# An Algorithm for Creating a Synaptic Cleft Digital Suitable for Further Numerical Modeling
## 1 Introduction
In the present study, a glutamatergic synapse is used as a prototype for the creation of a digital phantom (Figure 1). The mechanism of synaptic transmission consists of several steps. Initially, a presynaptic neuron is filled up with synaptic vesicles containing neuromediators (glutamate, Glu). These vesicles are attached to a presynaptic membrane. The presynaptic release machinery comprises mediating membrane fusion proteins (SNARE and Munc18-like proteins), enabling Ca\({}^{2+}\)-triggered fusion proteins (synaptaptagmines and corresponding complexin cofactors) and AZ scaffold proteins, which tether the synaptic vesicles and recruit Ca\({}^{2+}\) channels to neurotransmitter release sites (Rab3-interacting molecules, RIMs; RIM binding proteins; and Munc13-like proteins). The vesicular release machinery is canonical for all types of synapses and neuromuscular junctions, and just one feature distinguishes presynaptic terminals with different specializations--neurotransmitters and matching transporters, which fill vesicles, and synthetizing enzymes (if not glutamate or glycine, as usual cytoplasmic metabolites) [32]. The whole process of vesicular release is generally referred to as SNAR-mediated exocytosis. The released neuromediators diffuse to the postsynaptic membrane, then activate specific membrane receptors (NMDAR and AMPAR, in the considered case). Then, the molecules of Glu move out of the synaptic cleft due to convectional diffusion. This process is accompanied by Glu reuptake through specific transporters (EAAT(i)) on the membranes of astrocytes and neurons [33]. Then, Glu is converted to glutamine (Gln) by glutamine synthetase, as part of the glutamate-glutamine cycle. Gln can be transferred fluently to neurons, where it is converted back to Glu by glutaminase. The physiological functioning of the synapse is strongly regulated by the balance between Glu release, convectional diffusion, and EAAT-mediated reuptake. There are significant experimental evidences proving that the activity of glial glutamate transporters (EAATs) ceases under pathological conditions and aging, and the abundance of glutamate may damage the surrounding nervous tissue and hamper normal brain functions [34; 35]. The density and mutual distribution of EAATs on an astrocytic leaflet (Figure 1) are the main factors determining the balance of released Glu being included into the glutamate-glutamine cycle [35]. This process makes it possible to avoid excitotoxic damage to brain tissue in the CNS. Knowing the precise position of an astrocyte membrane relating to pre- and postsynaptic membranes makes it possible to correctly digitize a native synapse, which provides a powerful tool for the further modeling of the convectional diffusion of Glu during spillover. The introduced algorithm makes it possible to develop a digital phantom describing the properties of the synaptic cleft correctly. A numerical mesh of different parameters can be further applied to the phantom to solve a boundary convectional reaction-diffusion problem. The purpose of this solution is to provide information on how glutamate transporters are distributed on the surface of astrocyte membranes and how they relate to the neurotransmitter pool in the synaptic cleft. It is essential to note that, despite the existence of elaborate approaches for solving different boundary PDE problems, examples considering the application of these mathematical methods to real biological objects remain scarce. The explanation of this gap is the difficulties associated with reconstructing real-scale structures of biological prototypes which are suitable for further modeling. The digital model of the object should be three-dimensional with a clear indication of the boundary's orientation, and the most properties of the initial biological object must be taken into account.