Fast-and-flexible decision-making with modulatory interactions

The paper’s Theorem 5.2, for model (2), proves J(0) = −I + u0 S′(0) A, identifies u* = (S′(0)λmax)−1, shows stability loss at u0 = u*, branch tangency to vmax, and characterizes input sensitivity via wmax; it further states that if ⟨b,wmax⟩ ≠ 0 the bifurcation undergoes a universal unfolding. These match the candidate’s analysis, which uses Crandall–Rabinowitz and Lyapunov–Schmidt with the unfolding parameter β = ⟨wmax,b⟩ and spectral projection P = vmax wmax^T. The only substantive discrepancy is a likely slip in the paper’s statement of the sensitivity subspace: it asserts ⟨x*,vmax⟩ = 0, whereas the natural invariant complement is ker wmax (i.e., ⟨wmax,x*⟩ = 0). The proof text and LS reduction in the paper are otherwise consistent, and an apparent sign typo (“⟨b,wmax⟩ < 0” instead of “≠ 0”) does not affect substance. Overall, the arguments agree in content though they use different standard tools.