Spike-adding and reset-induced canard cycles in adaptive integrate and fire models

The paper rigorously establishes (i) existence of N-reset periodic cycles for arbitrary N via explicit piecewise-linear flows and carefully assembled return maps, (ii) existence of N-reset canard cycles at w0=(1+ε)(vres+I), and (iii) a canard-mediated 2→3 reset-adding transition, with precise ordering and small gap estimates, using local Poincaré maps and continuity arguments. By contrast, the model solution, while correctly reproducing the exact flows and slow manifolds and giving a plausible return-map viewpoint, makes two key mistakes: it misclassifies the plus-only itinerary by asserting it when w≤wc (the correct threshold is w≤vres+I); and it claims existence of N-reset cycles for arbitrary N by simply repeating a 1-reset orbit over an extended observation window (not the paper’s notion of a minimal N-reset cycle). Its contraction estimate is only justified on a subdomain where the plus-only itinerary holds, but this domain was not cleanly specified relative to ε. The model’s arguments for canard existence and the 2→3 transition are qualitatively consistent with the paper.