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Artificial neural networks (ANNs) struggle with continual learning, sacrificing performance on previously learned tasks to acquire new task knowledge. Here we propose a new approach allowing to mitigate catastrophic forgetting during continuous task learning. Typically a new task is trained until it reaches maximal performance, causing complete catastrophic forgetting of the previous tasks. In our new approach, termed Optimal Stopping (OS), network training on each new task continues only while the mean validation accuracy across all the tasks (current and previous) increases. The stopping criterion creates an explicit balance: lower performance on new tasks is accepted in exchange for preserving knowledge of previous tasks, resulting in higher overall network performance. The overall performance is further improved when OS is combined with Sleep Replay Consolidation (SRC), wherein the network converts to a Spiking Neural Network (SNN) and undergoes unsupervised learning modulated by Hebbian plasticity. During the SRC, the network spontaneously replays activation patterns from previous tasks, helping to maintain and restore prior task performance. This combined approach offers a promising avenue for enhancing the robustness and longevity of learned representations in continual learning models, achieving over twice the mean accuracy of baseline continuous learning while maintaining stable performance across tasks.