In volumes containing ions, the ions experience two effects in those two abstractions. When an invasion in the volume happens, electric potential, pressure, temperature, or concentration changes locally; dynamic changes begin to restore its balanced steady state. When the invasion persists, the system finds another steady state. If the invasion is local and affects only one macroscopic parameter, another macroscopic parameter(s) may change at the rest of the locations. An observer experiences that changing one macroscopic parameter of the system causes an unexpected (and unexplainable) local change in another macroscopic parameter. The microscopic world maps the changes from one abstraction to the other. Experimentally, the microscopic world maps the change from the world of electric abstraction to the world of thermodynamic abstraction and vice versa. Theoretically, we can do the exact mapping of macroscopic electrical and thermodynamical parameters using microscopic universal constants.
The phenomenon of invasion called ’electrodiffusion’ means that when a potential gradient is created in a volume with ions (while its thermodynamical parameters, such as its volume and temperature, are constant), it creates a concentration gradient. Conversely, a created concentration gradient creates a potential gradient. In a more inclusive form: in a physical system where multiple forces act on the body (specialized to an ion in electric and thermodynamic fields), dissipating energy and under constraint forces
| (2.7) |
That means when describing an ionic transfer process, we must not separate the electric current from the mass transfer: they happen simultaneously and mutually trigger each other. Furthermore, we must take into account the external force (say clamping), internal friction (moving in viscoseous fluid) and the possible conditional counterforces (such as open/closed ion gates). Notice that the thermodynamic term is ion-specific while the other terms are not. To be entirely balanced, the system must be balanced to all elements. In this way, changing one concentration implicitly changes all other concentrations and the electric field. Similarly, changing the electrical force by adding an external potential, changes the thermodynamic force, and so, the concentrations.
We can describe the equilibrium state (the mutual dependence of the spatial gradients of the electrical and thermodynamic fields on each other) using the famous Nernst-Planck transport equation
| (2.8) |
Unfortunately, Eq. (2.8) also comprises the issue specific for ions: it calculates the not-interpreted partial derivative without taking into account that at least the advection term modifies the local electrical gradient by changing the concentration. Assuming that the concentration is at equilibrium ( = 0; even if its calculation method and its value are questionable), without an external electrical field ( = 0), the change due to diffusion (in a statistical sense) is zero, and the flow velocity is zero (), results in the Nernst-Planck electrodiffusion equation in one dimension
| (2.9) |
describes the equilibrium state (the mutual dependence of the spatial gradients of the electrical and thermodynamic fields on each other). In good textbooks (see, for example, [24], Eq (11.28)), its derivation is exhaustively detailed. In the equation, is the spatial variable across the direction of the changed invasion parameter, is the gas constant, is the Faraday’s constant, is the temperature, the valence of the ion, the potential, and the concentration of the chemical ion. In simple words, it states that the change in concentration of ions creates a change in the electric field (and vice versa), and in a stationary state, they remain unchanged. However, in the classic science there is no way to take into account the field’s propagation speed. We must call attention to a nuance: to keep balance, the concentration and the potential must change in opposite directions. However, since they are implemented by ions, moving an ion’s mass changes the ion’s charge in the same direction. We note one more feature, that since the potential is linearly proportional to the concentration (and so are their derivatives), the function is of form , where, as an exception, the square of the first derivative equals the second derivative: .
It is one of the rare cases when the starting point was wrong, but the conclusion was right. In Eq (2.8), an identical speed for all interactions was assumed. The equation is a rearranged flux equation, where an identical speed for all interactions was assumed. Although the equation is not really applicable for describing the transport of ions (in its practical applications, the identical speed was calculated as a ”mean-field”, where the ”mean” stands for some average of interaction speeds differing by several orders of magnitude), in an equilibrium state, the actual value of both interaction speeds is zero, so they have the same value; furthermore, no advection takes place. This way, the wrong partial derivative is irrelevant.
It is a serious difficulty with using Eq.(2.9) that it relates entirely different quantities, so it is not easy to apply it. There exist attempts to interpret the task of transporting ions under the effect of several interactions with different speeds (for a review, see [114]). However, ”a mean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential” actually means averaging gradients propagating with speeds (electromagnetic interaction) and (ionic current), respectively, which is not appropriate for either (any way of averaging). The computational methods need position-dependent diffusion coefficient profiles, and in addition, they are generally quite limited for most confined regions such as ion channels. For this reason, they have joint issues, limitations, and high computational complexity; furthermore, biophysics [24] explains, ”while diffusion is like a hopping flee, electrodiffusion is like a flee that is hopping in a breeze”. This sentence is the complete mathematical description of a state change. The lack of notion of non-infinite interaction speed does not enable theory to say anything. The theory considers the process as just a momentary ”hop” between two states, although it admits that there are longer and much shorter moments. Classic theory has no idea what to do with non-infinite interaction speeds. This mistake is a significant obstacle, among others, when attempting to comprehend how the electrochemical charge handling implements neuronal computation and information transfer, furthermore, the life itself.
In ’ordinary’ physics, where the charge and mass are independent, changing one side changes the other in the opposite direction. In ’non-ordinary’ physics, valid for electrolytes, the two differentials must change in the same direction given that ions’ charge and mass cannot be separated. Correspondingly, when initiating an AP in a neuron, the rush-in of ions increases both concentration and potential simultaneously. Moreover, to restore equilibrium, the gradients of the two macroscopic parameters must have the same sign, since they refer to the same ion. The thermodynamic force acting on an ion is added to the electrical force. The two are inseparable; the magnitude of the force is ’falsified’.