Widespread neuronal chaos induced by slow oscillating currents

The paper builds and analyzes one-dimensional interval return maps with “arches” indexed by spike count and documents: (i) steepening near homSF and AHsub (vertical stretching) that enlarges expanding portions of the map, (ii) a broad chaotic zone organized by global Swiss‑roll mixing on MQ rather than the local Shilnikov map, and (iii) chaos-to-bursting transition boundaries governed by mergers of df/dVn=1 points that generate cusp points and unfold into SNPO curves; all of these are evidenced in the figures and discussion of §§VI–VII and the map geometry (arches, maxima at SF, minima at SD) in §VI.A (e.g., the return-map construction, arches/spike counts, and steepening near homSF/AHsub; the three slope‑one points α,β,γ and their cusp mergers; and the Swiss‑roll organization) . The candidate solution then supplies a standard two-branch expanding Markov (horseshoe) construction on two successive arches to get a Cantor invariant set with a semi-conjugacy to the full 2-shift (hence positive entropy), shows this persists on an open wedge to the right of homSF by C^1 continuity, and identifies the boundary via cusp mergers where slope‑one points collide, yielding SNPO in suitable iterates. This matches the paper’s phenomenology and boundary description while providing a more explicit dynamical-systems proof under stated hypotheses (H1)–(H3). The paper’s account is primarily qualitative/numerical and template-based, not a formal theorem; the model’s argument is a textbook horseshoe proof given the hypotheses. Hence both are consistent and correct by different methods.