Data-driven parameter estimation in stochastic dynamic neural fields by state-space approach and continuous-discrete extended Kalman filtering

Abstract

This work examines the state and parameter estimation problems for Stochastic Dynamic Neural Field (SDNF) models. Unlike other studies, we suggest a general approach based on the Galerkin method that reduces the SDNF model to a standard state-space representation. This naturally yields nonlinear continuous-discrete stochastic systems with complicated nonlinear dynamics and linear measurements of a high dimension because of a spatial approximation. This in turn paves a systematic way for a strong integration of nonlinear Bayesian filtering methods into mathematical neuroscience. Because of a high resolution utilized to keep an accurate spatial approximation, a key aspect of nonlinear filters to be derived for the SDNF models is their computational budgets. In this paper, we derive the sequential Itô-Taylor-based continuous-discrete Extended Kalman filter for state and parameter estimation of the SDNF models. The novel method processes each measurement vector in a component-like manner and, hence, avoids an inversion of a large matrix. Thus, the newly-derived sequential estimator is faster and, additionally, more numerically stable compared to the previously developed SDNF-oriented estimators. Besides, the novel filter is an adaptive method, i.e. it solves both problems, which are the state and parameter estimation, in parallel. Finally, the proposed method is applied to numerical examples to provide the comparative study and to show the effectiveness of our result.