## 7 Plasmon-Polariton Propagation in Linear Periodic Ionic System
Moreover, to accommodate the conductivity parameters that we calculated for a spherical geometry to the highly prolate geometry  
Figure 7: The exact solutions for the self-frequencies of the longitudinally and transversely polarized modes of plasmon–polaritons in an ionic chain (\(\omega\) in units of \(\omega_{1}\)) obtained by exactly solving Equation (42) at 1000 points in the region \(kd\in[0,2\pi)\) **(left)** and the corresponding group velocities for both types of polarization (**center)**; the functions \(ImF_{z}(k;\omega=\omega_{1})\) and \(ImF_{x(y)}(k;\omega=\omega_{1})\) for infinite chains of electrolyte spheres of radius \(a\) at separations of \(d=3a\), \(4a\), and \(6a\), the shift of the singularities toward band edges with lowering \(d/a\) is noticeable **(right)**. of the real oscillating ionic system, we must account for the fact that the Mie-type frequency of the longitudinal oscillations must be significantly lower than that for a sphere with a diameter equaled to the elongation axis. As a rough estimate, we assumed a correction factor of 0.1 [44; 45]. For the resulting Mie-type frequency, \(\omega_{1}\simeq 4\times 10^{6}\)\(1/\)s, one can determine the plasmon-polariton self-frequencies in a chain of spheres of radius \(a=50\)\(\upmu\)m (for a Schwann cell length of \(2a\)) and for small chain separations of \(d/a=2.01,2.1\), and \(2.2\) (corresponding to Ranvier node lengths of 0.5, 5, and 10 \(\upmu\)m, respectively) within the approach presented above (via solution of Equation (42)). The derivative of the obtained self-frequency with respect to the wave vector \(k\) determines the group velocity of the plasmon-polariton modes. The results for the model system (as in Figure 6) are presented in Figure 8. We observe that for the ionic system parameters listed above, the group velocity of the plasmon-polaritons reaches 100 m/s for longitudinal mode with polarization suitable to the prolate geometry assuming that the initial post-synaptic action potential or that from the synapse hillock excites predominantly longitudinal ion oscillations. Note that for \(\omega_{1}=4\times 10^{6}\)\(1/\)s and \(a=50\)\(\upmu\)m, the light-cone interference conditions \(kd-\omega_{1}d/c=0\) and \(kd+\omega_{1}d/c=2\pi\) are fulfilled for extremely small values of \(kd\) and \(2\pi-kd\), respectively (of the order of \(10^{-6}\) for \(d/a\in[2,2.5]\)), and thus are negligible with regard to the plasmon-polariton kinetics under these conditions (though were important for metals). The related singularities on light-cone induced by the far- and medium-field contributions to the dipole interaction are pushed to borders of the \(k\) wave vector domain and thus are unimportant in the case of considered ionic system. Hence, the quenching of the radiative losses (i.e., the perfect balance the Lorentz friction in each sphere by the radiation income from other spheres in the chain) for the plasmon-polariton modes in the axon model occurs practically throughout the entire \(kd\in[0,2\pi)\) region. Additionally, the above-mentioned singularities [40] are characteristic of infinite chains and therefore cannot fully develop because the nerve model electrolyte chains are of a finite length, whereas other effects, as quenching of irradiation losses, occur for finite chains due to very fast convergence of sums in Equation (43) with denominators \(m^{2}\) and \(m^{3}\) (practically, the chain consisting of only 10 elements exhibits almost same properties as the infinite one). Although the ionic system chain model for a myelinated axon appears to be a crude approximation of the real axon structure, it can serve for the comparison of the energy and time scales of plasmon-polariton propagation implied by the model with the observed kinetic parameters of nerve signals.