Extended Kalman filter (EKF)

The extended Kalman filter handles nonlinear dynamics or a nonlinear observation model by linearizing them at each step. It replaces the constant matrices A and C with the Jacobians of the nonlinear functions evaluated at the current state estimate, then runs the ordinary Kalman recursion on this local first-order approximation. This lets the state-space machinery accommodate, for example, a genuinely nonlinear (e.g. exponential or trigonometric) tuning function relating firing rate to movement.

The linearization is only first order, so the EKF is accurate when the functions are smooth and the posterior is well concentrated relative to their curvature, and it can be biased or even diverge when nonlinearity is strong, the initial estimate is poor, or the covariance is large. It also requires analytic or numerical Jacobians, which are error-prone and sometimes ill-defined. In BCI the EKF sees less use than the unscented variant precisely because neural tuning models are nonlinear enough that a derivative-free approximation is usually both easier and more stable.