed-form expressions are obtained for the two leading terms, for small , after the rapid transient is over. The remaining time dependence is only in the potential inside the cell, and is a simple exponential increase, independent of position within the cell. It is found that the transmembrane potential is insensitive to the location of the extracellular sink at long times, but not at short times. The dependence of the potential on location of source, sink, and observer is studied for long times after the quick transients are over. A uniqueness theorem is derived for the solution to Laplace's equation for the membrane boundary condition.This work was supported in part by NSF Grant No. GB-24965. Dr. Peskoff is the recipient of NIH Special Research Fellowship No. 1F03 GM 55849-01. Mr. Ramirez is a Ford Foundation Pre-doctoral Fellow.