Patterns of conductivity in excitable automata with updatable intervals of excitations

Abstract

We define a cellular automaton where a resting cell excites if number of its excited neighbours belong to some specified interval and boundaries of the interval change depending on ratio of excited and refractory neighbours in the cell's neighbourhood. We calculate excitability of a cell as a number of possible neighbourhood configurations that excite the resting cell. We call cells with maximal values of excitability conductive. In exhaustive search of functions of excitation interval updates we select functions which lead to formation of connected configurations of conductive cells. The functions discovered are used to design conductive, wire-like, pathways in initially non-conductive arrays of cells. We demonstrate that by positioning seeds of growing conductive pathways it is possible to implement a wide range of routing operations, including reflection of wires, stopping wires, formation of conductive bridges and generation of new wires in the result of collision. The findings presented may be applied in designing conductive circuits in excitable non-linear media, reaction-diffusion chemical systems, neural tissue and assembles of conductive polymers.