We propose a pulse-coupled neural network model in which one-dimensional excitable maps connected in a time-delayed network serve as the neural processing units. Although the individual processing unit has simple dynamical properties, the network exhibits collective chaos in the active states. Introducing a Hebbian learning algorithm for synaptic connections enhances the synchronization of excitation timing of the units within a subpopulation. The synchronizing clusters approximately exhibit a power-law size distribution, suggesting a hierarchy of synchronization. After applying a stationary signal to a subpopulation of the units with learning, the network then reproduces the signal. The learnable time range is much longer than the inherent time scale of the processing units, i.e., the synaptic delay time. Also, the network can reproduce periodic signals with time resolution finer than the delay time. Our present network model can be considered as a temporal association device which operates in chaotic states.