From a control theoretical point of view, the condenser geometry sets the ’always the same’ value and the currents serve to keep or restore that value. Forming membrane’s potential is an excellent example of the principle that nature uses ”one process to build and the other to correct” [72].
After introducing a finite-width membrane into an electrolyte, a potential difference between the two membrane surfaces is created; see Eq.(2.71). When adjusting the membrane’s potential, we must consider its ground and excited states separately, given that the perturbations in these two cases are vastly different. Nature employs various mechanisms to maintain the ground state and recover it after generating a spike. Although nature’s tools are remarkably similar (channels and pumps; here, we discuss only the ’downhill’ channels), the significant difference in current intensity needs a different discussion.
The potential’s and the electric field’s magnitudes depend on the concentration and the geometry (the finite width of the membrane). The ions’ chemical nature comes into play only if the polarizability can differ for different molecules. The thickness of the charged layer has a significant impact on the resulting electric field (i.e., the neuron’s ’setpoint’). If biological fragments are formed and settle down on the membrane’s intracellular side, due to the increased , the resting potential (and the threshold potential) may change, leading to neurological diseases.
Given that the same number of charged ions must be present on the two sides of the membrane, their surface density must be the same. Notice that the effect is purely electrostatic, resulting in an asymmetric ion distribution; no permeability is needed. If the membrane is (slightly) permeable, ions will move across the membrane until equilibrium is reached. The resulting potential difference is directly proportional to the concentration difference.
The classical condenser has an electric field, as the blue dashed diagram line shows in Fig. 2.12. No field is outside or inside the condenser plates (see Fig. 2.11). There is a sudden jump in the field on the conductor’s surface (the armatures) and a constant field between the plates. (The charged layer can be ideally thin if electrons are sitting on the surface but is of finite thickness on the opposite side where ions are on the surface. This latter effect is not discussed in classical electricity.)
The biological condenser behaves differently; see the red diagram line. The electrolyte (instead of an insulator) outside the condenser drastically changes the structure of the field. The electric field on the surface and between the plates changes by a factor of 3 to 4. Outside the plates, the polarization creates a field that changes logarithmically. An ion layer with finite thickness is built near the membrane’s surface. It is uniformly charged, so the field is linear. At the internal surface of the condenser, it takes the value of the field calculated for the gap; on the other side of the layer, it takes the value of the logarithmic field at that position. The dashed line represents the gap field, the continuous line represents the charged layer field, and the dotted line represents the electrolyte field. We can observe the ”correspondence principle”: the particle-continuous fields join seamlessly; furthermore, if the polarizability of the electrolyte decreases toward zero, the field value approaches the classical value.
As Eq.(2.51) shows, the ”thermodynamic electric field” increases as temperature increases, and raises the setpoint. As Eq.(2.35) shows, the current decreases as temperature increases, in line with the experimental evidence. The amplitude of action potential is decreased given that the setpoint increased and its duration is reduced given that the current decreases.
The thickness of the charged layer has a significant impact on the resulting electric field (i.e., the neuron’s ’setpoint’). If biological fragments are formed and settle down on the membrane’s intracellular side, due to the increased , the resting potential (and the threshold potential) may change, leading to neurological diseases.