Hierarchical synchronization of sleep oscillations establishes communication pathways to support memory reactivation, transfer, and consolidation. From an information-theoretical perspective, oscillations constitute highly structured network states that provide limited information-coding capacity. Recent findings indicate that sleep oscillations occur in transient bursts that are interleaved with aperiodic network states, which were previously considered to be random noise. We argue that aperiodic activity exhibits unique and variable spatiotemporal patterns, providing an ideal information-rich neurophysiological substrate for imprinting new mnemonic patterns onto existing circuits. We discuss novel avenues in conceptualizing and quantifying aperiodic network states during sleep to further understand their relevance and interplay with sleep oscillations in support of memory consolidation.