  A biophysical model of epimorphic regeneration based on a continuum
percolation process of fully penetrable disks in two dimensions is proposed.
All cells within a randomly chosen disk of the regenerating organism are
assumed to receive a signal in the form of a circular wave as a result of the
action/reconfiguration of neoblasts and neoblast-derived mesenchymal cells in
the blastema. These signals trigger the growth of the organism, whose cells
read, on a faster time scale, the electric polarization state responsible for
their differentiation and the resulting morphology. In the long time limit, the
process leads to a morphological attractor that depends on experimentally
accessible control parameters governing the blockage of cellular gap junctions
and, therefore, the connectivity of the multicellular ensemble. When this
connectivity is weakened, positional information is degraded leading to more
symmetrical structures. This general theory is applied to the specifics of
planaria regeneration. Computations and asymptotic analyses made with the model
show that it correctly describes a significant subset of the most prominent
experimental observations, notably anterior-posterior polarization (and its
loss) or the formation of four-headed planaria.
