| The synapse is at the heart of signal transfer between neurons. Multiple biological mechanisms are responsible for this unidirectional exchange of electrical signals. Their actions affect the shape of the transfered impulse usually referred to as postsynaptic potential. These potentials travel along the neuron's membrane, from the synaptic connections to the cell body. There, signals arriving at different times are superposed. When this superposition reaches a threshold value, the neuron fires an action potential through its axon. This action potential will itself trigger other synaptic transmissions.;The model is presented according to four successive steps. The first one is the release of the neurotransmitters from a vesicle on the presynaptic end. Following their secretion, the particles diffuse into a three-dimensional cleft. Once they have diffused, they reach the postsynaptic membrane on which a specific zone contains the totality of the receptors. If they fall into that zone they will certainly encounter a receptor, otherwise they are absorbed. In order to be in an active state, a receptor requires the fixation of two neurotransmitters. Following the activation, the postsynaptic membrane will be depolarized according to its electrical properties and the opening properties of the receptor.;The steps of release, diffusion and reception at the membrane are associated with different stochastic processes. For instance, the release initializes the random positions and velocities of the particles. They escape the vesicle according to a Poisson process.;Three diffusion processes are presented. The first one is based on an integrated Ornstein-Uhlenbeck process with random initial conditions. The second one includes a radial stochastic oscillator and the third includes a coloured noise instead of the usual white noise. The reception step depends on the release and the diffusion. Indeed, it defines the first passage time to the postsynaptic end. Furthermore, the probability of reaching the receptor zone is also established by this step. The activation of a receptor is a deterministic step as it depends solely on the electrical properties of the membrane and the receptor.;The aim of this research is to establish a mathematical model of such a chemical synapse in order to find an analytical function for the postsynaptic potential. This equation would be particularly useful in the field of artificial neural networks such as the spiking neuron and the radial basis function network. Moreover, the mathematical model itself would be of interest in the field of neuropharmacology and in cognitive sciences.;For each step, a theoretical framework is established. From this framework, simulations are performed in order to obtain the postsynaptic potential curves. Statistical analyses are performed on the simulations in order to determine how the model's parameters affect the potential curves. These analyses enable us to simplify the model and obtain an analytical representation of the postsynaptic potentials.;With a biologically plausible set of parameters, the simulations produce curves that are similar to the ones measured and published in earlier papers. Furthermore, the curves obtained with the analytical model fit the ones produced by the simulations. Thus, the aim of the research has been reached. |
