2.5.3 Peculiarities of ion current

On a microscopic scale, a charge creates a potential field and that field acts on another charge. In a conducting wire, filled with ions (the solids have a different current transfer mechanism), there are free charges, their number per unit volume is given by $n$, and $q$ is the amount of charge on each carrier. If the conductor has a cross section of $A$, in the length $dx$ of the wire we have charge $dQ=q\ast n\ast A\ast dz$. If the charges move with a macroscopic speed $v=\frac{dz}{dt}$, at macroscopic level, we define the current $I$ as the charge moved per unit of time as

$$I=\frac{dQ}{dt}=q\ast n\ast A\ast v$$

(2.25)

Notice that if any of the factors is zero, the macroscopic current $I$ is zero. Microscopic carriers must be present in the volume and have charge, the cross section must not be zero, and the charge carriers must move with a potential-assisted speed, which requires an external or internal force field. However, notice that the fellow charge carriers in the current also affect the speed, see also section 2.5.3. One of the fundamental mistakes by HH was to omit that effect (practically, neglecting the Coulomb force for ion’s electric interaction) and that the electrical potential is created by diffusion processes instead of ideal electric batteries.

When describing the macroscopic phenomenon ”current” in metals we apply a potential difference to a macroscopic piece of space (or measure it) and measure the statistical time course of the charge carriers which are electrons. In the abstraction we use, the external potential is constant (we use a ”voltage generator”) and the charge delivering has no ”side effects”. However, we must realize, that we have a hybrid circuit: in the electric half, electrons represent the current, while in the biological half, ions. We must convert the charge carrier there and back, furthermore, consider its possible side effects. When describing ”current” in entirely biological systems, it is represented by ions, and it is either a native current (without an external potential), or an artificial injected current (or potential generating a current). This way, the current is always producing or is accompanied by a change in concentration gradient, given that the moving ions represent mass transfer and charge transfer simultaneously. The potential and current are connected through the features of the medium (material) that hosts our measurement. One must not forget that ”Unfortunately, most measuring devices in neurophysiology are precise without being accurate” [2]. So are some definitions, too. Definitions and measurements, which are not accurate, conclude in wrong results.

According to Stokes’ Law, to move a spherical object with radius $a$ in a fluid having dynamic viscosity $\eta$, we need a force

$$F_d=6\ast \pi \ast \eta \ast a\ast v$$

(2.26)

The amount of current in a wire is not only influenced by the electric force field (specific resistance) but also by the number of charge carriers $n$. While the latter is commonly considered constant and part of the former, this is not necessarily the case for biological systems with electrically active structures inside. The medium’s internal structure introduces significant modifications. Applying an electric field to a wire can generate varying amounts of current as the number of charge carriers changes. For axons, we use a single-degree-of-freedom system, a viscous damping model, so the ions will move with a field-dependent constant velocity in the electric space; so it takes time while they appear on the membrane. The activity of potential-controlled ion channels in its wall may change $n$ in various ways; furthermore, that change can result in ’delayed’ currents during measurement, for example, in clamping, as physiological measurements witness.

If we have a concentration $C(z)$, in the volume $A\ast dz$, we have $dQ=C(z)\ast A\ast dz\ast q$ charge, resulting in another expression for the current

$$I=\frac{C(z)\ast A\ast dz\ast q}{dt}=C(z)\ast A\ast q\ast v$$

(2.29)

Combining equations 2.28 and 2.29:

$$v(z)=\frac{k\ast q}{C_k(z)\ast \eta \ast 6\ast \pi \ast a}\ast E(z)$$

(2.30)

We extend the original idea for living matter by using

	$E(z)$	$={\displaystyle \frac{\mathrm{\Delta }V}{\mathrm{\Delta }z}}$	$inlinearpotential$		(2.31)
		$={\displaystyle \frac{dV}{dz}}$	$ingradedpotential$		(2.32)
		$=E_{electric}+E_{(thermal)}$	$inelectrolytes$		(2.33)

Furthermore, correspondingly, here we introduce a speed gradient $v(z,t)$ which plays a vital role in the processes that occur in living matter.

The higher the resulting space derivative (gradient) and the fewer ions that can share the task of providing a current, the higher the speed. We hypothesized (it needs a detailed simulation) that in the case of this charged fluid, the electric repulsion plays the role of ’viscosity’ [22]. The higher the charge density, the stronger the force equalizing the potential; so $\eta$ is the lower, the higher the charge density (proportional to $C_k$). For the sake of simplicity, we assume that the speed is proportional to the space gradient of the local gradient. Recall that our equations refer to local concentrations only. The electric gradient can propagate only with the speed of the concentration gradient, given that only the chemically moved ions can mediate the electric field. The lower interaction speed limits the other interaction speed if the interactions generate each other.

The dependence of the diffusion coefficient on the viscosity can be modeled by the Stokes-Einstein relation:

$$D=\frac{k\ast T}{6\ast \pi \ast \eta \ast a}$$

(2.34)

so we can express the speed with diffusion coefficient

$$v(z)=\frac{D}{T}\ast \frac{q}{C(z)}\ast E(z)$$

(2.35)

or alternatively

$$v(z)=-\frac{D\ast R}{F}\ast C(z)\ast E(z)$$

(2.36)

In the equation, $z$ is the spatial variable across the direction of the changed invasion parameter, $R$ is the gas constant, $F$ is the Faraday’s constant, $T$ is the temperature, $q$ the valence of the ion, $V(z)$ the resulting potential, and $C(z)$ the concentration of the chemical ion. For the experimental evidence, see section 3.9.1.

Models in neuroscience (as reviewed in [118]) almost entirely ignore these aspects. In our physical model, we see that the measurable membrane potential and current change in the function of the ions’ speed, the concentration, and its time derivative; furthermore, all mentioned quantities depend on the effective potential.

It is important to remember for alternating current experiments: the different ions will move with different speeds under the effect of the same $E(z,t)$, which is the function of the local concentrations and the frequency of the alternating current. The conduction speed sensitively depends on the temperature, and through it, the shape parameters [119] of the AP and especially the length of the ”relative refractory” period [64].

As discussed in section 2.2.6, the overwhelming majority of physics phenomena can be described using the approximation that their interaction is instant; in other words, the interaction speed is infinitely high. In electricity it is a commonly used abstraction that the electric field and the current are ”instant”. Although we know that the field propagates with a speed near to the speed of light, and it is only an illusion (thanks to the ”free electron cloud”) that the current propagates at such a speed, the abstraction based on the approximation that those speeds are infinitely high, works. However, that abstraction has a range of validity and the biological systems not necessarily belong to it. It means that the laws of physics are valid, but in biology another approximations must be applied. The absolute value of speeds alone would not mean a problem, but as section 2.4 discusses, when force fields having different propagation speeds are mixing, special calculation methods must be used.

In electrolytes, two forces may act on the ions, and their balance may drastically influence the phenomena we can observe. When the two forces are balanced (either globally: no gradient acts across the volume; or locally: the two gradients differ by location, but at all places balance each other) no effective force acts on the ions. The electrolyte is in rest, and the thermodynamics entirely determines the speed of the ions: it is the diffusion speed. When some external electric gradient or concentration gradient applies (due to some internal charge-up process, external potential, external gradient change; or internal thermoelectric gradient change: a current inflow changes both concentration and potential gradient), the ions accelerate to their Stokes-Einstein speed (see Eqs. (2.35) and (2.36)). If the gradient does not change, they will move with that speed. This speed is much higher than the diffusion speed. On their path they may experience different gradients and they modify their speed correspondingly.

As we discuss in connection with the operation of ion channels, relatively small voltage (in the order of dozens of $mV$) may act on very short distances (in the order of $nm$), producing vast potential gradients (in the order of several tens of $kVcm/s$). Given that the ions travel short distances with vast speed, they may experience very different forces and they may travel with a speed differing by several orders of magnitude in a few $nm$ distance. In addition, since the moving ion causes change in the electric and potential gradients in a short distance, moreover due to the very low speed, the active volume on the departure and arrival sides are limited to a limited thickness.

In some cases, when precisely measuring the time course of current compared to the time course of voltage, one can experience a ’phase delay’ between them. Given that we are convinced that the charge conserves (does not appear/disappear) and the abstractions ’current’ and ’voltage’ are secondary abstractions and the manifestations of the primary abstraction ’charge’, the experience inspired further research, that led to inventing ’inductive’, ’capacitive’ and ’resistive’ currents. Science can describe how those currents combine and generate each other. In biology, similar delays are experienced, but the ’phase delays’ have been attributed to the media (membrane conductivity), without providing a clear reason how the same charge can produce its two different manifestations (potential and current) at different times.

Biology did not give an explanation (similar to the Mawell equations); instead, it claims that the living matter has a ’non-ohmic’ behavior, and that it cannot be described by the laws of physics. Including that some hidden power, for an unknown reason, changes the conductivity (meaning that charge can disappear/reappear; defying the law of charge conservation). However, it was not the case: in the living matter further interactions and their mixing speeds must be considered.

If we trigger at the same time two effects that propagate from one point to another, and they arrive at the target at different times, the penomenon may have different reasons. We may hypothesize that the triggering of one of them was delayed, or that their speed was different or that they took a break during their journey. Biology (without explaining or reasoning) assumes some ’delayed rectifier current’ and refuses the other two reasons.

As we discuss, there is a vastly different range of interaction speeds in science and during the electric processes in biological tissues, the charges may change their speeds when they pass from one biological object to another one. When sticking to mathematical formulas derived for pair-wise single speed interactions in homogeneous isotrop media in classic science, we miss the possibility to describe the true nature. The formulas representing a good approximation for one abstraction are not necessarily valid for another approximation. The abstraction ’metals’ differ sufficiently from the abstraction ’biological tissues and cells’, so we cannot hope that the notions, abstractions, approximations and laws describing the first one can describe the second one, despite that some initial resemblance exists.

There may be different reasons why a current appears apparently with a delay compared to the voltage, such as: the charge carriers of the current have finite speed, they are produced inside the media during the measurement, or they are stored for some reason for some time and released only some time later (as the conditions within the circuit change). In a limited way, one can imitate one effect with the other. As discussed in connection with Eq. (2.25), the current depends linearly on the number of the charge carriers $n$. The physical processes changing the number of charge carriers also define the intensity of the current, providing a way to imitate a slow current by a fast current, where the number of charge carriers is modulated by the spatiotemporal time course of the physical process of producing ions in the system.

Given the lack of mathematics describing the “slow” currents, it is a common practice to imitate a neuronal circuit with a simple electric $RC$ circuit having capacity $C$ and resistance $R$. Although in biology it is common to describe an ’electric equivalent’ of biological circuits, among others, biological oscillators one must not forget that instead of electrical processes (driven by an ideal voltage generator) electrochemical processes happen. The parallels have severe limitations.

The macroscopic equivalence is implemented at microscopic levels, among others, using ion channels huge electric gradients. The interplay of those biological objects can enormously change the speed of ions, that is the speed of ion current. Within the same phenomenon, the same charge carrier can have speeds differing by orders or magnitude.

From the structure of neuron’s membrane follows immediately that in the neuronal oscillator the capacity $C$ and resistance $R$ are connected serially instead of parallell. We assume a discrete equipotential membrane with capacity $C$ that leaks through a discrete resistance $R$. This also means we cannot apply Kirchoff’s Junction Law: the capacitive and resistive currents are not equal, because the condenser stores part of the charge that flows in through the membrane and the synapses.

Different damped oscillations can be produced depending on those parameters. The imitation is limited: the “rise time” gets smeared, and the output signals differ for the neural and the electric circuits. Instead of a step function, we expect for a “slow” current, we receive a smooth peak-like current time course (called a damped oscillator function). However, we can use the formalism developed for the “fast“ current. Adjusting its parameters allows the electric circuit to produce a behavior resemblant to a neuronal circuit.

Notice that in our imitated neuronal circuit, the peak of the “fast” current appears later. The “slow” current is seen, although the delay time is not explicitly present. If we use chained electric RC circuits, such as in the case of multi-compartment membrane models [45, 120], the second such circuit receives the output voltage of the first circuit at a later time, and so on. It is also described by a system of similar equations, but they are valid at different times. Handling the many equipotential compartments attempts to cover the fact that one imitates finite membrane size and slow currents.

However, in biology storing charge is implemented differently. The notion of storing charge can be used also in the sense that for the time of passing a finite-size element with finite propagation speed, the charge carriers spend the corresponding time in the element. That phenomenon resembles storing the charge, and that imitation enables us to describe a behavior resemblant to that of the biological circuit. Attempting to imitate the effects of biological “slow” currents using electric parallels hides that generating an AP is their native feature. No additional currents and sophisticated control mechanisms are needed: deriving-action-potential is a natural consequence of the interplay of the finite speed of the “slow” ionic current and the finite size of the neuronal membrane; furthermore, that slow currents may play a role also in cognitive functions.

As we detailed, the ions change their location during the observed potential changes. The currents described here flow in a thin layer on the top of the membrane

The ions (from any source) entering the layer with a high ion concentration in the segment with the lower bulk concentration will reside in the layer near the separating membrane; they are in thermal and electric equilibrium. They cannot diffuse inside their segment due to the attraction of the ions in the segment, so the mass current is zero. They cannot pass into another layer: the electric driving force is missing (or even, slightly opposite), so the charge current is zero. However, they induce the corresponding changes on the opposite side. As Eq.(2.9) describes, nothing changes.

The case fundamentally changes when a current drain appears in the layer. It decreases the local charge and potential, and the rest of the charge tends to be equipotentially distributed in the respective layer; a potential-assisted (slow) current will start. Given that the total charge in the layer decreases, its effect on the opposite side decreases, and the total amount of charge in the opposite layer also decreases, manifesting in bulk potential change. This charge ”redistributes itself” on the two sides of the membrane [24]. However, the circuit is closed through the drain and the extracellular space but not directly across the capacitor. Consequently, slow currents flow inside the two adjacent layers as well as in the bulks. In the high-potential layer, parallel to the membrane’s surface, and in the low-potential layer perpendicularly to the membrane, towards the bulk part of the segment. They are simple discharge-type currents (we consider only the one flowing in the layer in the segment with low concentration)

$$I_{Drain}=I_o\ast \exp (-\frac{1}{\beta }\ast t)$$

(2.37)

Given that the slow current, due to its finite speed, has a limited charge-delivering ability, unlike in electronics, no limiting resistance is needed in the circuit. The current generates voltage either on a capacitor, see axonal arbor [84, 85] in the case of axons (later on the membrane), or on a resistor, see the AIS [53]. If the delivered current can deliver more charge than that can flow away through the current drain, the effect of ’ram current’ (’impact current’) can be observed. Finally, as discussed in section 3.7.3, the AP is a direct consequence of the ’ram current’ due to the rushed-in ions.

Our equations call attention to the neglected aspects that the current evoking an AP on the AIS requires ions to be present in the electrolyte layer near the membrane; furthermore, that the rushed-in ions must propagate from the exits of the ion channels (and similarly, from the synaptic terminals) in the layer on the surface of the membrane to the AIS, AIS, which needs time. The potential changes observed at different membrane locations manifest the slow currents in the membrane. Recall the sizes of the measuring tip and that of the layer: the presence of the charged layer likely cannot be directly noticed However, its effects were noticed indirectly [95].

In the segment, external currents can also appear. Examples include synaptic inputs through the neuron’s synaptic terminals (with a time course of a PSP), the current from the AIS to the beginning of the axon (with a time course of an AP, and artificial currents with various time courses). In those cases, the external current delivers ions, generating the concentration’s and potential’s time course. As discussed, in our approximation the current increases the charge carriers on the arrival side and decreases it on the departure side. If the source is a potential-less current, a simple discharge function describes it

$$I_{source}=I_o\ast (1-\exp (-\frac{1}{\alpha }\ast t))$$

(2.38)

As evidence shows, the current provided by a population of ion channels depends only on their number and surface density, and the ion channels are distributed evenly over the surface. The charges appear everywhere on the surface, including near the drain. That means that the drain current starts immediately (the repulsion of the appeared charge creates the driving force), and an exponentially increasing current will flow in the layer with a potential-assisted speed. Its intensity will change due to the changing intensity of the source current and the changed potential drop in the drain. The two currents flow simultaneously, and its intensity is the product of the source current and drain current (this form, with different coefficients, seems to be valid for several biological systems comprising ion channels)

$$I_{out}=I_o\ast (1-\exp (-\frac{1}{\alpha }\ast t))\ast exp(-\frac{1}{\beta }\ast t)$$

(2.39)

The voltage’s time derivative describing the current in a system with source and current, needed for the biological law of motion (see section 2.4.5), is given by Eq. (3.6)

The channels in the membrane’s wall open quickly and the ions appear instantly; i.e., they produce a steep voltage gradient in the layer on the membrane (see Fig. 3.13). As discussed, because of the size of the measuring tip, this gradient is attributed to the membrane even though it has no charge carriers. As the local potential in the layer increases on one side, and decreases on the other, the driving force across the membrane in the ion channels decreases, and the rush-in current slows down; the ’ram current’ quickly produces a negative gradient. (The effect can also be interpreted as the effect of storing charge in the neural $RC$ circuit’s condenser.) The effect measured in [117] is reproduced in our Fig. 3.13. Later, the effect of the sudden change consolidates, and the gradient disappears (similarly to a damped oscillation) in a discharge-like way due to the intense current toward the drain. (The classic picture using fast currents would produce a simple discharge gradient with no AP-like form.)

As discussed, having charge carriers in the proximal layers of the membrane is a non-stationary stage, so the membrane tends to restore its steady state. In the classic model, simple equipotential surface (infinitely fast current) is assumed to provide only a static picture of the neuron. Our model uses slow current which can provide a dynamic picture: our equations can describe the time course of concentration and potential inside and outside the neuron.

Notice that our interpretation and equations excellently and naturally describe also the currents propagating without an external voltage, among others the axonal current and the membrane’s current. The AP arrives at the beginning of the axon in the form of a traveling wave of a slow current (an ion packet delivering ions). Recall that ions move in the ”skin” layer on the membrane, and they continue their way in the axon’s internal surface, creating a similar skin on the internal surface of the tube. There is really no ion current in the volume of the axon, as the classic physiology observed. However, the current is delivered in the atomic “skin” on the internal volume of the axonal tube, in full conformance with the laws of electricity, combined with the laws of thermodynamics.

The mechanism of the current transmission is the one we described above. We can subdivide the ion packet into $n$ pieces, and we choose a $dt$ time such that each piece travels a distance $v\ast dt$. That means that the pieces ”jump” in the position of their neighbor to the end of the time slot $dt$. The mutual repulsion is unbalanced at the edge of the spike (and recall that the rising edge of the current is exponential). The uneven distribution of ions in the first piece in the spike and the one immediately in front of it means a gradient. The ions are not in a stationary state and the forces due to the concentration and voltage gradients act in the direction of the spike propagation. The two gradients represent a driving force (see Eq.(2.9)) that moves the volume element to the position of the neighbor immediately in front of it. The case is described by Newton’s first law: the gradient acts on the charges by the force as described by Eq. (2.30). Given that the ($n-1$)-th element also moves with speed $v$, it leaves an empty volume element behind, so the gradient due to ions in the ($n-1$)-th element ”pulls” the ions after the rest of the spike. The potential-assisted speed is by orders of magnitude lower than the speed of the electric interaction, so the axonal current propagates in the tube at the potential-assisted speed. The charges can be observed as the potential they generate propagates along the axon and different changes [95] are accompanied to the primary change that the ions keep the maximal possible distance from each other while they are moving with a macroscopic speed $v$ along the tube (they cannot exit the tube). The electric repulsion of ions causes the observed travelling waves.

Notice that it is not a classic longitudinal current under the effect of some external potential: charge and voltage gradients represent an internal driving force. A very viscous electroscatic fluid represents the current where the ions do not lose their potential energy. (The classic modell for axonal charge propagation assumes a periodically changing in- and outflow of ions in connection with propagating a $10ms$ long spike at $10m/s$ speed requires the ion channels at distance of $1mm$ to concert the actions: at what rate to pump ions in at the beginning and the end to appropriately adjust the pumping intensity to accomodate to the spike’s current intensity at the places of the channels; given that the total charge delivered by the spike and the shape of the spike remains the same during the axonal delivery.)