Kalman gain

The Kalman gain K is the matrix that decides how much to trust the incoming measurement relative to the model's own prediction. It is K = P_pred C^T (C P_pred C^T + Q)^{-1}: large when the predicted state covariance P_pred is large (the model is unsure) or the measurement noise Q is small (the sensor is reliable), and small in the opposite regime. The corrected estimate is the prediction plus the gain times the innovation, and the posterior covariance shrinks as P = (I − K C) P_pred.

Intuitively the gain is a signal-to-noise weighting performed automatically and continuously. In a well-conditioned decoder it converges toward a steady-state value once the covariance recursion settles, so many real-time BCIs precompute a fixed steady-state gain to save computation, sacrificing little because P stops changing. Watching the gain also diagnoses the filter: a gain driven near zero means the decoder has stopped listening to the neurons and is coasting on its prior — often felt by the user as a sluggish, self-moving cursor.