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We study a complex-valued neural network (cv-NN) with linear, time-delayedinteractions. We report the cv-NN displays sophisticated spatiotemporaldynamics, including partially synchronized ``chimera'' states. We then usethese spatiotemporal dynamics, in combination with a nonlinear readout, forcomputation. The cv-NN can instantiate dynamics-based logic gates, encodeshort-term memories, and mediate secure message passing through a combinationof interactions and time delays. The computations in this system can be fullydescribed in an exact, closed-form mathematical expression. Finally, usingdirect intracellular recordings of neurons in slices from neocortex, wedemonstrate that computations in the cv-NN are decodable by living biologicalneurons. These results demonstrate that complex-valued linear systems canperform sophisticated computations, while also being exactly solvable. Takentogether, these results open future avenues for design of highly adaptable,bio-hybrid computing systems that can interface seamlessly with other neuralnetworks.

An exact mathematical description of computation with transient spatiotemporal dynamics in a complex-valued neural network