2.7 Controling potential 2.7.1 PID’s potential controling 2.7.3 Comparing equations

2.7.2 HH’s potential controling

One can reformulate the HH’s famous Eq. (1) in [9] to the form of the PID equation

\underbrace{I\times R_{AIS}}_{V_{M}^{OUT}}=\overbrace{\underbrace{I_{i}\times R% _{M}}_{ionic}}^{proportional}+\overbrace{\frac{1}{\underbrace{C_{M}\times R_{% AIS}}_{T_{i}}}\int_{0}^{t}\xcancel{(I_{bio}-I_{clamp})(\tau)}d\tau}^{integral,% \ clamping\ cancels\ this}+\overbrace{\underbrace{C_{M}\times R_{M}}_{T_{d}}% \frac{dV_{M}}{dt}}^{derivative,\ capacitive}

(2.74)

The proportional term comprises the ionic (and external) currents. At the time when Eq. (2.74) was set up, AIS was not yet known, so HH assumed that in the proportional term, the resistance is identical to the membrane’s resistance implemented by the ion channels in the wall (leading to the ideas of ”leakage current” and ”resting potential”, misleading physiological research.) The experimental conditions canceled the integral term entirely: the negative feedback from clamping precisely counterbalances the neuron’s internally generated current, so the current integrates to zero. (BTW: the ”integrate and fire” type models artificially revive the forgotten integral term.) The derivative term assumes that all currents are constants, so the voltage changes only due to the capacitive current. It entirely forgets that gradients are needed to operate an electrical system (leading to the ideas of constant-voltage batteries and magically operated resistors in the equivalent circuits, see Fig. 3.17). It was forgotten that a non-constant current contributes a gradient to the derivative term, although it was known from the beginning that the AP s comprise relatively steep rising and falling edges. HH’s formalism describes only the ’present’ (as can be expected when freezing the state), has no predictive power nor can give account of neuronal memories.

HH considered only the proportional term, plus in the derivative term, the voltage gradient due to the condenser (but not the capacitive current itself, directly leading to the wrong ad-hoc hypothesis of the presence of $K^{+}$ for explaining hyperpolarization). Given that clamping obscures the presence of gradient-like changes in the system, they missed that the input currents also produce a voltage gradient and that the output current through the AIS also produces a gradient. Furthermore, the equation shows that external currents (e.g., the rising and falling edges of step functions) also cause significant changes in the output voltage. Similarly, a ”foreign” invasion (changing mechanically the positions of charges by pressure, ultrasound, magnetic pulse, changing the concentration in one of the segments, and so on) generates a change in the position of charges on the neuronal membrane. This way, it generates a voltage gradient (as well as a pressure gradient and other changes), which may trigger neuronal spikes. Subthreshold excitations provide direct experimental evidence that the description of the resting state includes both serial and parallel contributions.