As we derived theoretically in section 2.5.3, in general case, the ion current generated by a potential gradient is proportional with the electric gradient (see Eq. (2.28)) and so the macroscopic current speed (see Eq. (2.35). By using the Stokes-Eintein relation relation we can also express the speed in the function of the potential gradient. According to that prediction, when we interpret that the clamping voltage is uniformly distributed along the axon, we have a (proportional to the clamping voltage), and the current proportional to ion speed, we can expect a linear dependence between them.
The mostly known and influencing axon current measurement has been published in 1952 [9]. They used a single-axon input and measured at different clamping voltages the neuronal membrane’s current (although they called it as conductance), which in this way was identical to the axon current. Their result is reproduced in Fig. 3.20 as the black bulbs and diagram lines.
As we discussed, ions diffuse across the axon’s wall, producing a saturation-type current in the axon and later on the membrane. As expected from our theoretical consideration in section 3.9.2, the experimental data are fitted with the theoretical function 3.1 to their experimental data (just reading back their graphically published measurement results). The parameters of the theoretical function are displayed in Fig. 3.21 in the function of the clamping voltage. Our simple model assumes that the time constant (through and ) depends on the clamping voltage). As expected, the potassium current, proportional to ion speed, changes linearly in function of the voltage gradient (proportional to the clamping voltage), see the blue dots. The similar dependence of the time constant on the clamping voltage (see the red circles) underpins that our theoretical discussion leading to Eq. (3.1) is correct. Even, the diffusion coefficient or viscosity can be derived. The diffused-in ions were transported towards the membrane as a ”slow” macroscopic ionic current (the speed of current HH [9] measured and also theoretically derived to be about ); it is in the order of magnitude we mentioned for the speed of macroscopic currents in metals and electrolytes.
A systematic discrepancy exists at the low time values of the time course function between the one fitted originally by [9] and the one fitted by us. The former one is a simple polynomial that is simply a wrong quasi-model; our fitting uses the correct model function. The dependence we use (a sudden and delayed exponential increase in membrane’s current) has been experimentally measured by [128]. The figure suggests that the saturation current depends linearly on the speed of ions (i.e., on the clamping voltage, see Fig. 3.20 and Fig. 3.21) in the tube.
We can spot two issues in connection with their measuring and one with the evaluation method of their excellent measurement. A fundamental problem to solve when measuring chemical electrolytes using electronic devices is their interfacing. At some point, the ionic charge must be converted to electrons (there and back), which usually happens in electrolyte electrodes. Interfacing the analyzed electrolytic wire and metallic wire in the measurement circuit introduces problems, not only the contact potentials but also the time delay due to the using electrolyte electrodes. These electrodes need to carry the ions to some distance, and that process is outside of the time scale of the primary measured process. The effect is noticed but not explained [9]: “the steady state relation between sodium current and voltage could be calculated for this system and was found to agree reasonably with the observed curve at 0.2 msec after the onset of a sudden depolarization.” Moreover, given that the speed of ions depends on the depolarizing voltage (see Eq. (2.28)), this time gap depends on the depolarizing voltage: the higher the voltage, the shorter the time gap, demonstrated in their Fig. 3. Actually, their fitted polynomial chooses a wrong time scale and adds the delaying effect of the electrolyte electrodes to the measured time. Since this delay depends on the clamping voltage, the measured time constant comprises a systematic voltage-dependent contribution, so it distorts the fitting and delivers wrong time constants.
The second issue is that they measured conductance, which measurement procedure (as we discuss it in section 2.3.3) means introducing a small voltage into the measured system and generating a small current in it, which – by biological mechanisms – may produce further charge carriers inside it. The generated small current contributes to the true current in the system, and the device measures their sum, that is, the true current plus that offset. As long as the true current is large (notice that the clamping voltage spans nearly two orders of magnitude), the contribution of the device causes only a negligible distortion. At small clamping voltages, however, that contribution is comparable to the measured effect, so it significantly distorts the measurement and shows much higher current than the true one.
The third issue is, that they fitted their data with a polynomial function, which draws a ”smooth” diagram line, at low time values contracting an initial ”no current” period with a period where the current grows exponentially. Given that the ”no current” period decreases as the clamping voltage increases, the polynomial fits a variable composition of current in those two periods.
We fitted our theoretical function (see Eq.(2.38)) to their measured data published in [9], omitting the delay period due to the electrolyte electrodes. This way we eliminated the first and third issues. We derived the timing constant and saturation current values using the clamping voltage as parameter. In Fig. 3.21, we compare our fitted data values with those derived by HH (displayed in their Table I).
In the case of using the right function for fitting the measured current value, we receive the theoretically expected conclusion, that the time constant depends linearly on the clamping voltage (that is, on the voltage gradient), while fitting the data with the wrong (polynomial) function, the time contants show an opposite dependence. The saturation current shows in both cases a linear dependence.
The wrong evaluation method led Hodgkin and Huxley to conclusions opposite to the real ones, from the correct measured data. Their measurement is precise but not accurate. It has very sever consequences: covers the presence of ”slow current” and disables understanding the physical process happening inside the neuron. Given that in the meantime the measurement technology developed (smaller electrolyte electrodes with much shorter delays furthermore conductance meters with higher internal impedance have been developed), and they coded those parameters (without the voltage-dependence we pointed out) into their polynomial coefficients (which are used in their differential equations), those issues significantly contributed to the difficulties their followers experienced when applying their equations.