• Mean field theory
• Morris-Lecar model
• Piecewise deterministic Markov process
• Poisson process

Neuroscience has always been an attractive and mysterious subject. In the last years the studies on the brain have increased massively and mathematics has played a key role in this research work.
This thesis project is an example of how mathematics is central to the comprehension and the modelling of biological processes (in this specific case neuronal processes).

This work is organised into three chapters: the first about a biological description of the neuron, the second about the modelling of the mechanism with which neurons communicate and the last about the analysis of a network of neurons.

More specifically, in the first chapter we describe the anatomy of the neuron and we analyse the mechanisms with which neurons receive and transmit signals from a biological point of view.

In the second chapter we model action potentials: processes which allow neurons to quickly transmit information. We start with the description of the Morris-Lecar model, a deterministic mathematical system developed by Catherine Morris and Harold Lecar to reproduce the transmission of signals in the giant barnacle muscle fiber. We construct then a stochastic version of this one in order to give to this base model a more realistic and random behaviour. To do that we use the piecewise deterministic Markov processes (PDMPs), a class of non-diffusion models introduced by Mark H. A. Davis in 1984. At the end of the chapter we show some simulations of action potentials obtained by using various methods such as the Gillespie method or the thinning.

In the third chapter we analyse a network of neurons. The aim of this part is to study the behaviour of a network of neurons and the spatial propagation of signals. We first introduce a network of interconnected neurons, with specific signal cumulation rules and show by numerical simulations how the signal propagates. Then, in the spirit of mean field theory, we prove that an empirical density of neurons converges to the solution of a partial differential equation (PDE), when the number of neurons tends to infinity.