3.10.6 ’Delayed rectifying’ current

As a consequence of using, by mistake, the integrator-type instead of the differentiator-type $RC$ circuit, the textbooks (see, for example [107]), explain that ’the membrane potential would have simply relaxed back to the resting value after the initial depolarizing stimulus if there had been no voltage-gated ion channels in the membrane’. This statement is wrong. The figure refers to an electric integrator-type circuit instead of a neuronal oscillator.

Unlike in the resting state, when generating an AP, there is no intense $K^{+}$ current. The explanation that ’the efflux of $K^{+}$ through $K^{+}$ channels, which open in response to membrane depolarization’ [107] is wrong. As we described, the $N{a}^{+}$ ions form for a short time (a small fraction of a millisecond), a thin $N{a}^{+}$-rich layer on the intracellular side of the membrane (this effect is misinterpreted as ions adsorption @cite Hodgkin-HuxleyAdsorption:2021), and, correspondingly, a $N{a}^{+}$-poor layer on the extracellular side. The strong repulsive force would prevent $K^{+}$ ions in the intracellular side from reaching their specific ion channels, even if the $K^{+}$ channels would ’know’ when to open. The driving force for $K^{+}$ would act in the opposite direction. Furthermore, an attractive force would act on the $C{l}^{-}$ ions. How big the driving force could be, can be understood from [107], chapter 11: ’The interior of the resting neuron or muscle cell is at an electrical potential about $50\mathrm{\dots }100mV$ more negative than the external medium. Although this potential difference seems small, it exists across a plasma membrane only about $5nm$ thick, so that the resulting voltage gradient is about $100,000V/cm$.’ The diameter of the ion channel is about $0.1nm$, and ’two $K^{+}$ ions in single file within the selectivity filter, separated by about $8A$. Mutual repulsion between the two ions is thought to help move them through the pore into the extracellular fluid.” [107]. Maybe, in biology, Newton’s third law in not active? We show a numeric calculation in section 2.5.7.

Fortunately, the correct differentiator-type circuit produces the ’hyper-polarized’ AP voltage time course (below the resting potential) alone, without needing to hypothesize some (unphysical) ’ghost’ current.

As discussed, the rushed-in $N{a}^{+}$ ions produce a ’traveling wave’ on the membrane. However, [107] shows that potential only on the axon. The textbook skips the conclusion that a traveling wave spreads over the membrane, because it would kill the starting hypothesis that the membrane is isopotential while generating an Action Potential.

The effect of the ion channels alone cannot produce a traveling wave. However, as we discussed, the rushed-in ions create a huge charge density on the membrane’s surface, and that charge can exit only through the AIS. That macroscopic ’slow current’ on the intracellular side of the membrane, on the differentiator-type $RC$ circuit, produces the ’traveling wave’ observed about the AIS and along the axon. When the book [107] was published, the structure of AIS [53] was not known. Now it is. It is high time to fix the neuronal circuit type and explain how to create the action potential with a correct model based on the first principles of science.

HH’s equations more or less accurately describe the features of the wrong oscillator type and those of the non-existing $K^{+}$ current introduced for compensating for the wrong oscillator selection. As the meticulous review [117] made clear, ”typically only a fraction of the various voltage-dependent potassium currents present in a neuron is significantly activated during normal action potentials”. That is, they might be significant in other periods, but not during generating normal APs.