Edward Vigmond <<a href=\"mailto:edward.vigmond@u-bordeaux.fr\"\
edward.vigmond@u-bordeaux.fr>
\n\n
Periodic boundary conditions\
\ connect the left edge of a sheet to the right, or the top to the bottom. This,\
\ for example models a cylinder very simply while allowing visualization of the\
\ whole domain. The intact atria or ventricles are better approximated by sheets\
\ with periodic boundary conditions and not the simpler no flux boundaries. Indeed,\
\ reentries last longer on periodic domains since they do not annilhilmate at boundary\
\ edges.
\n
The implementation of periodic b.c.'s is not straightforward since\
\ adding the equations equating nodes on the left boundary to those on the right\
\ boundary (or top and bottom) is mathematically trivial but destroys the symmetry\
\ of the system, and, thereby, invalidates many conjugate gradient preconditioners.\
\ A workaround is to define line connections linking the edges of the tissue, and\
\ assigning ridiculously large conductivity values to these links, effectively short\
\ circuiting the nodes. The mesher program will generate these links.
\n<h2 id=\"\
experimental-setup\">Experimental Setup\n<div id=\"fig-period-bcs\">\n\n\
<img src=\"07b_periodic/perCnnx.png\" class=\"align-center\" style=\"width:30.0%\"\
\ alt=\"\" />A sheet showing periodic boundary conditions in X implemented\
\ with linear elements (in pink) joining the left and right edges.\n\
\n\n\n<img src=\"07b_periodic/perbcs.gif\" class=\"align-center\"\
\ style=\"width:30.0%\" alt=\"\" />A point stimulus delivered off-center\
\ on a piece of tissue with periodic boundary conditions in both directions.\n\
\n<h2 id=\"initiating-reentry\">Initiating Reentry\n
The standard\
\ cross-shock method of reentry initiation will fail with periodic b.c.'s. The initial\
\ line stimulus will produce two wavefronts which collide, and the quarter sheet\
\ S2 will quickly go everywhere and annihilate itself. There are two simple ways\
\ to induce reentry.
\n<ol type=\"1\">\n
We define a mesh without the periodic\
\ connections and run an intial portion of the silulation on that. Just after the\
\ S2, we save the state and restart on the mesh with the periodic b.c.'s enforced.
\n\
We temporarily block propagation across an edge by clamping the transmembrane\
\ voltage and then turning off the clamp after the S2.