Recurrent switching LDS (rSLDS)

The recurrent SLDS (Linderman et al.) closes an important loophole in the plain SLDS: it lets the continuous latent state x_t influence the probability of the next discrete regime z_{t+1}, rather than treating regime switches as an independent Markov chain. The discrete transition distribution becomes a function (typically a softmax of a linear or neural map) of the current location in latent space, so the model partitions the latent space into regions, each with its own linear flow, and the boundaries themselves are learned.

This recurrence is what makes rSLDS a genuine tool for approximating nonlinear dynamics: a smooth nonlinear vector field can be represented as a set of locally linear pieces whose domains tile the state space, giving a piecewise-affine flow field that is both expressive and interpretable. For neuroscience it yields decompositions in which a decision or a reach is described by a sequence of local linear dynamics with data-defined switching boundaries. Caveats mirror the SLDS: results depend on regime count and priors, and the piecewise-linear boundaries are an approximation that can introduce artefactual switches where the true field is merely curved.