Theta Neuron Subject to Delayed Feedback: A Prototypical Model for Self-Sustained Pulsing

The paper derives, via the V = tan(θ/2) change of variables to the QIF form, an existence condition for (n+1)-spike-per-delay periodic orbits that exactly matches coth[(n+1)T − τ] = κ + coth[nT − τ] in the excitable case and immediately implies κ > 2 (paper eqs. (2), (9)–(10); see also their explicit κ>2 remark) . It then constructs the event-time map, linearizes it, and obtains the same companion-matrix Jacobian with parameter γ and characteristic polynomial g(λ) = λ^{n+1} − γλ^n − 1 + γ, together with the key properties summarized in their Proposition 1 (no Neimark–Sacker or period-doubling for 0<γ, and loss of stability only when a multiplier crosses +1 at γ=(n+1)/n) . The candidate solution reproduces these steps: the existence relation and κ>2 bound; the event-map linearization yielding δt_{k+1} = γ δt_k + (1−γ) δt_{k−n} with the same g(λ); and the full stability picture with the same threshold γ=(n+1)/n, adding standard complex-analytic proofs (Eneström–Kakeya, Rouché) for the location of roots. For the primary branch (n=0) the paper’s explicit T(τ) = τ + coth^{-1}(κ − coth τ) and the reappearance construction lead to closed-form saddle-node loci (eqs. (26)–(27)), identical to those in the candidate solution’s Part C . The paper also identifies superstable points at dT/dτ=0, consistent with the candidate’s observation that γ=1 implies F_τ=0 and thus dT/dτ=0 by implicit differentiation; furthermore, the paper shows γ>0 for both I<0 and I>0 cases and presents the cot/csc analogues for I>0 that the candidate also states . Overall, the methods match point-for-point; the model’s derivations mirror the paper’s argument and results, with minor presentational differences.