Characterising the Non-Equilibrium Dynamics of a Neural Cell

The paper posits the closure ∫^s ||x|| dt = α x(s) y(s) (their eq. (11)) and, combined with ẋ = a x − b ∫^s ||x|| dt, ẏ = c y − d ∫^s ||x|| dt (their eq. (10)), asserts the Lotka–Volterra system ẋ = a x − b′ x y, ẏ = −c y + d′ x y without a rigorous derivation; the key step identifying a time integral with an area is heuristic and not justified mathematically (see the eq. (10)→(11) transition and LV claim in the manuscript) . The manuscript further appeals to the existence of a Hamiltonian for LV but does not exhibit it nor reconcile this conservative structure with earlier dissipative-limit-cycle language . By contrast, the candidate solution correctly accepts the LV model as given and supplies a standard, complete analysis: positive equilibrium, explicit first integral H(x,y)=d′x−c ln x + b′y − a ln y, and the fact that all positive non-equilibrium trajectories are closed periodic orbits around a center (not an attracting limit cycle). Thus, the model solution is mathematically correct for the LV reduction the paper claims, while the paper’s derivation and stability claims are incomplete.