2.5.5 Imitating ion’s electricity

We have evidence that the membrane’s charge is proportional to the membrane’s voltage: the membrane has a fixed capacity $C$. We know that the arriving axonal currents (as well as the rushed-in ions after exceeding the membrane’s threshold voltage) cause massive transient changes [80, 81] (in other words: gradients) in the membrane’s voltage. We know that the charges on its surface can flow out only through the ion channels [50] in the neuron’s AIS, which we represent with a resistance $R$. We have evidence that the AIS only mediates the membrane’s changing voltage to the axon [50]. So, we have good reasons to assume that the membrane is not equipotential when generating an AP. Our hypothesis about ”slow” currents’ presence thoroughly explains the phenomena about the temporal behavior of neuronal processes mentioned. However, given that the classic picture also explains many phenomena, we must establish the connection between the two models and draw the borders within which the classic description can be used, and where our time-aware model must be used.

Some resemblance indeed exists in charge handling in electrical and biological circuits. However, the validity of parallels is limited. Assume an infinitesimally fast ion current on the surface will keep the membrane’s potential constant all the time. We can use the formalism developed for electricity, even in biology, if we want to use the point representation of a neuron. The price we pay is that we do not have access to the voltage and current of our finite-size membrane: they are confined in our fictive discrete elements (point representations within our point representation of neurons) despite the evidence listed above. This abstraction is appropriate for designing electric circuits and has a suitable formalism that describes their behavior. However, neither membrane nor AIS has the facilities to generate an AP in this approach.

The ”point representation” model results in a wrong parallel with the ”integrator-type” $RC$ electric circuits: by assuming discrete resistance and capacitance, we set up a fake hypothesis that the voltage drops within the resistor, and that current flows into the condenser and the charge is stored in it. A late consequence of the century-old idea of ”point representation” (which immedietely follows from the ’instant interaction’) is that we must omit the temporal dependence of axonal signal arrival; we try finding a correlation between neuron’s inputs; we do not see the role of fellow neurons in neural operation; we attempt to describe neuronal information with inappropriate representation and using inappropriate mathematical methods, etc.

The ”extended point” model reveals that all currents flow into the membrane, which is a distributed condenser, and the AIS with a resistor $R$ is a discrete output component of the circuit; see also Fig. 1 in  [50]. That is, the neuron shall be modeled as a ”differentiator-type” $RC$ circuit, having entirely different electric behavior from that of the commonly used ”point representation” model (with its implicit ”integrator-type” $RC$ circuit) predicted. From biological point of view, the vital difference is that this circuit type can produce an output voltage with opposite sign, enabling to describe to hyperpolarization, without needing any fake extra mechanisms, such as outflow of an intens $K^{+}$ current.

In our model, the neuron membrane is simply a two-dimensional elastic isolator surface (where needed, we imagine it as a thin, long, and narrow rectangular piece) that has current sources at different positions (axonal arbors), many concerted current sources in its body (the ion channels in the membrane’s wall) and a current drain (AIS) at the other. The input and output currents increase/decrease the voltage on the membrane. In our time-aware model, we assume that the ions on the membrane’s surface represent a kind of ”free ion cloud” (see also section 2.5.2), so we can interpret the capacitance $C$ (at least for our differential equation) in a classic way. However, charge carriers are not necessarily present on the surface. In the case of a neuronal membrane, the stable basic state is that there are no charges on the surface. If charge carriers (from an external source) appear, the potential increase that their appearance causes leads them to be removed. A “slow” current on the surface with a speed $v=\frac{dx}{dt}$ represents a current $I_{slow}=A\ast n\ast q\ast v$.

That is, the potential in the function of the distance will drop in the same way as if the membrane had a distributed resistance $R$. However, the resistance is located to the AIS, as if it were a discrete element. In electronics, the capacity $C$ is interpreted as opposite charges on the condenser’s plates. In biology, no similar stored charge exists. The charges spend some time on the surface, inversely proportional to the current’s speed (the inward positive current due to rushed-in ions and the outward positive current of the pumped-out ions has been observed, but not the corresponding negative currents). The distributed resistance and the specific capacitance are constant in the function of position over the membrane’s surface so that those values can be used in differential equations based on Kirchoff’s Laws. Notice, however, that currents joining the membrane at different points may spend different times on the surface (meaning different capacitance values), so the capacity changes in the function of the time, in this way distorting the time constant $RC$ and so the shape of AP.

The potential in the function of the time and the speed of the slow current mutually generate each other, as described by Eq. (2.40). In a steady state, no current flows. When some current arrives through the axons, or flows out through the AIS, a slow current starts to balance the potential difference created by the current. Changing the amount of charge on the surface transiently leads to a non-equipotential membrane. Notice the difference: if we assume instant interaction, we assume a constant membrane potential using discrete elements $R$ and $C$. The voltage drops on the discrete element $R$, and the charge is stored in the discrete element $C$. The voltage outside the discrete elements is constant, except for the voltage step, due to some incoming current (including the AIS, and there is no way to interpret how and why the AP is created. On the contrary, if the current is slow, it needs time to reach another position (we can change the membrane’s local ”charge storing” ability), and it can either increase or decrease the local voltage. When using a voltage generator with appropriate temporal behavior, the “slow” current explains why and how an AP in a biological neuron is evoked.

We can hypothesize that

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  ”making a hole” in the membrane [81] means that ”slow” ions are pressed into the membrane through the axon.

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  the inflection point is the turning point where the outward current exceeds the inward current, and it can be considered to be the time of the arrival of a spike (in the case of the first spike, it can be the signal ’Begin Computing’ [121]).

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  the inflow and outflow happen in parallel (the slopes of the PSP voltage course differ from those of the current pulse; see also their numeric time constants in Fig. 3.22); that is, we will see the difference between a “slow” and a “fast” current, with a particular temporal behavior.

The time needed to move a charge to a distance comprises two contributions. To move a charged particle in a piece of material (“the wire”), first, we must produce a force to accelerate the particle inside the wire at the position of the particle (the needed time is the distance to its location divided by the speed of propagation of the electric potential field). The second contribution is that the charged particle needs time to reach the other end of the wire (the distance to the external world divided by the particle’s drift speed). The object must be accelerated to that speed by that electric field; for the sake of simplicity, we consider the needed time negligible. To calculate the ”time delay”, we need to sum the field and charge propagation times. Let us suppose that the electric field’s propagation speed is infinite and the charge is in the immediate vicinity of the end of the wire. Fortunately, different physical mechanisms (such as “free electron cloud”) can produce the illusion of a much faster macroscopic current speed. In that case, the travel time of the charge is negligible. However, we can expect only in that case that the charge promptly contributes to the current, i.e., the current follows the voltage without delay.

We consider the cases of galvanic wire and electrolytic wire. There is no essential difference in the field propagation time: for our human senses (and even slower electronic tools), it is a good approximation that the electric field appears promptly along the wire, including the position of the charged particle. In galvanic wires, the electrons behave like an electron cloud: uniformly distributed in the wire. When the electric field appears (in the sense above: promptly), there are electrons in the infinitely small vicinity to the end of the wire. The field speeds them up immediately, so they exit the wire, and some other electrons enter the wire at the other end simultaneously. The charge carriers enter and exit immediately after an external potential is applied.

The phase change of voltage and current follow each other without a (noticeable) delay. Ohm’s Law is valid for this case: the derived entity connecting them (resistance or conductivity) is constant. The Law expresses the charge conservation: the same number of carriers passes the cross-section at any time. Remember that the essential conditions were that free charge carriers were present and uniformly distributed in the wire. Furthermore, they were moved by only one microscopic force (not considering the forces implementing an average macroscopic “resistance”). Even in metallic components, the derived material characteristics depend on many factors when we apply a step-like change in the voltage or the current. The so-called on-resistance is also known outside electrolytes and is influenced by various parameters such as temperature and supply voltage.

In an electrolytic wire, the ions in the electrolyte may be uniformly distributed (they form a kind of “free ion cloud” inside the electrolyte), i.e., after the electric field is applied, the ions can immediately exit the electrolyte and produce an electric current. In summary, the ions are very slowly moving charged objects (compared to the free electrons; BTW: the electrons move only slightly faster than ions; only the cloud provides the illusion of their high speed). However, they can create a prompt ionic current, provided that they are present in the corresponding volume and their concentration is isotropic. The living cell with its semipermeable membranes can produce situations where isolated structures do not fulfill that condition, and the less careful observer identifies the situation as non-ohmic behavior. As we discussed, the axonal tubes are empty (no charge carrier) at the beginning of a clamping experiment (see the measurement results in Fig. 3.24), and they are filled in their steady state (at beginning discharging), producing entirely different temporal behavior (”changing conductance”).