## 5 Damping of Plasmon Oscillations in Ionic Systems
#### Exact Inclusion of the Lorentz Damping to the Attenuation of Ionic Dipole Surface Plasmons
In order to solve the problem of the saltatory conduction we develop a new and original model for this mysterious conduction basing on the kinetic properties of collective plasmon-polariton modes propagating along linear and periodically modified electrolyte system, which in the case of an axon is the thin cord of the nerve cell periodically wrapped by Schwann cells creating periodic and relatively thick myelin sheath (Schwann cells myelinate axons in the peripheral system, whereas in the central neural system the white substance is built from axons myelinated by the oligodendrocyte cells [9; 11; 12]). Plasmon-polaritons were investigated and understood based on the well-developed domain of plasmonics [15; 16], especially of nano-plasmonics applied to long range low-damped propagation of plasmon-polaritons along metallic nano-chains [17]. The main properties of these collective excitations, occurring on the conductor/insulator interface [18; 19] by hybridization of the surface plasmons (i.e., the electron density fluctuations on the metal surface) with the electromagnetic wave, are as follows: (1) much lower velocity of plasmon-polariton in comparison to the light velocity, which gives much shorter wavelength of plasmon-polariton than the light wavelength with the same frequency, (2) the related strong discrepancy of momenta of plasmon-polaritons and photons with the same energy causes that the external electromagnetic wave does not interact with plasmon-polariton, i.e., photons cannot be excited or absorbed by plasmon-polariton due to momentum conservation constraint, (3) all the electromagnetic field associated with propagation of plasmon-polariton is compressed to the tunnel-volume of the chain, especially to the space between chain elements,  
Figure 5: Schematic presentation of the cable model of nervous signal transduction in dendrites and in non-myelinated axons [9; 12]; a capacity across the neuron membrane, \(c_{m}\), together with across-membrane resistivity, \(r_{m}\), create a series of coupled RC circuits; consecutive local charging and discharging of the capacitors \(c_{m}\) results in diffusion type [10; 13] transduction of ion current \(i_{i}\) inside the dendrite or non-myelinated axon; because of relatively large longitudinal resistivity \(r_{i(o)}\) of inside (outside) cytosol as well as of the value \(c_{m}\) and \(r_{m}\) across the cell membrane the velocity of the diffusive current is limited to 1–3 m/s [9; 10; 12]. (4) all the radiative losses are quenched and plasmon-polariton attenuation is only due to Ohmic losses of oscillating charge carriers (electrons in metal), which makes metallic nano-chains almost perfect waveguides for plasmon-polaritons, (5) long range and practically undamped propagation of plasmon-polariton is observed experimentally in metallic nano-chains [17; 20]. Plasmon-polaritons in metallic nanostructures are regarded as prospective for future applications in opto-electronics where conversion of the light signal onto plasmon-polariton signal allows for avoidance of the diffraction constraints strongly limiting miniaturization of conventional opto-electronic devices (as the nano scale of electron confinement inconveniently meets with several orders larger scale of the wavelength of light with energy similar to the electrons nano-confined) [19; 21]. Some of the unique properties of plasmon-polariton in metallic nano-chains are of high interest as they can be repeated in periodic linear alignments of electrolyte systems with ions instead of electrons as charge carriers. According to larger mass of ions in comparison to electrons and lower concentration of ions in electrolytes than of electrons in metals, the plasmon resonances in finite ionic systems (e.g., liquid electrolyte confined to the finite volume by appropriately formed membranes frequent in biological cell structures) occur in a different to metals scale of micrometers instead of nanometers and for several orders lower frequency (depending on the ion concentration).