## 2 Materials and Methods  
In this paper, we summarise the experimental confirmation of the results obtained from the thermodynamic approach [15]. To do so, we first summarise the previous thermodynamic results and, then, describe the experimental approach used. The living cell membrane is characterised by an electric potential difference \(\Delta\phi\) between the cytoplasm and the extracellular environment, with reference to the environment [26]. This membrane electric potential is related to different permeabilities of some ions (Na\({}^{+}\), K\({}^{+}\), Cl\({}^{-}\), Ca\({}^{2+}\), etc.). Let us consider the differential of the Gibbs free energy \(G\)[27; 28; 29]:  
\[dG=V\,dp-S\,dT+\sum_{i}\mu_{i}\,dN_{i} \tag{1}\]  
where \(V\) is the volume of the system considered, \(S\) is the entropy, \(\mu_{i}\) represents the chemical potential of the \(i\)-th chemical species, and \(N\) stands for the number of particles which flow across the boundary of the system. In relation to the distributions of the different ions, there exists an electric potential differences. In particular, cations (ions with positive charge) accumulate in low electric potential energy regions, while anions (ions with negative charges) present higher concentration at high values of electric potential energy regions [30]. The ion concentrations can be expressed as follows [30]  
\[c_{N}=c_{N,0}\,\exp\left(\frac{N\,e_{N}}{k_{B}\,T}\right) \tag{2}\]  
where \(c_{N}\) is the concentrations related to the number of molecules, \(e_{N}\) is the energy per molecule, \(k_{B}\) (\(=1.38\times 10^{-23}\) J K\({}^{-1}\)) is the Boltzmann constant, \(T\) is the temperature, \(c_{N0}\) is the reference value of \(c_{N}\) at \(e_{N}\) = 0 J molecule\({}^{-1}\), and \(k_{B}T\)\(\approx\) 4 \(\times\) 10\({}^{-21}\) J molecule\({}^{-1}\) for ordinary temperature [30]. The electric potential can be evaluated by using the Goldman-Hodgkin-Katz equation [27; 30]:  
\[\Delta\phi=\frac{R\,T}{F}\,\,\log_{10}\left(\frac{P_{\text{Na}^{+}}\,[\text{ Na}^{+}]_{\text{out}}+P_{\text{K}^{+}}\,[\text{K}^{+}]_{\text{out}}+P_{ \text{Cl}^{-}}\,[\text{Cl}^{-}]_{\text{out}}}{P_{\text{Na}^{+}}\,[\text{Na}^{+ }]_{\text{in}}+P_{\text{K}^{+}}\,[\text{K}^{+}]_{\text{in}}+P_{\text{Cl}^{-}} \,[\text{Cl}^{-}]_{\text{in}}}\right) \tag{3}\]  
where \(P\) is the permeability of the ion, [A] means concentration of the A-ion, \(R\) is the ideal gas constant, \(T\) is the temperature, \(F\) is the Faraday constant, and out stand for outside, while in for inside. This last relation points out how the membrane potential can be changed by alterations in the conductance of one or more ion. The ion channels and transporters provide different permeability to distinct ions, such as Na\({}^{+}\), K\({}^{+}\), and Cl\({}^{-}\). Any change in the ion concentration changes both the membrane electric potential and the related pH of the cytoplasm, because the concentration of a chemical species and the pH can be obtained by the following relations [27; 30]:  
\[c_{out}=c_{in}\,\exp\left(\frac{\Delta\phi}{R\,T}\right) \tag{4}\]  
and  
\[\Delta\phi=\Delta G_{H^{+}}+2.3\,\frac{R\,T}{F}\,\Delta\mathrm{pH} \tag{5}\]  
where \(G\) is the Gibbs potential, \(F\) is the Faraday constant, \(2.3\,\Delta\mathrm{pH}\) is the physiological concentration gradient, and H\({}^{+}\) is the Hydrogen ion which is used by the cells in order to modulate the membrane electric potential by changing the H\({}^{+}\) concentration. These changes in the ion transport modify the biochemical reactions inside the cells (the optic nerve in our analysis), with related modifications of the biochemical processes. Indeed, proteins play a fundamental role in ion transport. Proteins in the cytosol can be modified in their functions by phosphorylation or dephosphorylation. An ion actively crosses the membrane against its electrochemical potential, whereby the necessary energy is derived either from the hydrolysis of ATP, or from the movement of a co-transported, or coupled ion along its electrochemical gradient.