## 7 Plasmon-Polariton Propagation in Linear Periodic Ionic System
 \tag{43}\]  
Similarly as in the case of metallic nano-chains [38], \(ImF_{\alpha}(k,\omega)\equiv 0\) (for \(\alpha=z,x(y)\)), which indicates the perfect quenching of the irradiation losses at any sphere in the chain (it means that to each sphere the same amount of energy incomes from other spheres as outflows due to the Lorentz friction). One can easily verify this property as the related infinite sums in Equation (43) can be found analytically [38; 39]. Equation (42) is highly nonlinear with respect to the complex \(\omega\) and can be solved both perturbatively in the analytical manner [27] or numerically--more accurately [40]. The determined solutions for \(Re\omega\), \(Im\omega\) (i.e., for self-frequency and damping of plasmon-polariton) can be applied to the model of axon as the effective chain of electrolyte spheres with adjusted ion concentration to the real neuron parameters. The problem of propagation of the plasmon-polaritons along metallic nano-chains was developed, among others, by [PERSON] [25; 41] in order to assess the radiative losses of plasmon-polariton in metallic nano-chains in agreement with former observations related to radiation losses in one-dimensional and two-dimensional crystals [42]. The resonance frequency (\(Re\omega(k)\)), its \(k\)-derivative, i.e., the group velocity and the attenuation rate (\(Im\omega(k)\)) of dipole plasmon-polariton modes numbered by the wave vector \(k\), derived by solution of Equation (42) [27; 40] for ionic chains with exemplary concentration, chain size and ion parameters are plotted in Figure 7.  
Plasmon-Polariton Model of Saltatory Conduction--Fitting of Plasmon-Polariton Kinetics to Axon Parameters  
The bulk plasmon frequency \(\omega_{p}=\sqrt{\frac{q^{2}n4\pi}{m}}\) (we assumed for the model the ion charge \(q=1.6\times 10^{-19}\) C and the ion mass \(m=10^{4}m_{e}\), where \(m_{e}=9.1\times 10^{-31}\) kg is the mass of an electron) and for \(n=2.1\times 10^{14}\)\(1/\)m\({}^{3}\) concentration we obtain the Mie-type frequency for ionic dipole oscillations, \(\omega_{1}\simeq 0.1\frac{\omega_{p}}{\sqrt{3\epsilon_{1}}}\simeq 4\times 10^{6}\)\(1/\)s, where the relative permittivity of water is \(\epsilon_{1}\simeq 80\) for frequencies in the MHz range [43] (although for higher frequencies, beginning at approximately 10 GHz, this value decreases to approximately 1.7, corresponding to the optical refractive index of water, \(\eta\simeq\sqrt{\epsilon_{1}}=1.33\)). For the thin axon cord and thus, in the discrete model, strongly prolate inner ionic cord segment, we reduce the longitudinal Mie-type frequency by a factor of 0.1 [44; 45] compared with the isotropic spherical case. Let us emphasize that the axon consists of a cord with a small diameter of \(2r\), and this thin cord is wrapped with a myelin sheath of a length of \(2a\) per segment; however, for the effective model, we consider fictitious electrolyte spheres of radius \(a\). Thus, the auxiliary concentration \(n\) of ions in the fictitious spheres corresponds to a concentration of ions in the cord of \(n^{\prime}=\frac{n4/3\pi n^{3}}{2\pi r^{2}}\), which yields a typical concentration of ions in a nerve cell of \(n^{\prime}\sim 10\) mM (i.e., \(\sim\)\(6\times 10^{24}\)\(1/\)m\({}^{3}\)). This is because all ions participating in the dipole oscillation correspond in the sphere model to a much smaller volume in the real system, that of the thin cord portion (the insulating myelinated sheath consists of a lipid substance without any ions). The insulating, relatively thick myelin coverage creates the periodically broken channel (corrugated conductor-insulator structure) required for plasmon-polariton formation and its propagation. To reduce the coupling with the surrounding inter-cellular electrolyte and protect against any leakage of the plasmon-polaritons, the myelin sheath must be sufficiently thick, much thicker than what is required merely for electrical insulation.