Signal propagation in 2-dimensional threshold cellular space

A two-dimensional network of uniformly connected McCulloch-Pitts neurons is considered and signal propagations in the network are analyzed. The problems are set up in the framework of cellular space such that each cell is a copy of any given McCulloch-Pitts neuron and is connected to the nearest neighboring cells. It is assumed that the threshold value is positive and that there exists only one firing cell at the beginning. Then it is shown that essentially there are only four signal propagation patterns and a firing pattern at any time t can be obtained by such a superposition of the propagation patterns that includes the newly defined concept of dominance and assimilation. The exact formulae representing firing patterns at any time t are obtained for any finite rectangle cell space with constant 0 (i.e., non-firing) boundary condition and for the entire two-dimensional cellular space.