The Closed Forn Solution to the Periodic Core-Conductor Model Using Asymptotic Analysis

The distribution of the transmembrane potential along an infinite strand of cardiac cells generated by a point source under steady-state conditions has been calculated using the asymptotic analysis method. With the intracellular conductivity changing periodically in space, the problem can be treated as dependent upon two variables: the large scale variable x covering the whole strand, and the small scale variable y defined on the unit cell. The solution is given as a two-scale expansion in powers of the period length. Each term of the expansion can be determined by solving the differential equations derived by decomposing the original problem. These equations do not have to be solved simultaneously; moreover, the linearity of the problem allows the separation of the x and y dependence in the higher order terms. The series converges quickly, and for all practical purposes, the solution containing zero-, first-, and second-order terms has a negligible truncation error. The subsequent terms of the solution have the following physiological interpretation: The zero-order term is the solution to the classical core-conductor model obtained by the homogenization of the periodic model, the first-order term acts as the dipole sources located at junctions, and finally, the second-order term resembles the monopole sources arising at junctions.