I can't help feeling that the higher dimension theories here were created to solve a problem that didn't really exist in the first place. "Binding" is only a conceptual problem if the wave like nature of brain activity is ignored. There is ample evidence though that the broader scale events within the brain are indeed wave motions. There is no good reason for ignoring the obvious inference that wave motion measured by EEG is more than an epiphenomenon like the waves made by wind on a wheat field.
It makes far more sense to see that EM oscillations resulting from coincident firing of crowds of neurons must have a driving and constraining effect on the neurons of the crowd. This is loosely analogous to the way electric generators running in parallel are mutually constrained by their parallel connections to match speed.
This has the effect of maintaining a temporal structure or rhythm which through consistency of occurrence within its context, can be a signal or a structural element in something much bigger. I take this to be far more in line with Occam's razor than appeals to hypothetical dimensions in a parallel universe! If this be so then the only association with "higher dimensions" will be the use of mathematical objects to describe our brains' waves.
Mark - aka xodarap.