A pulse-coupled neural network is shown to contain invariant spatial information in the phase structure of the output pulse trains. Two time scales are identified. On the fast time scale the linking produces dynamic, periodic, fringe-like traveling waves. The slow time scale is set by the pulse generator, and on that scale the image is segmented into multi-neuron time-synchronous groups. These groups, by the same linking mechanism, can form periodic pulse structures whose relative phases encode the location of the groups with respect to one another. The time signals are a unique, object-specific and roughly invariant time signature for their corresponding input spatial image or distribution. The details of the model are discussed, giving the basic linking field model, extensions, generation of time series in the limit of very weak linking, invariances from the symmetries of the receptive fields, time scales, waves, and signatures. Multi-rule logical systems are shown to exist on single neurons. Adaptation is discussed. Hardware implementations, optical and electronic, are reviewed. The conjugate basic problem of transforming a time signal into a spatial distribution is discussed.