Does neuroscience still share the assumptions of Newton's classical physical science/mechanics and Descartes' bifurcated mind/matter dualism?

few neuroscientists take Cartesion dualism seriously. Most believe that consciousness is identical to neural activity. Newtonian dynamics is not really applicable since ionic conductances underly neural activity, and hence electrodynamics and electrochemistry are the underlying mechanisms, and not Newtonian dynamics.

Are there still not echoes such as cerebal mechanisms that affect attention, perception, memory and language processing?

.... would you say that current underlying force now steering neuroscientific thought to be packaged in relativity and quantum mechanics?

no

What role would you propose relativity, the special or general theories, to have in neuroscience?

if not, then what is? Is it now a rudderless fluctuating /oscillating ship?

what science does your avatar image derive out of?

electrodynamics and electrochemistry

.... has either established yet the reality of free will?

you now have me thinking of other possible models....

electromagnetic
thermodynamic
statistical
mechanical
hyperspatial
quantum mechanical
multi-dimensional

.... yet it seems that no simple application of existing physical theory is completely adequate.

what is "free will", and in what sense is it free?

ummm....yes?

Ok, so what are you really trying to ask here? The word "assumption" sets up a place from which to criticize.
I think I see where your going here. Each of the paradigms you mentioned may be at some point be based on some assumption that can be questioned. That's nice and I agree.

Of course you have the "standing on the shoulders of giants" effect. If we don't make some assumptions here, how can we possibly have progression? I'm ok if all the science of the world if found not true, as long as its applicable. Aspirin cures my headaches regardless of whether it's psychosymatic or not.

Pardon my jumping in here, but I was "unable" to resist. Free will is being able to want what you want to want. Will is free in the sense that undesired outcomes can be avoided.

For example, a heroin addict might some day see that his habit is ruining his life, so he might decide to stop wanting heroin so much. It's happened, you know. William Burroughs, for example.

Of course, those on the opposite side of the issue will say that the heroin addict changes because his brain chemistry changed, not because he resumed control of himself. It's just in how you look at it. One way has personal meaning, the other doesn't. Because the pro-free will approach has meaning (humanistically), I prefer it.