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Thank you for writing this so that I didn't have to.

One thing I might add: it's very difficult for me even to imagine how a network of interconnected neurons, some of which are spontaneously active, could fail to give rise to oscillations. This to me is the best reason to think that EEG signals don't have much explanatory power over and above the neuronal spikes upon which they supervene.

One last point of clarification that may be helpful to non-neuroscientists: it may be useful to distinguish "brain rhythms" at different levels of analysis. I can imagine someone reading this and thinking that hippocampal theta is pretty much like visual cortex event-related potentials (ERPs). These are completely different phenomena measured using completely different methods! Hippocampal theta is measured as a local field potential (LFP), which usually means actually sticking electrodes in the brain and then low-pass filtering the signal. One step up from that is intra-cranial EEG, in which electrodes are placed on the surface of the cortex (in neurosurgical contexts). And finally you have good old-fashioned EEG, where you wear a cap over the head, like in a psychology lab, and measure electrical signals that originate over many centimeters and are difficult/impossible to conclusively localize. This is where people see ERPs.

I'm not a reductionist at all costs by any means, but generally I think it's safe to say that the actual causal relevance of each of these "rhythms" decreases as you ascend the hierarchy. Hippocampal LFP stands out as potentially interesting because of phenomena like phase precession of place cells, but Justin L aptly points out that this is by no means a necessary feature of place coding in general.

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