(In physics.stack I have been suggested to post my question also here.)
In the classical theory of passive neurons (where the action potential is not yet excited), the voltage is successfully described by cable theory. The (unmyelinated) axon is modeled as a series of cylindrical sections in series. Each section is described by an electric circuit (in 1D) with an axial resistance, and the membrane is modeled as a capacitor.
If charges are injected at one end of the cylinder, the "cable equation" gives the transmembrane voltage at steady-state in the shape of an exponentially decreasing function of space. Here the characteristic 1/e decay length of the voltage can be long in large axons, of the order of mm.
My question is about the apparent lack of charge screening in this picture. At physiological conditions (where ions and counter-ions are present), I'd expect that an excess of charge, like the one injected by a micro-pipette, or generated by the opening of channels, is screened very quickly (microseconds), so a few Debye-lengths (nm) apart, the voltage should not be seen.
How can the theory and experiments show a voltage that persists at millimeter distances at physiological conditions where screening should be present?
EDIT from a comment: The electrotonic transmembrane potential is exponentially decreasing along the membrane. If we assume one side isopotential (say the interior), the other side should then have a voltage that decreases along the membrane. So measuring the voltage on two points along the membrane, one should see a drop in voltage, if the points are distant enough. Therefore on that side there is a current, and the charges on that side are not screened.
Is this description correct?