Parameter estimation in generalized fractional neuronal models

Abstract

We investigate a generalized stochastic fractional neuronal model combining fractional dynamics with correlated stochastic inputs. The proposed framework is described by a fractional differential equation driven by a latent stochastic process with stationary increments and mean-reverting structure. This formulation allows the inclusion of both short-range and long-range dependence structures and naturally produces non-exponential relaxation phenomena. The main goal is the development of a feasible parameter estimation procedure based on discrete observations of the neuronal state process. We propose a two-step methodology. First, the parameters governing the fractional dynamics are estimated by exploiting the asymptotic behavior of Mittag-Leffler functions near the origin. Subsequently, the latent stochastic input is reconstructed through fractional differentiation techniques, allowing the estimation of the parameters governing the hidden noise dynamics. We derive quantitative error bounds for the estimators and analyze the reconstruction error of the latent process under suitable regularity assumptions on the driving noise. In particular, the interplay between the order of the fractional derivative and the H\"older regularity of the noise process naturally emerges in the stability analysis of the reconstruction procedure. Finally, simulation studies illustrate the applicability of the proposed methodology and highlight the influence of memory effects and noise regularity on the quality of statistical inference. The results support the relevance of fractional stochastic analysis for the modeling and inference of neuronal systems with memory and correlated inputs.