Neurodynamical Role of STDP in Storage and Retrieval of Associative Information

The paper proves that, under single-frequency forcing b(t)=∑ sin(ωt−ξ_i)m_i, the STDP model ẋ=−x+Wx+b(t), Ẇ=−γW+ρ(xx_τ^T−x_τx^T) admits a periodic solution x*(t) confined to a 2D memory plane S with a constant, skew-symmetric W*(t)∈∧²(S) (Theorem 1 in the main text and its full statement/proof in the supplement). The proof shows b(t) lies in a 2D plane (Lemma A), constructs W* in the form α(vu^T−uv^T), and reduces the W-equation to a scalar root condition h(λ)=0 ensuring Ẇ=0, with xx_τ^T−x_τx^T time-independent and aligned with vu^T−uv^T . The candidate solution independently derives the same qualitative structure: it chooses an oriented orthonormal basis on S, treats S≅C so that A=vu^T−uv^T acts as i, computes the unique T-periodic response for fixed α, shows xx_τ^T−x_τx^T is constant and a scalar multiple of A, and then solves −γα+ρβ(α)=0 with β(α)=const·sin(ωτ)/(1+(ω−α)²), guaranteeing a real root. This mirrors the paper’s result after a different parametrization (α versus λ) and a simplification that, in the orthonormal case, collapses the paper’s Φ±-based expression to 1/(1+(ω−α)²). Both arguments are correct and consistent; the paper’s proof is more general in basis choice and bookkeeping, while the model’s proof is shorter via complex amplitudes.