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In EEG analysis, entropy and coherence are treated as opposites in a scale. Lower coherence is higher entropy. This is information entropy, not thermodynamic entropy.

In a thermodynamic system, how would coherent oscillations affect entropy? If the particles in the system, for instance, had a standing wave of different frequencies, would the entropy be decreased (since there are fewer valid microstates) or increased (since the coherence would be dependent on the influx of energy)?

Anybody know the relation between harmonic oscillations and entropy?

The field of neurodynamics studies the computational function of oscillating rhythms in the brain. As an example, here is an article that describes how neural systems communicate through periodic synchrony. When a neural system fires at a frequency that is a close harmonic of another system, a large proportion of neural firings will have periodic synchrony. On the other hand, firing frequencies with the ratio of the golden mean will have (mathematically) the least periodic synchrony. That is, neural processing is multiplexed -- it is partitioned into temporal bands proportioned according to the golden ratio. Why? Because that is where the least regular synchrony occurs.

Hard to believe that the golden ratio plays such an essential part in the brain. But really, this effect emerges only from the mathematical necessity of it: http://journal.frontiersin.org/article/10.3389/neuro.03.001.2008/full

Schrödinger described how living things rely on the production of "negative entropy". How does coherence in the brain relate to Schrödinger's negative entropy? Consider: "Neurons that fire together, wire together; Neurons that fail to sync, fail to link". In the brain, synchrony is a fitness function that drives the evolution of neural systems. (e.g., Hebbian Learning, Neuronal Group Selection). The mathematics and physics of coherent oscillations, synchrony, harmonics seem essential for being able to understand the brain.

Which is why I ask, again: is coherence the opposite of entropy?

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Not sure about EEG analysis, but in quantum theory coherent oscillations have close to zero entropy, with only the quantum fluctuations contributing to some entropy. In the article at

https://arxiv.org/abs/quant-ph/0104083

It discusses some examples, for open but well defined systems, and shows that even in the ground, lowest energy state (maybe the vacuum) it still can be greater than zero. Temperature as well.

Still, this whole question of entropy has to be well defined to make it make sense. Entropy is generally and conceptually defined as unknown or hidden information or information loss. For Black Holes that does very well, for systems of many particles it is the possible number of states of the system that for specific macroscopic known variables (like volume and pressure and temperature for gases, for instance) it is the number of possible microscopic states consistent with that. That's the so called hidden information, or lack of information. And when you take the log of that number you get something that depends on temperature, and that's the entropy

Now, coherence is its own concept, and it basically refers to oscillations, or waves, that have and maintain a frequency and phase. Or, two waves are said to be coherent if they maintain the same frequency and a constant phase difference. That's the physics definition, but it gets more complex for quantum systems. See the article on coherency at Wikipedia at https://en.m.wikipedia.org/wiki/Coherence_(physics)

And you can see the similar but quantum definition of information entropy at Wikipedia at https://en.m.wikipedia.org/wiki/Coherent_information. It's not too different, conceptually, though the math and details can look complicated.

For EEG, there are various definitions that people have used for entropy. A couple of them are described in what I could easily Google, and it generally has to do with what they call the 'irregularities' of the waves. See for instance the specific article http://uknowledge.uky.edu/cgi/viewcontent.cgi?article=1205&context=gradschool_theses

These are best brought up in a biology or biomedical site, though yes,it is biophysics also. Possibly somebody in this site can comment or answer. I would guess that one uses a similar but different prescription than in physics: define the abstract space of EEG oscillations, and define some measure of regularity (which may be coherency, in the sense of wave coherency as stated above), and estimate it.

If you can post a good reference for the treatment of coherency and entropy for EEG, and you can do a bit of homework and try to specifically post the definitions and maybe a simple example, people in this site might be able to answer on the consistency with the physical defitnions or the information theory definitions, which in essence are pretty much the same in different contexts.

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  • $\begingroup$ This was my first question on physics overflow and this is a beautiful answer! I apologize I'm only seeing it months later. I will prepare some specific definitions and examples from neuroscience. Thank you so much for thinking about this -- I find the relationship between information entropy (shannon entropy) and thermodynamic entropy a fantastic mystery. Shrödinger claimed that living organisms need to generate "negative entropy". I am looking at how neural systems use phasic synchronization to line up different brain rhythms, to support communication. This will take a further post. $\endgroup$ Apr 7, 2017 at 5:53
  • $\begingroup$ "Thermodynamics of Coherent States and Black Hole Entropy" -- It is so funny how the article is about both of these topics! One would be enough :) But they connect them thus: "mean number of quanta and entropy of a black hole are of the same forms as the relevant values for coherent states of the harmonic oscillator or quantum field in a vicinity of the static source....It is therefore quite natural to expect that a coherent state of virtual excitations is formed in a vicinity of the black hole and, as a result, the black hole’s entropy is the thermodynamic entropy of this coherent state." $\endgroup$ Apr 7, 2017 at 6:11
  • $\begingroup$ You say, "in quantum theory coherent oscillations have close to zero entropy, with only the quantum fluctuations contributing to some entropy" -- So when system gains coherence, it must lower entropy? Is coherence always involving less entropy? I understand entropy as "fewer system configuration possibilities". Boltzman entropy is volume of the phase space of the possible microstate configurations making up a macrostate observation. Gibbs is equivalent, but instead of the volume of the phase space, it is the surface area. Somehow, Boltzman wins out because it permits negative temperature. $\endgroup$ Apr 7, 2017 at 6:18
  • $\begingroup$ You probably have already seen this article on coherency for improved inter-neuronal communications. ncbi.nlm.nih.gov/pmc/articles/PMC4706353 . I am not familiar with this area of research, but would look first for a review article of the whole idea before trying to assess any sense on these kind of articles. It looks a little too much like looking for something that may or may not be there, pulling at straws. There must be some basic thing that allows two neurons to communicate, and a simple way to see if phase locking makes a difference. $\endgroup$
    – Bob Bee
    Apr 8, 2017 at 2:10
  • $\begingroup$ Neurons have incredible diversity, so any model describing their communication requires significant caveats. However, the classic description of Hebbian Learning is "Neurons that fire together, wire together; Neurons that fail to sync, fail to link." While this is a gross simplification, synchrony between neurons can be viewed as a competitive "fitness function" that drives the evolution of cortical columns and general connectivity in individuals brain. Up to 1/3 of all neurons are "pruned"; lack of synchrony with their peers is a key factor. Buzaki's "Rhythms of the Brain" is a strong review. $\endgroup$ Apr 11, 2017 at 19:13

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