2.8.3 Neuronal membrane

They could “find equations which describe the conductances with reasonable accuracy and are sufficiently simple for theoretical calculation of the AP and refractory period”. However, their equations cannot explain the delay experienced by a sudden change; furthermore, they explained that AP is created because of, for some secret reason, the membrane’s conductance changes in time (although they noticed the presence of a “slow” current that behaves differently from the “fast” currents that their equations describe). The primary issue with their model is that it concludes, as they admitted, a wrong description (irrealistic delay) of sudden changes, such as the arrival of a spike, of making clamping measurements, or of interpreting the mechanism of neuronal information transfer.

Their followers modified both the form of their mathematical description (without assuming any physical model, using ad-hoc equations) to achieve minor improvement in the temporal behavior of the description. For a review of ideas, see [75]. This latter work attempted to introduce “a physiologically, physically and chemically viable model” that had to assume a physically not plausible ion-adsorption buildup mechanism to be able to explain the mentioned delay, see their Eq. (45). Those attempts, however, did not change what HH noticed [9]: “there is the difficulty that both sodium and potassium conductances increase with a delay when the axon is depolarized but fall with no appreciable inflexion when it is repolarized”. Without admitting a “slow” current exists, we must presume that sodium and potassium concerted their actions, and conductance is indeed misinterpreted in both cases. HH concluded [9] (presumably after many unsuccessful attempts) that ”there is little hope of calculating the time course of the sodium and potassium conductances from first principles”. It is correct: the existence of such a time course itself is against the first principles of science. However, if we make correct (physically plausible, instead of ad-hoc) assumptions, we can derive a ”time course” (well, not of the conductance because it is a misinterpretation of the physical phenomena, see section 2.3.3; instead) of the ionic current from first principles although we must mix microscopic and macroscopic parameters.

It is a long-standing enigmatic phenomenon that ”the emergence of life cannot be predicted by the laws of physics” [13] (unlike the creation of technical systems). Still, we can provide a complete description of the biological phenomena from first principles if we consider the finite interaction speed instead of using the idea of “prompt interaction” taken from classic physics, which is a fake abstraction for that goal. Models in neuroscience (as reviewed in [118]) almost entirely leave the mentioned aspects out of scope. We introduce a finite interaction speed without introducing either twisted mathematical handling or obscure physical (for example, adsorption) mechanisms. In our straightforward physical model, we see the measurable membrane potential and current change in the function of the speed of ions $v$.

The commonly used physical picture behind the process is that the membrane, as if it were metal, is equipotential, and the “fast” axonal current flows directly to the membrane. This assumption is why we expect an instant appearance of the axon’s current in the membrane’s current (instead, we experience a “time-dependent conductance”).

This axonal charge-up current, a phenomenon we are exploring from an abstract perspective, flows into the membrane. It causes transient changes [80, 81] in its voltage, providing direct evidence that the membrane is not always equipotential. The ions on the membrane’s surface can propagate at a finite speed. The membrane attempts to remain isopotential, the ions move freely on its surface.

After the membrane reaches its threshold potential, the voltage-controlled ion channels open, and many ions from the extracellular space rush into the intracellular space, as we explained in section  2.8.2. The ion channels open and close themself autonomously and quickly. There is no way or no need to simultaneously open other ion channels in the opposite direction. As we discussed above, the charged ions immediately in front of the membrane generate an electric gradient in the order of $100,000V/cm$.

The sudden membrane potential change in the charge-up period acts as a valve. Given that the ions in the axonal arbor need to enter the membrane against the actual membrane potential, the potential stops the ion inflow to the membrane for the period while the membrane’s voltage is above the threshold: it effectively inhibits further inflow through all axons. This behavior naturally explains the absolute refractory period. After the membrane’s voltage drops below the threshold value, the ions can enter the membrane again (see Figures 3.22 and 3.15), but they need time to reach the AIS later (see Figure 1.6) when in the meantime the membrane’ voltage proceeded toward its hyperpolarized state; so they seem to appear dozens of microseconds later at the AIS, explaining the relative refractory period (measuren in [64].

The inflow charge generates a ”potential wave” (a solid current outflow) through the AIS; see the discussion in section 3.7. The decreasing charge causes the membrane’s potential to decrease toward its resting potential, so it falls below the threshold voltage of the axonal gate at some point. If ions are still waiting on the other side, stopped when the membrane’s charge-up process started (recall that they cannot exit the axon of the presynaptic neuron, and previously they could not enter the membrane), or newly arrived while the gate was closed, they can enter the membrane again. The ions travel a finite distance on the surface of the membrane with a finite speed, so there must be a delay between their entry and exit times. Furthermore, the inflow current must equal the outflow current. As discussed in section 2.5, charge conservation is not necessarily valid in all contexts of biological operation. If we measure the input and output currents, they may differ (see Fig. 1. in [50]); see section 3.7.

Notice that, to some measure, the case of switching a clamping voltage on is analogous to the arrival of a spike. Initially, the axon contains no ions. The front evoked by a step function is linear because of the slow current. In the classic picture, the axonal current flows into the membrane with capacity $C_m$ and increases the membrane’s voltage $V_m$ with a time constant discussed after Eq.(2.77)

There is no voltage-dependent conductance [112]. Instead, the finite speed of ions and the wrong assumption that conductance is a primary entity misleads physiological research. With wording that ”conductance changes”, one states that charge carriers appear/disappear/reappear; that is, the charge conservation is not fulfilled (with nonphysical consequences listed in connection with the model in [9]). The physics background of the phenomenon is that the number of charge carriers changes (ions are “created” in the axon, and they appear on the membrane, as we detailed above).

In contrast, when the clamping voltage is switched off, the axon is still filled with charge carriers (but not filled after); the resting potential reaches the end at the membrane “instantly”. The driving force disappears, the ion stream stops, and no more ions enter the membrane. The lack and the presence of ions in the axon when switching clamping on and off, respectively, produce the difference that “conductances increase with a delay when the axon is depolarized but fall with no appreciable inflexion when it is repolarized” [9]. The potential is equalized by the AIS current, producing a net exponential decay:

During the regular operation of a neuronal membrane, after opening the ion channels, a vast amount of ions flow into the intracellular space from the extracellular space, imitating the effect of switching a clamping voltage. The essential difference is that the ions arrive through the axon to the joining point in clamping. In contrast, through the membrane’s ion channels, they directly contribute oon the membrane’s entire surface. The membrane’s size is finite, so with a finite current speed, it takes time until the charges on the membrane’s surface arrive at the AIS, the same way as we discussed for the axonal current. These findings have significant implications for our understanding of the operation of neurons, including their signal processing and memory.

From a computational point of view [121], a persisting significant deviation from the resting potential (the arrival of the first spike from one of the upstream neurons) provides the signal ’Begin Computing’, opening the ion channels in the membrane provides ’End Computing’. After that, we will be in the ’Signal Delivery’ phase until the end of the charge-up process. After that, ’Signal Transmission’ follows. Our simple neuronal condenser can only perform one operation, to integrate the current it receives. Its result is the integration time itself. It cannot distinguish its operands (which synaptic inputs provided the current it integrates). Furthermore, not all operands must be present at the beginning of the computation process. The membrane potential slowly returns to its resting value; furthermore, the current arriving during the ’relative refractory period’, represent a (time-dependent) memory, see section 1.7.1. Notice that the content of that memory may depend on the neuronal environment.