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I realized that research describing the different stages of sleep all rely on the assumption that we can accurately determine the stage of sleep using an EEG. I wondered if brainwaves are anything more than arbitrary patterns that appear within EEG data and it's due to our tendency to look for patterns in data rather than these being intrinsic phenomena that actually "mean something". For example like inferring personality from facial structure (which while potentially possible doesn't hold value as personality isn't determined by facial structure).

Can we even say with certainty that extrapolations made from these measurements actually determines the presence of real phenomena? For example: That someone is truly asleep rather than their neurons electrical firings matches a pattern that we associate with sleep. As our only way of determining that is using this device?

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Yes, EEG measurements are meaningful. The "slow" oscillations of sleep, for example, are tied to "UP" and "DOWN" states in neocortical neurons, the UP state being a period of coordinated activity, and the DOWN state being a period of coordinated inactivity. These have been measured in single neurons and groups of neurons simultaneously with EEG. For example from Steriade et al 1993:

Slow oscillation

From Steriade, M., Nunez, A., & Amzica, F. (1993). A novel slow (< 1 Hz) oscillation of neocortical neurons in vivo: depolarizing and hyperpolarizing components. Journal of neuroscience, 13(8), 3252-3265.

The top is a single-cell recording, the bottom is EEG.

Sleep staging is more complicated than just using the EEG as it uses other features as well (EMG, for example). Sleep has many other characteristic features, as well, such as reduced thresholds to response to stimuli and molecular signatures, that allow us to identify sleep in other animals, including those that are quite different from humans. Some references to go deeper:

Cirelli, C. (2009). The genetic and molecular regulation of sleep: from fruit flies to humans. Nature Reviews Neuroscience, 10(8), 549-560.

Zimmerman, J. E., Naidoo, N., Raizen, D. M., & Pack, A. I. (2008). Conservation of sleep: insights from non-mammalian model systems. Trends in neurosciences, 31(7), 371-376.

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