Transcritical symmetry breaking in a minimal three-process model with dynamic memory

AbstractWe study a minimal model of three processes with self-excitation and mutual inhibition, coupled to a symmetric Hebbian memorythat relaxes at rate λ toward βxixj. In the nine-dimensional physical state space the flow is of gradient type: there is an explicitLyapunov functional, and at every equilibrium the Jacobian factorizes as J = −MH, where H is the Hessian of the functional andMis a positive-definite diagonal mobility. Three exact consequences follow: all eigenvalues are real, local stability reduces to theinertia of a 3×3 reduced Hessian, and no static threshold depends on λ. For gain G = 8 and threshold θ = 0.2, attractors with three,two, and one active process coexist. The fully active configuration is stable for β > βtr = 0.2275429 and loses stability through atransverse transcritical bifurcation of multiplicity two, before the longitudinal fold of its branch at βSN = 0.2162648. The normalform has a nonzero cubic coefficient and predicts a cubic scaling of the Lyapunov difference between this state and the asymmetricsaddle, ∆F ≃ 1.34×103(β−βtr)3, which we verify with 50-digit arithmetic. The two- and one-process states exist without memoryand lose stability at β ≈ 1.012 × 106 (a fold) and β ≈ 678.08 (a pitchfork that breaks the symmetry of the inactive pair); asymptoticformulas reproduce both values. Although λ does not shift the thresholds, it sets the relaxation rates and the threshold observedwhen β varies in time.Keywords: transcritical bifurcation, permutation symmetry, Lyapunov function, Hebbian plasticity, multistability, gradient flow