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Line attractors in neural networks have been suggested to be the basis of many brain functions, such as working memory, oculomotor control, head direction, locomotion, and sensory processing. I will discuss how, by incorporating pulse gating into feedforward neural networks, graded information may be propagated. This propagation can be viewed as a line attractor in the firing rate of transiently synchronous populations. I will show how pulse-gated graded information transfer persists in spiking neural networks and is robust to intrinsic and extrinsic noise. Then, using a Fokker-Planck approach, I will show that the gradedness of rate amplitude information transmission in pulse-gated networks is associated with the existence of a cusp catastrophe, and that the slow (ghost) dynamics near the fold of the cusp underlies the robustness of the line attractor. Understanding the dynamical aspects of this cusp catastrophe allows us to show how line attractors can persist in biologically realistic neuronal networks and how the interplay of pulse gating and synaptic coupling can be used to enable attracting one-dimensional manifolds and thus, dynamically control graded information processing. Finally, I will demonstrate how pulse-gating in combination with Hebbian learning can be used to construct predictive neural circuits and how such circuits generate oscillations with a characteristic frequency spectrum.