During rem sleep there are observed eye movements, which seem like they would correlate with the person visualizing their dreams.
Do their eye movements actually try to follow what they're dreaming about? If so, do the eyes ever actually focus on something?
When a person closes their eyes and are thinking, are the eyes actually being used. By this I mean are the muscles moving in accordance with focusing. How is the focal point determined?
In regards to actively trying to control your eyes while they are closed, like attempting to do visualization like NLP, I believe that you choose what to focus on and where to focus at, exactly the same as my "pitch black" analogy whereby if you look into, say, a pitch black room, then although you know it's only 15 feet towards the wall, your vision does not.
Therefore, if you believe that it extends for millions of miles, then you can choose quite easily to focus into the darkness that way.
In regards to if you close your eyes and just kind of forget about them without going to sleep...like "resting your eyes", then I believe Chuck in another answer answered this best in his bottom two paragraphs. Quick link: https://cogsci.stackexchange.com/a/4467/3433
Now in regards to during REM sleep, this gets tricky.
I did not think I'd find a study that answers your exact question about "how is focal point determined during REM sleep?". I'll just start with that study first, then throw in a few others that are related to something called the "scanning hypothesis" which will help to unravel some of your other questions more specifically if they're about REM sleep specifically.
Let's start off with a definition of the "scanning hypothesis":
According to psychologydictionary.com,
The scanning hypothesis is a concept hypothesizing that rapid eye movements identified in the course of the REM stage of sleep correlate to subjective gaze adjustments of the dreamer casting their gaze about in the dream with fixations in particular places.
At first I was worried that all the studies would just focus on this "scanning hypothesis," then I came across this gem:
Exp Brain Res. 1997 Oct;117(1):153-60. Binocular eye movements not coordinated during REM sleep. Zhou W, King WM.
Department of Neurology, University of Mississippi Medical Center, Jackson 39216, USA. wuz@vor.umsmed.edu
Abstract
Rapid eye movements (REMs) are a defining characteristic of REM sleep during which vivid dreams occur. It has been suggested that REMs may be binocularly coordinated and related to "watching" dream images. For the first time, binocular eye movements were recorded during natural REM sleep in monkeys to test the conjugate nature of the oculomotor system and the "scanning hypothesis" of REMs during sleep. During REM sleep, the lines of sight of the two eyes are frequently misaligned up to 30 degrees horizontally and/or vertically. Since the lines of sight usually don't intersect, there is no fixation point. Instead, each eye is aimed at a different part of the visual field during REM sleep. Furthermore, REMs are not usually conjugate, but are disjunctive or even monocular in horizontal or vertical directions. These data argue against the idea that REMs actually "track" dream images, unless each eye is watching its own dream! Binocular misalignment and disjunctive (even monocular) REMs during sleep suggest that separate left eye and right eye pathways generate saccades in each eye and control the position of each eye. Binocular coordination cannot be the passive result of anatomical connectivity as has been argued previously, but instead must result from a high-level process associated with the awake state that coordinates activity in left-eye and right-eye pathways. Hering's law of equal innervation is not consistent with these data. PMID: 9386014 [PubMed - indexed for MEDLINE]
Apparently also, part of the "scanning hypothesis" is that your eyes are trying to track down something...in other words fixate upon it...which would cause them to focus. This results from this study say no to that.
So, as far as...
"By this I mean are the muscles moving in accordance with focusing. How is the focal point determined?"
...during REM sleep, this study says that there is no fixation upon one thing and thus there is no focal point. So, no, according to this study the muscles are not moving in accordance with focusing.
As a precursor to this study, a definition of "rapid eye movement disorder" is good to know. So here is wikipedia's definition:
Rapid eye movement disorder is a sleep disorder (more specifically a parasomnia) that involves abnormal behaviour during the sleep phase with rapid eye movement (REM) sleep. It was first described in 1986. The major and arguably only abnormal feature of RBD is loss of muscle atonia (paralysis) during otherwise intact REM sleep. This is the stage of sleep in which most vivid dreaming occurs. The loss of motor inhibition leads to a wide spectrum of behavioural release during sleep. This extends from simple limb twitches to more complex integrated movement, in which sufferers appear to be unconsciously acting out their dreams. These behaviours can be violent in nature and in some cases will result in injury to either the patient or their bed partner.
It's important to note the difference between the first study which was done in 1997 and this one done in 2010. Perhaps the overall definition of the "scanning hypothesis" changed, or there is slight subjectivity to it.
For instance, in the first study the author says,
"It has been suggested that REMs may be binocularly coordinated and related to "watching" dream images. For the first time, binocular eye movements were recorded during natural REM sleep in monkeys to test the conjugate nature of the oculomotor system and the "scanning hypothesis" of REMs during sleep."
So he's trying to test the "scanning hypothesis" by looking to see if the monkeys actually do focus on something by looking to see if the eyes are working together (ie. trying to triangulate distance and thus focus) are actually working together.
Now these French people (they always seem to have creative ideas actually) admit that the "scanning hypothesis" has been testing in this way...as in...by monitoring the muscles in the eyes. So they're taking a different approach and (cleverly might I add) exploiting people who essentially have a disorder that is very similar to sleepwalking...but they lay in bed...THEN they are mixing in control patients who are normal so that they can compare the two and use that mixture to extrapolate it to the normal population as well & noone can criticize them for only using patients with that 1st specific disorder.
They're looking to see how their eyes align/what direction they move to when they are acting out certain motions.
Brain. 2010 Jun;133(Pt 6):1737-46. doi: 10.1093/brain/awq110. Epub 2010 May 16. Do the eyes scan dream images during rapid eye movement sleep? Evidence from the rapid eye movement sleep behaviour disorder model. Leclair-Visonneau L, Oudiette D, Gaymard B, Leu-Semenescu S, Arnulf I. Source Sleep Disorder Unit, Pitié-Salpêtrière Hospital, Assistance Publique - Hôpitaux de Paris, 75013 Paris, France.
Erratum in Brain. 2011 Jul;134(Pt 7):1286.
Abstract
Rapid eye movements and complex visual dreams are salient features of human rapid eye movement sleep. However, it remains to be elucidated whether the eyes scan dream images, despite studies that have retrospectively compared the direction of rapid eye movements to the dream recall recorded after having awakened the sleeper. We used the model of rapid eye movement sleep behaviour disorder (when patients enact their dreams by persistence of muscle tone) to determine directly whether the eyes move in the same directions as the head and limbs. In 56 patients with rapid eye movement sleep behaviour disorder and 17 healthy matched controls, the eye movements were monitored by electrooculography in four (right, left, up and down) directions, calibrated with a target and synchronized with video and sleep monitoring. The rapid eye movement sleep behaviour disorder-associated behaviours occurred 2.1 times more frequently during rapid eye movement sleep with than without rapid eye movements, and more often during or after rapid eye movements than before. Rapid eye movement density, index and complexity were similar in patients with rapid eye movement sleep behaviour disorder and controls. When rapid eye movements accompanied goal-oriented motor behaviour during rapid eye movement sleep behaviour disorder (e.g. grabbing a fictive object, hand greetings, climbing a ladder), which happened in 19 sequences, 82% were directed towards the action of the patient (same plane and direction) (Taal Note: Note what they just said in parenthesis). When restricted to the determinant rapid eye movements, the concordance increased to 90%. Rapid eye movements were absent in 38-42% of behaviours. This directional coherence between limbs, head and eye movements during rapid eye movement sleep behaviour disorder suggests that, when present, rapid eye movements imitate the scanning of the dream scene. Since the rapid eye movements are similar in subjects with and without rapid eye movement sleep behaviour disorder, this concordance can be extended to normal rapid eye movement sleep.
So, 1) Although behavioral movements and what the eyes are doing do not always seem to correlate:
"Rapid eye movements were absent in 38-42% of behaviours."
When they do correlate at least during goal-oriented motor behavior, they do "imitate the 'scanning' of the dream scene." And that seems to happen about 82-90% of the time when they do this sort of behavior.
The math confuses me there as well though in contrast to how 38-42% of movements were absent of eye moments as well, but I believe the discretion is due to using the control vs. disordered group and lumping different group(s) into different calculated percentages.
Now also note above that when the goal-oriented behavior occurred and the eyes reflected this, they did so in the "same plane and direction." This was peculiar as the word "plane" could mean horizontal or vertical plane I believe...it could also mean focal plane. I wish they distinguished that better - I'm going to hypothesize it was just either the vertical or horizontal plane so that
In essence, a large part of the time people are acting out their behaviors when asleep the eyes correspond highly to the direction, but only to one plane, which means there wasn't really focal point - as you'd need to correspond to both the horizontal and vertical planes then.
It has been theorized that it has to do with "visualizing" dreams, but the movements themselves are by virtue of the pattern of electrical activity as the waves travel between the Pons (in the brainstem), Geniculate nuclei (in the thalamus), and Occipital lobe.
From PGO Waves
PGO waves and REM sleep
PGO waves are an integral part of rapid eye movement (REM) sleep. As stated earlier, the density of the PGO waves coincides with the amount of eye movement measured in REM sleep. This has led some researchers to further theorize about the usefulness of PGO waves for dreaming.
One key use of REM sleep is for the brain to process and store information from the previous day. In a sense, the brain is learning by establishing new neuronal connections for things that have been learned. Neurophysiological studies have indicated a relationship between increased P-wave density during post-training REM sleep and learning performance. Basically, the abundance of PGO waves translates into longer periods of REM sleep, which thereby allows the brain to have longer periods where neuronal connections are formed.
The importance of PGO waves during REM sleep also aids the idea of PGO waves as a signal that a person is dreaming. Since dreaming occurs during REM sleep, the PGO waves are theorized to be the signals that make the brain start to recount the experiences from the previous day. This, in turn, allows us to "see" our dreams since our visual sense is quickly going through the information it has stored.
In other words, the eyes are moving by virtue of the brain activity, not the other way around.
So, the fact that we are "seeing" is only in the abstract sense, and there's not really a specific point of focus that the eyes are following.
As far as the second point goes, you are speaking of the concept of "vergence", where the eyes are moving to attend to a stimulus. It turns out that in the dark, according to this lecture, the eyes are reflexively brought parallel:
Simply being conscious, even without a visual stimulus (i.e., in the dark, or with the eye closed), will cause innervation to the muscles that brings the eyes to a nearly parallel (or slightly eso) orientation. This is known as tonic vergence. It is also called dark vergence
