We recently demonstrated that Mott memristors, two-terminal devices that exhibit threshold switching via an insulator to conductor phase transition, can serve as the active components necessary to build a neuristor, a biomimetic threshold spiking device. Here we extend those results to demonstrate, in simulation, neuristor-based circuits capable of performing general Boolean logic operations. We additionally show that these components can be used to construct a one-dimensional cellular automaton, rule 137, previously proven to be universal. This proof-of-principle shows that localized phase transitions can perform spiking computation, which is of particular interest for neuromorphic hardware.