Joint estimation of states and parameters of Hodgkin–Huxley neuronal model using Kalman filtering

Fitting biophysical models to real noisy data jointly with extracting fundamental biophysical parameters has recently stimulated tremendous studies in computational neuroscience. Hodgkin–Huxley (HH) neuronal model has been considered as the most detailed biophysical model for representing the dynamical behavior of the spiking neurons. The unscented Kalman filter (UKF) has been already applied to track the dynamics of the HH model. In this paper, we extend the existing Kalman filtering (KF) technique for the HH model to another version, namely, extended Kalman filtering (EKF). Two estimation strategies of the KF, dual and joint estimation strategies, are derived for simultaneously tracking the hidden dynamics and estimating the unknown parameters of a single neuron, leading to four KF algorithms, namely, joint UKF (JUKF), dual UKF (DUKF), joint EKF (JEKF) and dual EKF (DEKF). Detailed derivations of these four methods and intensive simulation studies are provided. Our main contribution is in the derivation of the EKF-based methods, JEKF and DEKF, for tracking the states and estimating the parameters of the HH neuronal model. Since these EKF-based methods are faster than UKF-based versions, they can particularly be employed in dynamic clamp technique, which connects artificial and biological neurons in order to assess the function of neuronal circuits.