Membrane current from transmembrane potentials in complex core-conductor models

Core-conductor models, used to integrate the behavior of the longitudinal currents with the distributed voltages of electrically active tissue, have evolved for over a century. A critical step in the use of such models is the computation of membrane current from the set of distributed transmembrane potential values that exist at a given moment, where the potentials are obtained either experimentally or computationally. Over time, interest has developed in a number of substantial extensions of the original model to include such features as nonuniform spatial resistances, loop instead of linear structure, and multiple sites of extracellular stimulation. This paper concisely restates and extends the equations for calculation of transmembrane currents with the systematic inclusion of alternative cases, noting how they reduce to the standard forms. An important issue is how complex the calculation of membrane current has to be. Thus, the paper goes on to show criteria (based on the uniformity of resistance and the presence of stimulation) for deciding when membrane currents can be obtained with a relatively simple calculation with a single equation involving local variables versus a more complex calculation involving the simultaneous solution of a (possibly large) set of equations.