Fixed-point and attractor analysis

Once a dynamics model (typically an RNN or neural ODE) has been fit, fixed-point analysis interrogates its computation by finding states where the flow vanishes or nearly vanishes (slow points), classifying their stability from the local Jacobian, and mapping how they organize behavior. This is the standard route from a trained black-box recurrent model to a mechanistic account: a memory can appear as a line attractor, a binary decision as a pair of stable fixed points separated by a saddle, an oscillation as a limit cycle, and a ring attractor as a continuous circle of fixed points (as for head-direction coding).

The method (Sussillo and Barak) is powerful but has honest limits. Fixed points are properties of the model, and are only as good as the fit and the identifiability of the latent coordinates beneath it; slow-point finding is an optimization that can miss structure or return spurious points; and finding a line attractor in a network is evidence about the model, which becomes a claim about the brain only when corroborated by perturbations and by replication across fits. Used well, it turns fitted dynamics into testable circuit hypotheses; used loosely, it over-reads geometry the data do not pin down.