Fixed point and attractor

A fixed point of a dynamical system is a state z* at which the dynamics vanish, f(z*) = 0, so the system left there stays there. Whether nearby states are drawn in or pushed away is set by the eigenvalues of the Jacobian of f at z*: all decaying modes make z* a stable attractor, at least one growing mode makes it a saddle or unstable point. Richer objects are built from these primitives — a line attractor is a continuum of marginally stable fixed points along which the state drifts freely, giving a mechanism for graded persistent activity such as neural integration of evidence or holding eye position.

Fixed points are the vocabulary in which trained recurrent networks are reverse-engineered: locating a model's fixed points and linearizing around them typically reveals how it stores memories, makes decisions, or times outputs. In motor cortex the same language describes an attractor-like preparatory state that the system settles into before movement and a subsequent flow away from it during execution. The caution is that biological data rarely sit at true fixed points for long, and attractor structure inferred from short, input-driven epochs is model-dependent, so claims of a specific attractor should be checked against alternative dynamics that fit equally well.