## 7 Plasmon-Polariton Propagation in Linear Periodic Ionic System
### Possible Linkage with the Topological Model of Neuron Web  
The topology characteristics of entangled filament webs, like neuron webs, may be described in terms of homotopy reflecting the complexity of trajectories of many particle systems [54]. The first homotopy group of the space \(D\) marked with \(\pi_{1}(D)\) is the collection of topologically non-equivalent (non-homotopic, i.e., one cannot be continuously deformed to another) classes of closed trajectories in the space \(D\). If this space is a configuration space of a system of \(N\) particles, each of which runs its own trajectory on a manifold \(M\), then the appropriate \(\pi_{1}\) homotopy group is called a braid group. The configuration space for \(N\) identical particles located on \(M\) manifold (e.g., \(R^{n}\)) is defined as follows, \(F_{N}(M)=M^{N}\setminus\Delta\), for distinguishable identical particles; \(M^{N}\) is a \(N\)-fold normal product of \(M\) manifold, \(\Delta\) is a set of diagonal points (where the coordinates of two or more particles coincide), which has to be subtracted in order to conserve the number of particles in the system (particles cannot hide one behind another). The braid group is the first homotopy group [51; 54]\(\pi_{1}\) of the configuration space of a system of \(N\) particles. Braid groups represent topologically different available trajectories of a system of \(N\) particles (not referred to any specific dynamics). The pure braid group which we will utilize to neuron web is thus defined as  
\[\pi_{1}(F_{N}(M))=\pi_{1}(M^{N}\setminus\Delta). \tag{44}\]  
When trajectories (represented as the neuron filaments) are mutually entangled in the multi-line braid form, the initial ordering of particles must be unchanged. By \(l_{ij}\) the elementary generators of the pure braid group can be assigned (Figure 12) [49], they correspond to the simplest entanglement of two trajectories of a particle pair, \(i\)-th and \(j\)-th, while the rest of the particle trajectories remain not-tangled. The generators \(l_{ij}\) can be expressed with the use of \(\sigma_{i}\) (exchanges of neighboring particles \(i\)-th and \(i+1\)-th not conserving their positions [49]),  
\[l_{ij}=\sigma_{j-1}\cdot\sigma_{j-2}...\sigma_{i+1}\cdot\sigma_{i}^{2}\cdot \sigma_{i+1}^{-1}...\sigma_{j-2}^{-1}\cdot\sigma_{j-1}^{-1},\ \ 1\geq i\geq j\geq N-1. \tag{45}\]  
Pure braid group can be considered as an abstract mathematical group generated by \(l_{ij}\) defined by the following relations [49; 52] (cf. Figure 13),  
\[l_{rs}^{-1}\cdot l_{ij}\cdot l_{rs}=\left\{ \begin{array}{ll}l_{ij},&i<r<s<j \\ l_{ij},&r<s<i<j\\ l_{rj}\cdot l_{ij}\cdot l_{rj}^{-1},&r<i=s<j\\ l_{rj}\cdot l_{sj}\cdot l_{ij}\cdot l_{sj}^{-1}\cdot l_{sj}^{-1}\cdot l_{rj}^ {-1},&i=r<s<j\\ l_{rj}\cdot l_{sj}\cdot l_{rj}^{-1}\cdot l_{sj}^{-1}\cdot l_{ij}\cdot l_{sj} \cdot l_{rj}\cdot l_{sj}^{-1}\cdot l_{rj}^{-1},&r<i<s<j.\end{array} \right. \tag{46}\]One can propose to identify a fragment of the neuron web connected via active synapses predefined by a prior record of some information in filament web as some pattern from the pure braid group. In particular, one can use the binary code to record in a three-filament bundle of any message (two generators of the corresponding \(N=3\) pure braid group serve to code 0 and 1 bits and the third generator encodes the end of a word). It is possible to code of \(N\)-letter alphabet message in \(N\)-order pure braid group [53]. We have demonstrated that the ratio of an information efficiency of \(N\)-order pure braid group to resources needed to organize a physical web (like a neuron web with \(N\) entangled filaments) attains its maximum at \(N=20\)-\(30\), which agrees with the number of phones in majority languages [52]. This might suggest that words are cadded in relatively small number of filaments in entangled bunches. In the braid scenario we propose to code information in entanglement patterns in pieces of the neuron web in the brain selected by activation of some synapses.