Circadian oscillation provides selection advantages through synchronizationto the daylight cycle. However, a reliable clock must be designed through twoconflicting properties: entrainability to synchronize internal time withperiodic stimuli such as sunlight, and regularity to oscillate with a preciseperiod. These two aspects do not easily coexist because better entrainabilityfavors higher sensitivity, which may sacrifice the regularity. To investigateconditions for satisfying the two properties, we analytically calculated theoptimal phase-response curve with a variational method. Our result indicates anexistence of a dead zone, i.e., a time period during which input stimulineither advance nor delay the clock. A dead zone appears only when inputstimuli obey the time course of actual solar radiation but a simple sine curvecannot yield a dead zone. Our calculation demonstrates that every circadianclock with a dead zone is optimally adapted to the daylight cycle.
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