ELECTRIC-POTENTIAL IN 3-DIMENSIONAL ELECTRICALLY SYNCYTIAL TISSUES

The problem of calculating the potential induced in an electrical syncytium by a point source of current is studied. The interiors of the many interconnected cells are treated as one continuum with resistivity ρ i . The interdigitated extracellular space is treated as a second continuum of resistivity ρ e, occupying the same overall volume, coupled to the first via the resistance R m and capacitance C m of the cell membranes. The intra- and extracellular potentials are then solutions to a pair of coupled partial differential equations. The equations are uncoupled, yielding a cable equation for the transmembrane potential and a Poisson equation for a second auxiliary potential. For an unbounded syncytium the potential for a step function source is obtained in terms of error functions. For a spherical syncytium of radius a, bounded by a membrane with surface resistance R a, and capacitance C a, expansions are obtained in spherical harmonics and spherical Bessel functions. For e{open}=ρ ia/R a and β=ρ i /ρ e small, an asymptotic expansion of the potential is developed. The results are compared to earlier results for a spherical cell as well as to microelectrode measurements of the lens of the eye. © 1979 Society for Mathematical Biology.