Neurons, although one dimensional, are observed to follow the same three stages of activity. Basically, the neuron is inactive, but a stimulus can trigger a depolarization wave along the long tubular cell membrane. Slowly acting chemical processes restore the electrical potential across the cell wall, ready for another discharge.
Contact between neurons occurs at the far ends of the tubes - the synapses, various conditions governing whether a polarization wave crosses the bridge. The structure is neither as regular nor as local as a cellular automaton.