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Art by Dalmathio Frau
Q: So what is the reticular formation up to during the hypnagogic phase of falling asleep, when people can perceive fleeting hallucinations in various sensory modalities?  Or are those more likely to be perceptual glosses on disorganized activity in the cortex?

I don't know what's going on, but that sounds plausible.  Maybe some kind of noisy activation of the locus coeruleus?

Q: So if I'm interpreting this correctly, the RF is where the brain would incorporate a pseudo-environment as a means of satisfying the still-functioning higher cortical regions that suggest we are still "awake"?

No, I think memory constructs of an imaginary or remembered environment would be in higher cortical regions.  The RF is more like a central switchboard that filters incoming and outgoing messages, and selects what areas of the brain can be active.  A dream, as an imagined scenario, would not be played out as a pattern of activity in the brain stem, but as a complex pattern in the cortex.

Back to the RF and its functions:

4) Integrating visceral reactions. The most commonly used example of this function is the vomiting reflex, which is managed by the RF.  Think back to the last time you felt really, really sick: all these different reactions were coordinated, from initial sweating, feeling of distress, and sudden intake of breath, to the violent contraction of muscles all up and down the gut and body.  All of it is quite neatly timed -- you wouldn't want to take that deep breath while your esophagus is busy in the process of reverse peristalsis.  Basically, if an event is going to involve the semi-conscious interaction of gut, respiration, heart, and motor activity, the RF is going to be the central switchboard.  (Those feelings that something is sitting on your chest?  Activating certain regions of the RF can do that.)

5) Arousal states. I mentioned before this idea that the RF has a spotlight function, selecting regions of the brain and telling them to increase or decrease their activity.  To picture how this works, imagine what things are like when you first get up in the morning.  You're sluggish, not thinking very clearly, and you just kind of trudge your way to the bathroom.  This is a result of the RF -- it has sort of "shut down" higher cortical centers while you are asleep, since you don't need them.  They are still operating at a lowered level when you first get up.

Then you splash cold water on your face.  This constitutes a rapid change in sensory input, which the RF detects, as it carefully monitors ascending sensory information from the trigeminal sensory nuclei.  The RF is very sensitive to transitions, and it immediately sends a general alert to the cortex, telling it to perk up and start paying attention.  It also prioritizes all the sensory information from your face, so you get a somewhat prolonged tingling sensation, even after you've toweled yourself off.

Let's finish this!
I'm feeling sick right now!!