The Ubiquity of Three Steady States: Minimal Multistable Zero-One Reaction Networks

What the math gives to ML

The paper provides a constructive multistability mechanism in three-dimensional quadratic zero-one mass-action networks: among minimal six-reaction motifs, positive steady states occur in an alternating stability pattern, with consecutive Jacobian determinants having opposite signs and exactly half of the states stable. It also establishes a sharp maximum of three positive steady states for the (3,6,3) class, so the minimal bistable motif has two stable equilibria separated by one unstable equilibrium. A transferable neural-network construction is a constrained continuous-time recurrent cell whose vector field is assembled from such quadratic nonnegative reaction terms, yielding certified attractor-based memory rather than relying on a generic RNN to learn a suitable phase portrait.