I like this point. There seem to be a number of different perspectives on or descriptions of the activity between neurons. Those phenomena produced from the interactions should in principal, arise from the local interactions between neurons as described by Hodgkin, Rall or whoever, regardless of which way you want to look at or describe the interactions. If the interactions between neurons are classical in nature, they are separable, so phenomena such as “coherent neuronal oscillations” should emerge regardless of which description you choose of those interactions.You say "coherent neuronal oscillations", I say "electrical activity of neurons" - but we are essentially talking about the same thing. Chemical synapses aren't the only thing involved - hormones from the bloodstream, genetic processes affecting receptor density and distribution, electrical synapses which allow for much faster coupling than chemical synapses, second messenger systems that change the electrical properties of the neuron by increasing the conductance through a receptor or changing the kinetics of the votage response of the receptor.
Self organization is something which arises from local interactions. It’s also defined by “weak emergence”. Assuming neurons don’t exploit any of the special features of quantum mechanics, then downward causation can’t influence the brain as a whole. Since there’s no reason to accept that the brain as a whole is some kind of quantum computer, there’s no reason to accept that local interactions are insufficient to produce all of the phenomena produced. The interactions between neurons are 'calculated' using software like "Neuron" and "Genesis" and if you look at those software, you will find they are consistent with weak emergence and self organization so they should produce all the same phenomena (in principal) as actual neurons regardless of how you want to describe those interactions.