Our brain contains many neurons. These neurons can be compared to capacitors that store an electrical charge. When the charge of one or two neurons is measured, it is almost impossible to notice. But, changes charge in many neurons are more noticeable and we can detect it.
For this using electrodes attached to the head. The science that studies how these charges change is called Electroencephalography (EEG).
Using this, we can observe how the charge of neurons behaves during sleep and observe the change in sleep phases visually.
To try and tie in the couple of sort of incomplete answers here: the main method of recording sleep is the electroencephalogram (EEG), as described by u/NonStandardCode. It was first used almost 100 years ago. The electrical activity of the brain is strong enough that electrodes on the scalp can pick up signal that is the sum of activity of large groups of neurons.
Researchers noticed using this method that during sleep, the signal went through distant patterns. As we fall through gradually deeper non-rapid eye movement (REM) sleep, we see large, ‘slow’ (low frequency) waves – this is thus called slow wave sleep. REM sleep, during which the eyes twitch rapidly (hence the name) but the muscles of the body are paralysed, is characterised in the EEG by activity at a certain higher frequency called theta (which is also present during some activity when you’re awake). By recording for a whole night, we can easily see the pattern of slow wave sleep and REM as they cycle.
Beyond that, we’ve learned a lot about what is happening in the brain during sleep through more sophisticated methods (looking more closely at certain regions in animals for example, which isn’t possible with human EEG). But the exact function of sleep is still not conclusively understood.
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