Can anyone explain the neurochemical process of a bipolar mood swing?

Very complex...: But we know that mood swings beget more mood swings. That is, the more mood swings one has, the more likely it is one will get one in the future. A common model for this phenomena is the kindling model for the development of seizures. Once one has a seizure, it takes less of a stimulus to induce another one. Kindling happens because of changes in cell membrane potentials. Very complex.

To some extent: The "high" and the "crash" of amphetamines and Cocaine are a good model. Presumably mania increases Dopamine and/or norepinephrine levels which are decreased during depressive episodes. Exactly why this happens spontaneously in the brain is not fully understood.