Oscillations in neuronal activity: a neuron-centered spatiotemporal model of the Unfolded Protein Response in prion diseases

Elliot M. Miller, Tat Chung D. Chan, Carlos Montes-Matamoros, Omar Sharif, Laurent Pujo-Menjouet, Michael R. Lindstrom

incompletemedium confidence

Category: math.DS
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper formulates and simulates a PDE–delay model with a saturating, delayed Neumann flux and defines observables such as havg and Wavg, but explicitly leaves well-posedness and rigorous analysis for future work. The candidate model provides a solid roadmap toward rigorous results (global well-posedness, steady states, Hopf for large delay, slow oscillations, and monotone sensitivities), yet key analytical steps are only sketched: handling the delayed boundary operator in an abstract semigroup setting, compactness on an exterior domain for the steady-state fixed point, and a justified center-manifold reduction to a scalar retarded equation. As written, both are incomplete.

Referee report (LaTeX)

\textbf{Recommendation:} major revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript presents a biologically well-motivated PDE with delayed, saturating boundary feedback and explores it numerically across parameters and dimensions. The qualitative insights into oscillations and UPR intensity are compelling and consistent with expectations from delay feedback systems. However, all rigorous analysis is deferred; statements about solution existence, bifurcation, and monotonicity are not backed by proofs. The accompanying candidate solution outlines a credible analytical path but omits technical lemmas essential for correctness. Substantial analytical development or explicit re-scoping of claims is required.