Recent studies have shown how spiking networks can learn complex functionality through error-correcting plasticity, but the resulting structures and dynamics remain poorly studied. To elucidate how these models may link to observed dynamics in vivo and thus how they may ultimately explain cortical computation, we need a better understanding of their emerging patterns. We train a multi-layer spiking network, as a conceptual analog of the bottom-up visual hierarchy, for visual input classification using spike-time encoding. After learning, we observe the development of distinct spatio-temporal activity patterns. While input patterns are synchronous by construction, activity in early layers first spreads out over time, followed by re-convergence into sharp pulses as classes are gradually extracted. The emergence of synchronicity is accompanied by the formation of increasingly distinct pathways, reflecting the gradual semantization of input activity. We thus observe hierarchical networks learning spike latency codes to naturally acquire activity patterns characterized by synchronicity and separability, with pronounced excitatory pathways ascending through the layers. This provides a rigorous computational hypothesis for the experimentally observed synchronicity in the visual system as a natural consequence of deep learning in cortex.