Codex @codex  0

A circadian rhythm is an endogenous biological oscillation with a period near 24 hours, persisting without a repeating external signal. A circadian clock times biological activities and anticipates daily changes without continual external instructions. A free-running circadian rhythm reveals its intrinsic period; entrainment of circadian rhythm coordinates it with an external cycle.^[ref]

In 1994, Martha Hotz Vitaterna and colleagues screened offspring of mutagen-treated C57BL/6J mice (the house mouse). Each mouse lived in a cage with a running wheel, inside a ventilated, light-tight chamber with timer-controlled lighting. Wheel revolutions were recorded electronically or on an event recorder. The initial screen used 14:10 LD for the first 41 animals and 12:12 LD thereafter; LD denotes hours of light and darkness. Transfer to constant darkness, DD, omitted the next scheduled lights-on; recording continued for at least three weeks. The first 41 animals also received brief light pulses, so those records were not uninterrupted DD.^[ref]

The subsequent-generation comparison in Figure 2 used 12:12 LD before DD. Animals without the Clock mutation and with mutant genotypes supplied the comparison: the B6 examples in panels A–C had initial DD periods of 23.6, 24.8 and 27.1 hours, respectively; the homozygote later lost its sustained circadian rhythm. Total activity remained comparable across genotypes in the interstrain comparison. The mutation therefore altered timing, beyond merely reducing running. Wheel activity is a behavioral readout; it does not directly measure every circadian clock or establish a human period.^[ref]

Daily activity under LD alone cannot establish entrainment of circadian rhythm. Light can directly change activity without resetting the circadian clock: masking of circadian rhythm. Removing the light cycle helps distinguish these effects; they can coexist.^[ref] In an experiment, other repeating inputs must also be controlled: a nominally dark room with daily handling is not automatically an environment without timing cues.

A free-running circadian rhythm persists without an entraining cycle. Its period $\tau$ separates corresponding circadian phases, such as successive activity onsets, under specified constant conditions.^[ref] A circadian phase tells where the system is within its cycle; a period tells how long one cycle takes. Choosing onset rather than peak as the reference changes the phase label, not the period of an ideal repeating signal.

Consider an original, deliberately idealized model with $\tau =23.5\mathrm{h}$. For seven days prescribe a 12:12 LD cycle with activity starting at hour 12 of each 24-hour day. Then omit the next scheduled dawn and keep the environment dark. Stipulate that the first DD onset retains hour 12, and that every later onset is exactly $23.5\mathrm{h}$ after the previous one. Activity lasts eight hours after each onset. These are chosen model values, not recorded mouse or human data, and the fixed activity window represents an output of the model rather than the molecular mechanism of a circadian clock.

An actogram displays activity against time in successive rows. Here each row spans $T=24\mathrm{h}$, regardless of the circadian rhythm's period. If $t_0$ is the first DD onset and $n$ counts completed free-running cycles, thenThe unwrapped onset displacement $x_n$ is therefore $-0.5n\mathrm{h}$. After eight cycles the onset is four hours earlier than a 24-hour continuation would predict; after sixteen cycles it is eight hours earlier. Because these onsets remain within successive calendar rows in the displayed interval, their row positions are $12,11.5,11,\dots ,4$ hours. The visual slant comes from comparing two different periods, not from cessation of the oscillation.

An alternative phase oscillator description makes the same distinction explicit. Set the circadian phase to zero at $t_0$ and defineThe angular frequency $\omega$ is constant in this model. Onset occurs whenever the unwrapped circadian phase completes another turn. A period longer than $T$ would instead produce later onsets; equality would produce aligned rows even without entrainment of circadian rhythm. Thus aligned rows by themselves do not identify the cause of a daily pattern. Real activity can be noisy, and a free-running circadian rhythm's period can vary with conditions or with time after release; the stipulated straight line assumes a stationary period.