## 9 Effect of Mesh Quality on the Algorithm Performance and Results of Simulations  
The algorithm, basically, consists of two modules. The first one is responsible for calculations initialization. Among others, it initializes the vector of the unknown variables i.e., the density vector \(\rho\). The second one performs iterative calculations over time and space mesh. In the second module, the iterative one, the calculations in each iteration are divided into three steps: the first one, during which the equation system is solved without taking release and synthesis into consideration, the second one when the values of the vector \(\rho\) are updated with the synthesis, and the third one, that involves the diminishing the value of the neurotransmitter density by the amount resulting from the exocytosis. Within the second said step, solving the equation system must be repeated until the status of each tetrahedron\(-\)\"producing\" or \"non-producing\"\(-\)is established between iterations. This means, among others, that calculations on a single tetrahedron may cause the program get stuck in the loop in case when the tetrahedron has one vertex in the release zone and another one in the synthesis zone. Such vertices exist in the worst, unimproved mesh in the real model. In general, the comparison of calculations for different meshes is a difficult task. The reason is that vertices of different meshes covering the same object, are usually not identical. Therefore it is not possible to compare results obtained in the same mesh points. In the case of modeling the dynamics of transmitting signals in synapse, however, the situation is convenient because the total release of neurotransmitter into synaptic cleft in the time unit is a direct effect of the run of the processes occurring in the presynaptic bouton. This directness is guaranteed by the fact that the presynaptic bouton, considered as a cybernetic system, has the observability property [6]. Referring to the conducted simulations, the phenomena mentioned above are visible clearly. First of all, for the worst unimproved mesh, the calculations with the synthesis turned on got stuck in a loop because of existence of tetrahedra that intersect both the release area and the synthesis area. Such an example tetrahedron is marked bright line in Figure 7.  
In the case of simulation without synthesis, the total amount of neurotransmitter released into the synaptic cleft differs by a factor of around two, for the worst mesh (Mesh 1) compared to the middle-quality one (Mesh 2) and the best mesh (Mesh 3) which both give practically the same result\(-\)see Figure 13 and Table 11. The remarkable difference in the simulated amount of released neurotransmitter means, among others, that even in the case of a simple dynamic model\(-\)diffusive-reactive PDE model\(-\)the use of a mesh of a poor quality produces results which do not reflect the real solutions. It must be admitted that the shape of the impulse is maintained, and that can be confirmed by comparing impulses conducted via calcium channels measured by other authors [2,3] with the result of our research. In all the papers, including this one, the impulses were V-shaped. It should be also noted that the amount of the released NT, used in this paper, is a very accurate indirect measure of the postsynaptic potential (see Figure 1A in the related work [3]). The peak of the impulse, however, is higher for the worst mesh (Figure 13) which would lead to false conclusions. In genetic research this could even favor the object which produces higher output amplitude as more heterozygous since homozygosity usually means impairing life processes (see a related paper [4], especially Figure 3C therein). The performed simulations, furthermore, shed\(-\)to some extent\(-\)light on the problem of the mesh quality in the context of dynamical numerics applications.