CH4 and CO2 levels show correlation throughout the Pleistocene:
What are the reasons of this coupling?
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This question is addressed in a Nature paper on "Understanding the glacial methane cycle" (Hopcroft et al. 2017).
There conclusion is that changes in methane concentrations are driven by changes in the biosphere associated with the global temperature changes. The dominant factor is the reduction in methane emissions from wetlands in a colder climate. There is also a reduction in emissions associated with forest fires in the cold phases (colder climate means less forest fires; warmer climate means more forest fires).
Note that the model used by Hopcroft et al. has the CO2 levels prescribed. In this situation the temperature changes in response to the CO2, and then the methane changes in response to the temperature.
Just a guess, not a specialist on the topic. Pleistocene was characterised by multiple glaciations and interglacial periods. The increase in CO2 concentration and the increase in temperature normally leads to thawing of large ares of permafrost and formation of swamp areas in its place. These swamps release methane in the same way they do today.
Obviously, the reduction in CO2 levels and subsequent drop in temperature would lead to the opposite process of swamps freezing over and release of CH4 in the atmosphere stops.
These charts in the original post make it hard to confirm this, but with better resolution I'd expect CH4 levels to be lagging slightly behind CO2 and temperature graphs.