Dependencies between spike times of a couple of neurons modeled via a two-dimensional LIF model

There exists a large literature on the Stein’s model. However, the largest part of these studies performs a diffusion limit on Stein’s equation to get a mathematically tractable stochastic process. Use of these continuous processes has allowed the discovery of various neuronal features that are hidden in the original Stein’s model, for instance the stochastic resonance. In this work, we consider a diffusion limit of two or more neuronal dynamics governed by Stein’s model to describe dependencies between their spike times. For this reason, we separate the PSPs impinging on each neuron into two groups, one with the PSPs coming from a common network and the other one with those typical of the specific neuron. We study the diffusion limit of these equations, superimposing a common threshold S and we describe the interspike intervals as first passage times of the bidimensional diffusion processes through the boundary. The introduced dependency between the two Stein’s processes is maintained in the diffusion limit. The aim of this work is to relate the introduced dependencies on the processes with those obtained on the spike times of the two neurons, through their joint law.