This is the sixth post in a series about mechanism of entrainment, running all day today on this blog. In order to understand the content of this post, you need to read the previous five installments. The original of this post was first written on April 12, 2005.

A Phase Response Curve (PRC) can be made in three ways:

One can construct a PRC for a single individual. If you have a reasonably long-lived organism, you can apply a number of light pulses over a period of time. The advantage is that you will always know the freerunning period of your organism, and you will know with absolute certainty that the conclusions you draw from the PRC will apply to that individual - information that may be important if the goal is to ensure the maximal health (or reproductive state) of that individual in captivity. The disadvantage is that you have a sample size of one (N=1), so you may need to repeat the experiment with a few more individuals in order to do statistics. Unless, of course, your organism is a blue whale, or platypus, or something endangered, so even data on a single organism are more valuable than none.

Usually one works with a group of animals, each kept isolated from the others and each being exposed to one or more pulses. The resulting PRC is a combined PRC of the individuals in the group. N is greater than one, but the interpretation of results may not apply to each inidividual equally - it is an averaged curve and outliers may behave differently than the group mean suggests. Still, the freerunning period (&tau - tau) for each individual is known, so additional analysis for each individual is possible.

Finally, one can construct a group (or population) PRC. For instance, eclosion (hatching out of eggs) is a single developmental event in the life of each individual insect (as well as in some birds, like chickens), yet it is timed by the circadian clock, and the rhythm can be seen if one observes a population. A bunch of eggs ecloses almost simultaneously (e.g., at dawn, or subjective dawn), another bunch about 24 hours later, another bunch another 24 hours later and so on. While individual &tau is not known, the population &tau is known and can be used for analysis.

It is important to keep in mind, during analysis, if the PRC is individual, combined or population PRC.

We can use the Phase Response Curve to determine if a particular animal will be able to entrain to a particular LD cycle. First, we need to determine what kind of daily shift is needed to entrain the animal. If an individual's freerunning period (&tau) is 23 hours, and the period of the entraining cycle (T) is 24 hours, this means that this animal requires a daily phase-shift (&Delta &Phi - delta Phi) to be a delay of 1 hour. The simple formula is &tau - T = &Delta &Phi .

Now, we can see at what phase of the circadian rhythm must the onset of light fall in order for stable entrainment to occur. We look at out PRC to find where (on X-axis) can we find a delay (on Y-axis) of 1 hour. We see that this happens at two places: on the negative slope portion in the early subjective night and a couple of hours later on the positive slope of the curve. This is a place to remember: **Stable entrainment cannot occur if the light onset falls on the positive-slope portion of the curve**.

So, we pick the phase at which the 1-hour delay is on the negative slope of the curve. That is the phase at which light has to come on every day for stable entrainment to occur. If errors occur and shifts the cycle a little bit in either direction (advance or delay) the nature of the curve ensures that, via a feedback mechanism, the cycle falls back to the correct phase. So, if an error phase-advances the cycle by 10 minutes, the light (next day) will fall on the portion of the curve that delays the rhythm by 1 hour + 10 minutes (70 minutes), bringing the rhythm back to the most stable phase. If an error phase-delays the rhythm by 10 minutes, the light (next day) will fall on the portion of the curve that delays the rhythm by 1 hour - 10 minutes (50 minutes), again bringing the rhythm back to the most stable phase.

If the light falls on the positive-slope portion of the curve, the effect is feed-forward: an error resulting in a slight advance leads to a greater advance which leads to greater advance etc. Eventually the animal will hit the correct phase again and stably entrain, but it may take weeks to get there.

Occasionaly one sees a PRC in which a **portion of the negative slope is very steep** - greater than 2. Stable entrainment is **not possible** on this portion either, as very small errors in the phase cause very large shifts of the curve to the left or to the right. The correction (in the next cycle) is then much bigger than the original error leading to unstable shifting back and forth until the light finally hits the correct phase.

Of course, there is another portion of the curve on which **entrainment is impossible - the dead zone**. Since light has no effect, the rhythm will keep freerunning (ignoring the light cycle) until it encroaches into a phase-delaying or phase-advancing portion of the curve.

**Determining the Phase Angle**

Let's assume that we are dealing with a nocturnal animal that starts its daily acitivity at CT12. Let's also assume that its freerunning period is 23 hours. From the formula we calculate that it needs a daily delay of 1 hour. From the PRC we find the phase of the rhythm at which 1-hour delay is on a negative slope, e.g., at CT14. From this (14-12=+2h) we calculate that this animal has a positive phase angle of 2 hours. This means that, when stably entrained, it will always start its activity two (circadian) hours before the onset of light-pulse.

**Determining the Range of Entrainment**

Using the above formula and the PRC one can calculate the lower and upper limits on entrainment of that animal to that type (duration, intensity, quality) of light pulse. Let's assume that the animals' mean &tau is 23 hours. From the PRC we see that the greatest phase-delay is 3 hours and the greatest phase-advance is 2 hours. Let's plug in the numbers:

&tau - T = &Delta &Phi

23 - T = -3; Tmax = 26h

23 - T = +2; Tmin = 21h.

Thus, this animal cannot entrain to cycles with a period shorter than 21 hours, or longer than 26 hours. If the duration of the pulses used for the construction of the PRC was 3 hours, the lower limit of entrainment is LD3:18 and the upper limit of entrainment is LD3:23. Outside of this range, one is likely to observe relative coordination, scalloping, phase-jumping, or freerunning, but no stable entrainment. And, the coolest thing of all, one can predict from the PRC using very simple math, exactly how the animal will behave in such cycles outside of its range.

**Entrainment by Skeleton Photoperiods**

Likewise, one can predict the behavior of the animal if exposed to the skeleton LDLD cycles. The first pulse acts as "dawn" and the other one as "dusk". The shift produced by the first pulse determines the phase on which the second pulse will land and the resulting shift. If the sum of the two effects is equal to the daily shifting requirement for stable entrainment, the animal will entrain to the skeleton photocycle. In other words, &tau - T = &Delta &Phi 1 + &Delta &Phi 2.

Organisms tend to interpret skeleton photoperiods as shorter of the two possiblities. For instance LDLD 0.25:13.5:0.25:10 can be interpreted as a full photoperiod of LD14:10, or as LD 10.5:13.5. The latter one is shorter, thus prefered by the animal. A phase-jump is likely to occur. However, in some cases, a skeleton photoperiod will be interpreted depending on the phase of the rhythm at which the first light hits at the beginning of the experiment. If it illuminates a phase close to CT0 it will be interpreted as the "dawn" pulse, and if it falls around CT12 as the "dusk" pulse. This ability to entrain both ways to a mid-length (i.e., not too long) skeleton photoperiod is called **bistability phenomenon**.

I will next plunge into the posts about the use of understanding of entrainment in the study of photoperiodic time measurement (measuring seasonally changing daylengths), but PRC is not done yet. What I have presented so far is the most basic stuff taught in intro college courses. There is a lot more arcane stuff to discuss, but I will have to come back to it later.

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This April 16, 2005 post gives you links to further online resources and literature on entrainment and Phase-Response Curves, as well as a link to a database of PRCs so you can play with them yourself.

One of the most useful chronobiological databases available online is the PRC Atlas. Compiled by Dr.Carl Johnson of Vanderbilt University, it contains hundreds of published and unpublished Phase-Response Curves. One can sort the Curves by species or by type of stimulus (e.g., light pulses, pulses of varius chemicals, dark pulses on constant-light background, etc.) and one is also able to manipulate (i.e., re-plot) the data to one's own liking.

The page contains links to four important papers/reviews of the utlility of PRC-construction in studies of circadian rhythms as well as a list of further references. The files are available for PC and Mac and one can use them even on very old operating systems as the files were prepared more than a decade ago.

There is also a useful little page of comments on the way PRCs are plotted in the Atlas and are usually plotted (or should be plotted) in the literature.

Unfortunately, the database has not been updated for at least the past 6-7 years, so some more recent PRCs are not included. Still, if one is interested in performing a meta-analysis (e.g., correlating particular circadian properties with ecological niches or phylogenetic histories - I wish someone would actually do this), the data are freely available on this website.

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