Data-Driven Inference of High-Accuracy Isostable-Based Dynamical Models in Response to External Inputs

The paper’s Appendix B proves by induction that the steady-state jth-order isostable component under u(t)=ε sin(ωt) is a finite Fourier series with harmonics up to jω and gives the frequency-by-frequency response formula (B6). The candidate solution follows the same inductive structure: expand In,A(ψ), collect O(εj), show the forcing’s harmonics are bounded by j via product-to-sum, and solve the linear ODE harmonic-by-harmonic, yielding the same filter (ikω−λn)^{-1}. Minor discrepancies are present: the paper explicitly allows a DC (k=0) term (e.g., at second order), whereas the candidate at one point asserts a mean-zero normalization “used in the paper.” This does not affect the main claim about the maximum harmonic and the recursion, which matches the paper’s B6.