Jump-Diffusion processes as models for neuronal activity.

Aiming at an improvement of the existing neuronal models, we consider a mixed process ensuing from the superposition of a continuous diffusion and of Poisson time-distributed sequence of impulses and focus our attention on the firing time. We consider three different instances: the large jumps model in which each jump causes the neuron firing, the reset model characterized by jumps towards the resting potential and a more general model with constant amplitude excitatory and inhibitory jumps. By resorting to analytical argument and to numerical computations the main differences of the considered models are outlined.