2.5.7 Ions’ dynamics

Notice an important difference. The acceleration of an ion is unbelievably large. It is sufficiently large to keep the potential at the same value, provided that the ions must follow a small change, such as due to the ”leaking” current through the AIS. Similarly, the ions can ’instantly’ follow quick changes such as a square wave gradient. However, if many similar ions are ahead, their repulsion decreases the acceleration, and the ion travels only at a few $m/s$ speed. The huge forces and accelerations means that a potential change acts immediately. However, the force decays quickly. The ions start to move ’instantly’, but the charge carriers can move only with a limited speed, much below the interaction speed of EM interactions. The effect can propagate only with that lower speed.

See the case of axon: there exist a mechanical constraint that the ions cannot spread through the wall (they must keep the direction), and the ion package propagates as Equs.2.13 and 2.16 describe it.

As we described in section 2.4.7, this sudden concentration change provokes a voltage gradient as described by Eq. (2.24). From that point on, gradients $\frac{dV}{dt}$ and $\frac{dC}{dt}$ excite each other, as described by Eqs. (2.10) and (2.11). In simple physical picture, the rush-in ions appearing on the neuron’s membrane are confined in the volume formed in an atomic layer on its surface. They attempt to be uniformly distributed. The electric repulsion enables a hyper-viscous behavior for the electric fluid, so the surface – without invasion – would be equipotential. The membrane is an excellent isolator, except at the AIS. Here the ion channels represent a resistance (also limits the current), so a current starts, see section 3.7.4, creating a potential gradient.

Here comes into play the limiting effect of the interaction speed. Given that the electric repulsion is mediated by the concentration, the decrease of the potential can happen with the speed of the slow current in the proximity of the AIS and the ions from the distant region can increase it with their potential-assisted speed. The potential gradient creates a speed gradient in proximity of the AIS, AIS, while it remains zero at larger distances. The speed gradient propagates with the speed given by Eq. (2.17) and its delayed effect creates a ”ram current” effect, which – in a good approximation – is resemblant to the effect of an $RC$ circuit. The charge is ”stored” (cannot flow out because of the limiting resistor) for some time and at approprite parameters the neuron behaves as a damped $RC$ circuit.

Let us suppose we have a simple electrodiffusion, say, positive ions rush-in into the intracellular space of a neuron through the membrane’s ion channels, in one packet, about ${10}^5$ ions per channel. For all channels, in the order of ${10}^2$ channels on the membrane, the number of ions appearing (suddenly, in $psec$ time per channel (see section 2.5.7) and in $nsec$ time per membrane) on the surface is only ${10}^7$. This charge appears as current at the beginning of the AIS and moves (the neuron’s membrane discharges) without an external potential. The huge difference in the charge density (and concentration) on the membrane’s surface and the no-charge at the beginning of the axon can explain why a current flows. (The other way round: ions escape into the volume and they repulse each other; so they will move in the direction of the drain).

The Nernst-Planck equation could explain that the concentration gradient causes a potential gradient, i.e., explain why do we see a current without voltage. However, the number of ions is not surely sufficient to apply thermodynamics. At the same time, the ions exist in a volume having thickness about tenths of nanometer, i.e., the density of the ions can be sufficient to behave as a macroscopic charged fluid in that limited volume. This case is neither ”net” macroscopic nor microscopic. On the one side, the Nernst-Planck equation refers to large (on this scale) volume. On the other, evidence shows that dozens of $pA$ current flows, and behaves in a macroscopic way.

When we assume ${10}^7$ rushed-in ions for evoking a single action potential, it means $1.6\ast {10}^{-12}Cb$ charge. We take the typical resistance and capacitance values from [24]: we assume $100pF$ capacitance and $100M\mathrm{\Omega }$ resistance (i.e., $\tau ={10}^{-2}s$). If we assume that the charge flows out in a period of $10ms$, we should measure a $160pA$ current on the axon, furthermore, that current causes a $160mV$ voltage drop (aka Action Potential) on the resistance. See also section 3.9.3: $300pA$ flows in a $5ms$ period. Similar value can be derived from [117]. All those values are in the order of the measured values.

So, the charge from the micro-world can be excellently mapped to the measured macroscopic current. On the other way round, due to the Nernst-Planck equation (2.9), it could be expressed as a consequence of the sudden increase in the concentration (a concentration gradient $\frac{dC}{dx}$) of the chemical ions on the surface. Provided that thermodynamics can be applied to a volume of area of ${10}^{-2}m{m}^2$ and thickness say up to $10\ast {10}^{-9}m$. With those numbers we arrive at that in that layer the ion density during generating an action potential is ${10}^{23}{m}^{-3}$, which is not far from the numbers where thermodynamics can be applied; so we assume that the Nernst-Planck equation (and the ones we described as ”laws of motion of electrodiffusion”) can be used to describe why an AP evokes in a neuron.

We assume that in a balanced state, a $N{a}^{+}$ and a $K^{+}$ ion are placed at the entrance of the ion channel. According to the calculations above, $N{a}^{+}$ ions pass the ion channel in $15ps$. At the same time, since the mass of a $K^{+}$ ion is $m_K=6.492\ast {10}^{-26}kg$, and the same force accelerates it, its acceleration

With this acceleration, the $K^{+}$ ions traverse in the the ion channel a distance only $d=\frac{1}{2}\ast 0.246\ast {10}^{14}\ast {(1.54\ast {10}^{-11})}^2=2.92\ast {10}^{-9}nm$. That means that when the $N{a}^{+}$ ions already arrived at the low-concentration layer, at that time the $K^{+}$ ion passed only $\frac{2}{3}\ast d$ distance. If the potential on the arrival side increases due to the already arrived ions, they will experience a repulsive potential, and they will not be able to arrive: they cannot pass the uphill potential barrier and they turn back, with the downhill gradient.

Given the current intensity above, that the ion channels’ size is not much larger than an ion, furthermore that the time available for interaction is shorter than the chemical transition time for the molecules,