{title2=$\tau$}

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.<Overview of Circadian Rhythms by Martha Hotz Vitaterna and others>[pp. 86–87, Figure 1A; p. 92, Free running]{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>.

\Image[/circadian-free-running-actogram.png]
{title=An idealized <circadian rhythm> under 12:12 LD followed by constant darkness}
{id=image-circadian-free-running-actogram}
{description=Original stipulated <actogram>: each row is a fixed 24-hour calendar day, with successive days running downwards. Yellow denotes light, pale gray denotes darkness, and teal denotes the chosen activity window. The DD <period> is $23.5\,\mathrm h$, so successive activity onsets move earlier on the plotting grid. No biological measurements are plotted.}

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, then
$$
t_n=t_0+n\tau,\qquad t_{n+1}-t_n=\tau,\qquad x_n=t_n-(t_0+nT)=n(\tau-T).
$$
The 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 $12,11.5,11,\ldots,4$ 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 define
$$
\theta(t)=\frac{2\pi(t-t_0)}{\tau}\pmod{2\pi},\qquad \omega=\frac{2\pi}{\tau}.
$$
The <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>.


 Free-running circadian rhythm (source code)