## 7 Plasmon-Polariton Propagation in Linear Periodic Ionic System
The plasmon-polariton kinetics in the periodic structure of the axon may be a convenient explanation for high performance nerve signaling by providing quick signal propagation despite low ordinary conductivity for the practically unlimited range when the energy is persistently supplied to cover only relatively small plasmon-polariton Ohmic losses (lower than for ordinary current). Moreover, plasmon-polaritons do not irradiate any energy. The energy supply is provided residually by the ATP/ADP mechanism in neuron cell, which energetically contributes to the signal-dependent opening and closing of \(Na^{+}\) and \(K^{+}\) channels in the Ranvier nodes and then to the restoration of the steady-state conditions, i.e., via the active transport of ions across the membrane against the ionic concentration gradient which needs the external energy supply. Moreover, the coincidence of the micrometer scale of the axon's periodic structure of Schwann cells (of approximately 100 \(\upmu\)m in length) with the typical requirements for the size of ionic chains supports the suggested plasmon-polariton concept as the explanation of the transduction of the action potential along the axon in agreement with the tentative quantitative fitting of the model. It must also be emphasized that plasmon-polaritons do not interact with external electromagnetic waves, or, equivalently, with photons (even with adjusted frequency), which is a consequence of the large difference in the group velocity between the plasmon-polariton and the photon velocity (\(c/\sqrt{\epsilon}\)). The resulting large discrepancy in wavelength between the photon and the plasmon-polariton of the same energy prohibits the mutual transformation of these two types of excitations because of momentum-energy conservation constraints. Therefore, plasmon-polariton signaling by means of collective wave-type dipole plasmon oscillations along a chain--plasmon-polariton--can be neither detected nor perturbed by the external electromagnetic radiation. This also well fits with neural signaling properties in the peripheral nervous system and in the white myelinated matter in the central system. It is also worth noting that the ionic surface plasmon frequencies are independent of temperature, although the ionic volume plasmon frequencies are temperature dependent, as has been proved in the former paragraph. However, the temperature influences the mean velocity of the ions, \(\
u=\sqrt{\frac{3 kT}{m}}\), thereby causing an enhancement in Ohmic losses with increasing temperature (cf. Equation (26)), which, in turn, strengthens the plasmon-polariton damping. Hence, at higher temperatures, higher external energy supplementation is required to maintain the same long-range propagation of plasmon-polaritons with a constant amplitude. This property is also consistent with experimental observations. Note, however, that energy costly is an active ion transport through the membrane to recover the steady state of each of Ranvier nodes. Only residual part of this energy is sufficient to eventually support and maintain undamped plasmon-polariton propagation, which can be regarded in the scheme of the steady solution of the forced and damped oscillator applied here to plasmon-polariton dynamics similarly as in the metallic chain activated by a coupled external source (e.g., a coupled quantum dot system [48]). Note that the energy supply from the consecutive excited Ranvier nodes continuously covering plasmon-polariton Ohmic losses results also in a convenient narrowing of the plasmon-polariton \(k\)-wave packet shifting it toward the long-wave limit (small \(k\)), which follows from the Fourier picture of the Dirac delta (the larger number of the chain segments contributes to the excitation of the plasmon-polariton, the narrower plasmon-polariton \(k\)-wave packet is formed). This favorable property allows for the precise ignition of sequential Ranvier nodes by unlimitedly propagating relatively narrow plasmon-polariton wave packet triggering consecutive Ranvier nodes.