Description and numerical study of a kinetic Fokker-Planck equation in neuroscience

The well-known Integrate and fire model in a partial differential equation form has been at the center of many mathematical developments since Brunel's work and the seminal paper by Caceres, Carrillo and Perthame in 2011. Its descriptive shortcomings such as the inability to see sub-threshold oscillations or to obtain resonances gave birth to the resonate and fire model (Izhikevich, 2001), phenomenologically complex enough but computationally not costly.
However, there has been no deep mathematical study of it yet. In this work, we first establish a PDE corresponding to the mean-field limit of a population of resonate and fire neurons.
The obtained formulation corresponds to a non-linear kinetic Fokker-Planck equation, with a non-local linearity and a measure source term, studied on a half plane. Even though the obtained operator has properties of hypoellipticity, the theoretical study is tedious, encouraging the pursuit of a numerical study to obtain information about the behaviour of the solutions.
We thereby will describe the positivity and mass preserving finite differences scheme of experimental order one we developed, which allows us to observe all the properties we were expecting from the original single neuron model, and even giving birth to some conjectures which we shall detail.