  We study a complex-valued neural network (cv-NN) with linear, time-delayed
interactions. We report the cv-NN displays sophisticated spatiotemporal
dynamics, including partially synchronized ``chimera'' states. We then use
these spatiotemporal dynamics, in combination with a nonlinear readout, for
computation. The cv-NN can instantiate dynamics-based logic gates, encode
short-term memories, and mediate secure message passing through a combination
of interactions and time delays. The computations in this system can be fully
described in an exact, closed-form mathematical expression. Finally, using
direct intracellular recordings of neurons in slices from neocortex, we
demonstrate that computations in the cv-NN are decodable by living biological
neurons. These results demonstrate that complex-valued linear systems can
perform sophisticated computations, while also being exactly solvable. Taken
together, these results open future avenues for design of highly adaptable,
bio-hybrid computing systems that can interface seamlessly with other neural
networks.
