Dynamics of an age structured neuron population with the addition of learning processes.

In this thesis I studied a model which shapes the dynamics of a population of neurons modeled through the time elapsed since their last discharge. The system results in a renewal equation with the addition of the spatial extension and some learning processes. The network is assumed to be non-homogeneous, and the Hebbian learning rule counts for the adaptation of the communication channels between the neurons. In the weak interconnection regime it is proved progressively the well-posedness of the problem, the existence and uniqueness of a stationary solution and the exponential convergence of the system to it. The analysis is conducted both in the linear and non-linear case and makes use of common tools of the mathematical analysis combined with more sophisticated instruments as the Doeblin's theory.