State-space model

A state-space model describes a system by a hidden state that evolves over time (the state or process equation) and a set of noisy measurements generated from that state (the observation or measurement equation). The state is Markov: given the present state, the future is independent of the past. In BCI the hidden state is typically the kinematic intention (position, velocity, sometimes force), and the observations are binned firing rates, spike trains, or band-power features. Decoding is then the problem of inferring the hidden state from the observation stream.

Two ingredients define the model: a dynamics/prior on how the state moves (e.g. a smoothness constraint that velocity changes gradually) and an observation/encoding model relating state to neural activity (e.g. cosine-like directional tuning). Separating these two is what lets state-space decoders combine a physiologically motivated prior with the current neural evidence, rather than mapping neural activity to output in a single memoryless step. The Kalman, extended/unscented Kalman, particle, and point-process filters are all recursive estimators for specific choices of these two equations.