Rhythmic sharing: A bio-inspired paradigm for zero-shot adaptive learning in neural networks

The paper reduces ϕ̇ = ω0 + ε sin(Ψ0 − ϕ) to θ̇ = ω0 − ε sin θ by a constant shift and, via the tangent half‑angle substitution, presents the closed-form solution ϕ(t) = 2 tan⁻¹[ ε/ω0 + (Δ/ω0) tan((Δ/2) t + ξ0 ) ] with Δ = √(ω0² − ε²). It then argues that for |ω0| < |ε|, Δ is imaginary and the tan becomes tanh, yielding convergence (oscillation death), whereas for |ω0| > |ε|, one gets a genuine tan and no limit exists . The candidate solution follows the same method, gives a unified formula (also keeping the Ψ0 shift explicit), computes the limiting equilibrium, and provides an independent monotonicity argument for the nonconvergent regime. Minor issues: the paper’s definition of ξ0 contains an extraneous factor of 2, and both paper and model omit the boundary case |ω0| = |ε| (which also converges).