# Voltage propagation in neurons

Context

In the classical theory of passive neurons (where the non linear action potential is not excited), the voltage is successfully described by cable theory. The axon is modeled as a series of (infinitesimally small) coupled cylindrical sections. Each section is described by an electric circuit (in 1D) with an axial resistance, and the membrane is modeled as a capacitor. Here and here are more details.

If charges are injected at one end of the cylinder, the "cable equation" (close to the diffusion 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 $$mm$$. All this is functional to the "saltatory transmission" of the action potential.

Question

My question is about the apparent lack of charge screening in this picture. At physiological conditions (ions and counter-ions present), I'd expect that an excess of charge, like the one injected by a micro-pipette, 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?

• I think when you inject charge (or ions through and ion channel), it is still a net charge present despite the additional screening effects. In the ion channel case the opening and shutting and the movement of charge gives the voltage transient. If you inject charge or ions physically you also get a transient. The cable equation with the capacitance and conductance determine the velocity of signal (wave) that travels down the transmission line. Jun 20, 2022 at 23:54
• Related: Telegrapher's equations. We are dealing here with a lumped-circuit description of a continuous properties - in particular, the screening is understood in the values of capacitances. The question is essentially whether these values are correct and/or how they are estimated. Jun 21, 2022 at 7:35
• @UVphoton interesting, are you saying that is the continuous replenishment of charges that prevents screening from killing the potential? Maybe you can post an answer? Jun 21, 2022 at 15:18
• @RogerVadim thanks. I don't see well how screening can be modeled by capacitance. After the transient, screening lowers the potential down to zero. Can a capacitance do it? Maybe with resistances? Could you expand more on "the screening is understood in the values of capacitances." ? Maybe in an answer if possible? Jun 21, 2022 at 15:26
• Capacitance is screening: there's charge in neuron and charge outside, and change in one is compensated by change in the other. Jun 21, 2022 at 16:15