Reconstruction of phase-amplitude dynamics from electrophysiological signals

The paper proves that the Laplace average of g(t)=r(t)−γ(r)(θ(t)) recovers the inverse amplitude Σ up to a constant q^{(r)}_{1,0} by (i) expanding r via the forward Fourier–Taylor map K(r)(ϕ,σ), (ii) cancelling the n=0 block with K(r)(ϕ,0), (iii) evaluating along ϕ(t)=ωt+ϕ0, σ(t)=σ0 e^{λ t}, and (iv) showing only (n,k)=(1,0) survives the time-average, yielding g^*_{λ}(x0)=q^{(r)}_{1,0}σ0=q^{(r)}_{1,0}Σ(x0) (their Eq. (26)). This follows immediately from their definition of the Laplace average (Eq. (23)), the series representation (Eq. (20)), and the reduced flow (Eqs. (24)–(25) in the text), with the large-T limits made explicit in the derivation leading to Eq. (26) . The candidate solution reproduces the same steps (explicitly computing the per-term factor (1/T)∫ e^{[(n−1)λ+ikω]t}dt and invoking λ<0), adds a brief justification for termwise integration via uniform convergence, and reaches the identical conclusion and normalization comment. Hence, both are correct and essentially the same argument.