Control in triphasic rhythmic neural systems: a comparative mechanistic analysis via infinitesimal local timing response curves

The paper rigorously defines the lTRC, derives the generalized first-order duration-shift formula with entry, exit, and in-domain terms (including a Dirac-delta representation), and applies it to three models; it also proves instability of the intrinsic-escape equal-duration rhythm via a 2D map Jacobian. The candidate solution reproduces these results with a slightly different derivation (via a transport PDE for the exit-time functional) and reaches the same qualitative conclusions for all three model applications. Minor issues in the candidate write-up: (i) in Part B it momentarily suggests f'(f(y)−y)=f'(y), which is unnecessary and not generally true (the determinant computation does not rely on this equality); (ii) in Part C for the heteroclinic model it assumes fixed boundaries (contradicting the paper’s shifting-boundary treatment) and does not state the small negative sign for phase 1 that the paper derives. Overall, the conclusions align.