## 4 Geometrical Models of the Bouton
### Geosphere\(-\)The Simplest Model of the Sphere  
The two-dimensional geosphere surface mesh, often used, for example in [18], to depict the surface of a sphere, consists of triangles, which are almost equilateral. Their characteristics exceed the ideal values by 1.3% to 25.5% which constitutes a good reference point for comparison of the quality of 3D meshes. The analyzed surface (Figure 1, left) had 15,680 triangular faces and 7482 vertices. The angles of the triangular mesh considered in Table 1 vary between the values of 48.53\({}^{\circ}\) and 71.98\({}^{\circ}\). The range of triangle area was from \(S_{min}=0.001903\)\(\mu\)m\({}^{2}\) to \(S_{max}=0.002051\)\(\mu\)m\({}^{2}\). Therefore, the ratio \(S_{max}/S_{min}\) was 107.8%. The surface mesh was then submitted as the input data to the TetGen program which produced the tetrahedral mesh of the sphere (Figure 1, right). The parameters of the sphere were as in Table 2. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Name** & **Definition** & **Ideal** & **Min** & **Max** & **Max/Ideal** \\ \hline \(PS2\) & \(P/\sqrt{5}\) & \(2\sqrt[4]{27}\approx 4.55901\) & 4.55903 & 4.6104 & 1.011 \\ \(ER2\) & \(E_{max}/R\) & \(2\sqrt{3}\approx 3.4641\) & 3.4684 & 3.9245 & 1.133 \\ \(EH2\) & \(E_{max}/H_{min}\) & \(2/\sqrt{3}\approx 1.1547\) & 1.1575 & 1.4527 & 1.258 \\ \(MAXE\) & \(E_{max}/P\) & \(1/3\approx 0.3333\) & 0.3337 & 0.3702 & 1.111 \\ \(MINE\) & \(E_{min}/P\) & \(1/3\approx 0.3333\) & 0.2913 & 0.3325 & 1.144 1  \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geosphere model surface mesh quality. Figure 1: The surface (**left**) and the tetrahedral mesh (**right**) of a geosphere. The input parameters for the meshing software were as follows: maximum volume of a single mesh element (tetrahedron) (\(VT_{max}\)). Another two important parameters are the maximum aspect ratio (\(EH_{max}\)) which is not standardized but computed directly as the ratio of the longest edge and the shortest height of a tetrahedron. The last restriction is that of the minimum dihedral angle (\(DA_{min}\)). It should be stressed that the last restriction is the most strictly observed in the program. The 2-dimensional mesh is generated in such a way that each spherical polygon is replaced by a flat polygon with vertices at the same points. The quality of the globe mesh was tested in the following subspace of the above parameters:  
\[EH_{max}\in[4.0;10.0]\]  
\[VT_{max}\in[0.0006;4.0]\]  
\[DA_{min}\in[10^{\circ};19^{\circ}]\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Name** & **Definition** & **Ideal** & **Min** & **Max** & **Max/Ideal** \\ \hline \(SV\) & \(\sqrt{S}/\sqrt[3]{V}\) & \(\sqrt[4]{3}\sqrt[4]{22}\approx 2.6843\) & 2.6844 & 4.7147 & 1.756 \\ \(ER\) & \(E_{max}/R\) & \(2\sqrt{6}\approx 4.8989\) & 4.9301 & 37.4427 & 7.643 \\ \(EH\) & \(E_{max}/H_{min}\) & \(\sqrt{3}/\sqrt[2]{2}\approx 1.2247\) & 1.2398 & 14.628 & 11.944 \\ \(MAXS\) & \(S_{max}/S\) & \(1/4=0.2500\) & 0.2503 & 0.4434 & 1.774 \\ \(MINS\) & \(S_{min}/S\) & \(1/4=0.2500\) & 0.0639 & 0.2498 & 3.912  \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mesh quality of the geosphere 3D model. Figure 2: Wireframe structure of the surface of a 3D globe mesh. Bouton membrane—**green**, neurotransmitter release area—**blue**, supply zone boundary—**red**. The example results of these tests are presented in Figure 3:  
1. for the fixed values of \(EH_{max}=5\) and \(VT_{max}=0.001\) and for \(DA_{min}\) ranging from \(10^{\circ}\) to \(19^{\circ}\);
2. for the fixed values of \(EH_{max}=5\) and \(DA_{min}=13^{\circ}\) and for \(VT_{max}\) ranging from \(0.0006\) to \(0.0015\). The best quality was achieved when the imposed restrictions were: \(EH_{max}=10.0\), \(DA_{min}=13^{\circ}\) and \(VT_{max}=0.0012\). The input 2D mesh and the results of its tetrahedralization are presented in Tables 3 and 4.