Linked Gauss-Diffusion processes for modeling a finite-size neuronal network

A Leaky Integrate-and-Fire (LIF) model with stochastic current-based linkages is considered to describe the firing activity of neurons interacting in a (2×2)-size feed-forward network. In the subthreshold regime and under the assumption that no more than one spike is exchanged between coupled neurons, the stochastic evolution of the neuronal membrane voltage is subject to random jumps due to interactions in the network. Linked Gauss-Diffusion processes are proposed to describe this dynamics and to provide estimates of the firing probability density of each neuron. To this end, an iterated integral equation-based approach is applied to evaluate numerically the first passage time density of such processes through the firing threshold. Asymptotic approximations of the firing densities of surrounding neurons are used to obtain closed-form expressions for the mean of the involved processes and to simplify the numerical procedure. An extension of the model to an (N×N)-size network is also given. Histograms of firing times obtained by simulations of the LIF dynamics and numerical firings estimates are compared.