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A number of investigations have implicated poising at critical points of the dynamics in brain function, and the maintenance of such critical modes has been experimentally associated to the awake state in primates and humans. A diagnostic of such critical modes is the presence of large numbers of close-to purely imaginary eigenvalues in the Hessian of the effective dynamics of the system. Here we show that when such couplings are in force, simple dynamically systems, such as chains of nonlinear oscillators or iterated maps of the interval, acquire complex spatiotemporal dynamics, including regimes in which glider-like coherent excitations move about and interact, “rewriting” local complex states, strongly reminiscent of universal computation CAs.  These models illustrate the interplay of two properties: circuits with a complex internal dynamics, such as multiple stable periodic solutions and period doubling bifurcations, and coupling with a “critical” synaptic matrix, i.e., having purely imaginary eigenvalues. It is now explicit that such critical couplings are volume-preserving in the sense of Liouville’s theorem.  Our results suggest that if the units in isolation are capable of featuring multiple dynamical states, then local critical couplings lead to a wide variety of emergent spatiotemporal phenomena.