2.8.6 Axon and axonal arbor ‣ 2.8 Neuronal operation ‣ Chapter 2 Physics for neurons ‣ Dynamic Abstract Neural Computing with Electronic Simulation (DANCES) Version 0.2.5

We model the axons as electrolyte-filled semipermeable membrane tubes with ion channels in their walls. The axons do not passively follow the potential’s time course, but they mediate the changes in their internal volume by using an ion pool available in their extracellular volume. The applied potential (including that of the mediated ions) opens the ion channels in the axon’s wall.

In their native mode of operation, the three modes of ion channels define the ’direction of the time’ [62, 127, 26] (the direction of the current that transmits the spike). The layer that the front of the spike creates on the surface (on both sides of the tube) propagates in both directions, but it cannot open the ion channels on the side where the spike arrived from, and the ion channels are still inactivated.

Clamping sets up an artificial working regime for the ion channels: the permanent electric field on the outer surface enables ions to enter the inner volume where formerly no ions (and no potential) existed. The rushed-in ions will flow away from the place of their entrance (recall that the current removes part of the ion layer on the surface), and a slow current toward the membrane can start. Under clamping conditions, the experimenter sets the voltage instead of the transmitted signal and in a static way instead of an autonomous dynamic one.

Initially, the membrane, the clamping point on the axon, and the intracellular and extracellular fluid maintain their resting potential. When an external potential is applied suddenly to some point of the axon, an electric field $\frac{dV}{dx}\propto(V_{membrane}-V_{clamp})$ appears on the outside surface of the axon. The extracellular space with its high ion concentration $C_{k}^{ext}$ represents an ”ion cloud”.When the clamping voltage is switched on, a “fast” current instantly delivers the potential along the outer surface of the axon. However, this is not the case (at least not in the initial moment) on the inner surface. There is no charge present that could change the potential: ’the intracellular concentration at rest is around five orders of magnitude less than that in the extracellular space’ [24]. The physical picture that the clamping potential instantly appears at the end of the axon at the membrane (i.e., if (apparently) they have an infinitely large propagation speed) is valid only if charge carriers exist in the axon.

As described above, the charge gradually increases the potential along the axon (starting from the position of the clamping electrode) until the clamping potential reaches the axon’s end at the membrane. (We could see the effect when measuring voltage instead of conductance on the axonal tube instead of the membrane, shown in our Fig. 3.24.) At that point, the driving force gradually disappears: the potential at the end of the axon and that on the membrane becomes the same. The macroscopic streaming of ions inside the tube only slightly complicates the process: the local internal concentration can saturate only later, given that part of the inflowing ions is delivered to another place within the axon. Notice that the current (and the voltage) on the axon increases in the function of the time exponentially instead of linearly or step-wise, which would be expected when assuming instant interaction or no “slow” macroscopic current.

In this model, we assume that during the time $d t$ , in the volume $d x$ , we have a constant ion inflow $I_{wall}$ through the axon’s wall, which increases the charge and concentration already in the volume. The charges in the tube experience the field $\frac{dV}{dx}$ , and they move with speed $v$ inside the tube (see Eq. (2.28)). The ionic fluid with velocity $v$ (proportional to $\frac{dV}{dx}$ ) transfers the ionic charge in the volume to the neighboring element at a distance $v*dt$ , and delivers the charge and concentration from the neighboring element at a distance $-v*dt$ into this element. At the time $t$ , the concentration at $x$ will result from the inflow at the place $x-v*t$ (see also the general discussion around Eq. (2.2)).

As described above, the charge gradually increases the potential along the axon (starting from the position of the clamping electrode) until the clamping potential reaches the axon’s end at the membrane. (We could see the effect when measuring voltage instead of conductance on the axonal tube instead of the membrane, shown in our Fig. 3.24.) At that point, the driving force gradually disappears: the potential at the end of the axon and that on the membrane becomes the same. The macroscopic streaming of ions inside the tube only slightly complicates the process: the local internal concentration can saturate only later, given that part of the inflowing ions is delivered to another place within the axon. Notice that the current (and the voltage) on the axon increases in the function of the time exponentially instead of linearly or step-wise, which would be expected when assuming instant interaction or no “slow” macroscopic current.

The unusual physical situation in making electric measurements in biological systems is that, in the metallic half of the circuit, the electrode at the membrane (and, if being equipotential, the membrane itself, too) takes ”instantly” the external voltage. However, in the biological half of the circuit, the voltage $V$ at the end of the axonal tube initially remains the same: inside the tube, there is no charge around to produce a potential (actually, without charge inflow, it is a piece of insulator).