International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 862813, 19 pages
Research Article

On the Sodium Concentration Diffusion with Three-Dimensional Extracellular Stimulation

Departamento de Matemática e CMAF, Faculdade de Ciências da Universidade de Lisboa, 1749-016 Lisboa, Portugal

Received 3 December 2010; Revised 1 March 2011; Accepted 20 May 2011

Academic Editor: Brigitte Forster-Heinlein

Copyright © 2011 Luisa Consiglieri and Ana Rute Domingos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We deal with the transmembrane sodium diffusion in a nerve. We study a mathematical model of a nerve fibre in response to an imposed extracellular stimulus. The presented model is constituted by a diffusion-drift vectorial equation in a bidomain, that is, two parabolic equations defined in each of the intra- and extra-regions. This system of partial differential equations can be understood as a reduced three-dimensional Poisson-Nernst-Planck model of the sodium concentration. The representation of the membrane includes a jump boundary condition describing the mechanisms involved in the excitation-contraction couple. Our first novelty comes from this general dynamical boundary condition. The second one is the three-dimensional behaviour of the extracellular stimulus. An analytical solution to the mathematical model is proposed depending on the morphology of the excitation.