Paying attention improves performance, but attention is metabolically costly, so how should a resource-efficient agent allocate it? We study optimal allocation strategies using a normative model of a signal detection task in which attention comes at a cost. The model reveals that optimal attention is temporally structured in one of two patterns depending on task conditions: when attention costs penalize intense focus, optimal attention ramps up as evidence for the signal accumulates, but with less prohibitive attention costs, optimal attention fluctuates rhythmically. In cases with rhythmic attention, allocation frequency increases with task parameters such as reward magnitude for successful detection, signal brevity, and signal frequency. We argue that rhythmic attention emerges naturally in an agent whose belief updates are dominated by a stable temporal prior and not by the likelihood of new sensory observations. Our results characterize the conditions under which rhythmic vs. ramping attention is optimal and offer a normative account of attentional fluctuations observed in sustained attention tasks.

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