What are brain oscillations?
I think it is first important to recognize what brain oscillations refer to: they are small, somewhat localized fluctuations in voltage that are often measured by EEG (electroencephalogram), though they can also be measured inside the skull or inside the brain.
Most of these oscillations are also seen in the membrane potentials of individual neurons. Fast oscillations, like gamma oscillations, can also be seen in membrane potentials but are best associated with spiking activity. The cause of the skull-measured oscillations is the coordinated activity of many many neurons acting in concert.
Why certain frequencies?
Many brain oscillations are created by coupling of excitation and inhibition. The gamma oscillation, for example, is most associated with interactions between excitatory neurons and particular inhibitory neurons called fast-spiking and/or parvalbumin-positive cells. The frequency of the gamma oscillation is due to the time constants and conduction times between the reciprocal circuitry between excitatory cells and fast-spiking cells. If you were to maintain the same activity, but change these time constants, the gamma frequency would change.
Other frequencies may be slower because they involve circuits over a longer distance, such as interactions between thalamus and cortex.
Some of the slowest oscillations, called delta oscillations, come about due to long periods of high and low activity, sometimes called "UP" and "DOWN" states or "ON" and "OFF" periods. Oscillations are often nested within each other, referred to as cross-frequency coupling, or more specifically often phase-amplitude coupling. During the "DOWN" phase of a slow delta oscillation, amplitude in higher frequencies is low because there is little spiking activity; during the "UP" phase the amplitude in higher frequencies increases.
What is the purpose of brain oscillations?
We don't know. There are many theories that certain oscillations are important for certain functions, but most of the evidence for these theories is either theoretical or correlational. In very general terms, slower oscillations tend to be associated with rest whereas higher frequency oscillations are associated with active processing.
For example, gamma oscillations are possibly involved in 'binding' different types of information across different cortical areas: synchronous gamma activity could be the way a brain region primarily processing sounds can associate that information with a brain region primarily processing the visual object producing that sound.
It is also possible that many oscillations are simply epiphenomena: they result from particular types of activity, rather than being the cause of certain functions. Gamma, for example, seems to increase simply whenever overall neuronal activity increases.
The problem is that there is really no way to study an oscillation independent of the rest of the brain activity: anything that would impact an oscillation is necessarily going to be influencing neural activity in other ways.
Probably the best evidence for an actual functional role of brain oscillations is in phase coding in the theta band (and associated gamma oscillations), best-studied in the navigational networks of the hippocampus in rodents. The timing of spikes relative to the phase of an ongoing oscillation can carry more information than the spikes in solitude: the oscillation provides a reference signal.
What happens if brain oscillations were to magically change?
It sort of depends on what the mechanism of the magic is: you could certainly change oscillations by doing extreme things like stopping all neuronal activity: that would go quite badly. Epileptic seizures are a more naturally-occurring but similarly damaging example of oscillations gone wrong. Absence epilepsy is of particular interest since the mechanisms are a bit more complex than grand mal seizures.
However, you can also "magically" change oscillations as simply as closing your eyes: increases in alpha power are well-known to occur when you just briefly close your eyes. Changes in brain oscillations are a normal part of brain function: oscillations are very different during sleep and wake, for example. The amplitude and frequency of certain oscillations can be modulated by how alert or attentive you are. Depending on the specific type, anesthetic agents tend to produce brain oscillations that are at least qualitatively similar to sleep.
There are also correlations between different oscillation patterns and psychological diseases or disorders. People with schizophrenia show different patterns of gamma activity than non-schizophrenics, for example. However, it is unclear if it makes sense to think about oscillation differences as causal versus as a symptom. Autism, for example, is associated with reduced synchrony over long distances in the brain. This reduced synchrony is thought to be due to differences in connectivity. Depending on your perspective, you could highlight the change in synchrony as an important feature, or you could treat it as a symptom of the underlying difference in connectivity.
Summary, Conclusions, and Caution
Brain oscillations are a key window to understanding nervous function, especially in humans where they can be recorded more easily than any other type of neuronal activity. However, we need to be careful about thinking about oscillations as specific entities.
Oscillations in one frequency band could come about by completely different mechanisms in different situations. Changes in oscillations measured outside the skull can reflect changes in synchrony over long distances rather than changes in local oscillatory activity in individual columns. As an analogy, consider ripples in a pond: the amplitude of the ripples might tell you a bit about the size of an object that disrupted the water's surface, but if that's all you have you can't know whether the object came initially from above or below the surface, whether it was a bird, fish, or rock, etc.
Many papers are published that tie oscillations to certain states, conditions, or diseases, and these can be pathways to understanding but they are unlikely to directly identify underlying mechanisms.
For the references below, I've tried to use review articles where possible that are fairly digestible without too much additional information. In particular, I bolded a couple that I think are great starting points.
Bazhenov, M., Timofeev, I., Steriade, M., & Sejnowski, T. J. (2002). Model of thalamocortical slow-wave sleep oscillations and transitions to activated states. Journal of neuroscience, 22(19), 8691-8704.
Buzsáki, G., & Wang, X. J. (2012). Mechanisms of gamma oscillations. Annual review of neuroscience, 35, 203-225.
Cohen, M. X., Elger, C. E., & Fell, J. (2008). Oscillatory activity and phase–amplitude coupling in the human medial frontal cortex during decision making. Journal of cognitive neuroscience, 21(2), 390-402.
Goldman, R. I., Stern, J. M., Engel Jr, J., & Cohen, M. S. (2002). Simultaneous EEG and fMRI of the alpha rhythm. Neuroreport, 13(18), 2487.
Harris, K. D., Henze, D. A., Hirase, H., Leinekugel, X., Dragoi, G., Czurkó, A., & Buzsáki, G. (2002). Spike train dynamics predicts theta-related phase precession in hippocampal pyramidal cells. Nature, 417(6890), 738.
Hasselmo, M. E., Bodelón, C., & Wyble, B. P. (2002). A proposed function for hippocampal theta rhythm: separate phases of encoding and retrieval enhance reversal of prior learning. Neural computation, 14(4), 793-817.
Kwon, J. S., O'donnell, B. F., Wallenstein, G. V., Greene, R. W., Hirayasu, Y., Nestor, P. G., ... & McCarley, R. W. (1999). Gamma frequency–range abnormalities to auditory stimulation in schizophrenia. Archives of general psychiatry, 56(11), 1001-1005.
Lisman, J. E., & Jensen, O. (2013). The theta-gamma neural code. Neuron, 77(6), 1002-1016.
Steriade, M., McCormick, D. A., & Sejnowski, T. J. (1993). Thalamocortical oscillations in the sleeping and aroused brain. Science, 262(5134), 679-685.
Tallon-Baudry, C., & Bertrand, O. (1999). Oscillatory gamma activity in humans and its role in object representation. Trends in cognitive sciences, 3(4), 151-162.
Uhlhaas, P. J., & Singer, W. (2006). Neural synchrony in brain disorders: relevance for cognitive dysfunctions and pathophysiology. Neuron, 52(1), 155-168.