Inference on the effect of non homogeneous inputs in Ornstein-Uhlenbeck neuronal modeling

A non-homogeneous Ornstein-Uhlembeck (OU) diffusion process is considered as a model for the membrane potential activity of a single neuron. We assume that, in the absence of stimuli, the neuron activity is described via a time-homogeneous process with linear drift and constant infinitesimal variance. When a sequence of inhibitory and excitatory post-synaptic potentials occurres with generally time-dependent rates, the membrane potential is then modeled by means of a non-homogeneous OU-type process. From a biological point of view it becomes important to understand the behavior of the membrane potential in the presence of such stimuli. This issue means, from a statistical point of view, to make inference on the resulting process modeling the phenomenon. To this aim, we derive some probabilistic properties of a non-homogeneous OU-type process and we provide a statistical procedure to fit the constant parameters and the time-dependent functions involved in the model. The proposed methodology is based on two steps: the first one is able to estimate the constant parameters, while the second one fits the non-homogeneous terms of the process. Related to the second step two methods are considered. Some numerical evaluations based on simulation studies are performed to validate and to compare the proposed procedures.