Bifurcation of the neuronal population dynamics of the modified theta model: transition to macroscopic gamma oscillation

The paper’s Theorem 6.3 matches the model’s goal, but key proof steps are under-justified or inconsistent: (i) at μ=0 the paper asserts extra generalized eigenvalues from poles of G̃, yet the analytically continued eigen-equation (6.2) collapses to λ = −1/τ when μ=0; the given argument multiplies both sides by (λ−iω*)^p and effectively shows 0=0, not that λ=iω* solves the equation. The model repeats this claim and additionally assumes T is block diagonal at μ=0, which is not literally true. (ii) Both the paper and the model give plausible but sketch-level arguments for a transverse crossing under small variance, relying respectively on delta-approximation and a Plemelj-based system without full control. (iii) For large μ, the paper’s sign argument excludes Re λ > −1/τ but does not treat λ = −1/τ; the model closes this gap cleanly, but still relies on unstated uniformity assumptions. Overall, both proofs need tightening: the paper to resolve the μ=0 inconsistency and boundary case; the model to justify several analytic claims.