Spike-timing-dependent plasticity (STDP) provides a biologically-plausible learning mechanism for spiking neural networks (SNNs); however, Hebbian weight updates in architectures with recurrent connections suffer from pathological weight dynamics: unbounded growth, catastrophic forgetting, and loss of representational diversity. We propose a neuromorphic regularization scheme inspired by the synaptic homeostasis hypothesis: periodic offline phases during which external inputs are suppressed, synaptic weights undergo stochastic decay toward a homeostatic baseline, and spontaneous activity enables memory consolidation. We demonstrate that this sleep-wake cycle prevents weight saturation while preserving learned structure. Empirically, we find that low to intermediate sleep durations (10-20\% of training) improve stability on MNIST-like benchmarks in our STDP-SNN model, without any data-specific hyperparameter tuning. In contrast, the same sleep intervention yields no measurable benefit for the surrogate-gradient spiking neural network (SG-SNN). Taken together, these results suggest that periodic, sleep-based renormalization may represent a fundamental mechanism for stabilizing local Hebbian learning in neuromorphic systems, while also indicating that special care is required when integrating such protocols with existing gradient-based optimization methods.