Switching linear dynamical system (SLDS)

An SLDS models a nonlinear or piecewise process as a small library of linear dynamical regimes together with a discrete latent state that says which regime is active at each moment. A discrete Markov chain z_t indexes the parameters (A_z, b_z, Q_z) of a continuous linear-Gaussian latent evolution x_t = A_{z_t} x_{t-1} + b_{z_t} + noise, with an observation model mapping x_t to spikes or field potentials. The result is globally nonlinear but locally interpretable: each regime is a linear approximation to the dynamics valid in some part of state space.

SLDS are attractive for neural data because behavior is often organized into discrete phases (rest, preparation, movement, or the syllables of birdsong), and the inferred discrete states frequently line up with those phases without supervision. Inference couples the discrete and continuous states and is intractable in closed form; practitioners use structured variational inference, message passing with Laplace/Gaussian approximations, or particle methods. The main caveats are that the number of regimes is a modeling choice that strongly shapes the story, and that switch times can be smeared by observation noise.