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Classification of bursting patterns: A tale of two ducks

Fig 7

Extended classification.

(Top part) Main idea of the Rinzel/Izhikevich, Bertram/Golubitsky and colleagues and folded-node bursting classifications. ((ai)-(di), i = 1,2,3) Exemplary “folded-node/homoclinic” bursting, presented in the full 4D system and in its 2D fast and slow subsystems (resp.), showing that both subsystems are required to fully understand this bursting profile; all equations are given in the left column (a1)-(a3). Top row (b1)-(d1), full system bursting solution in 2 different 3D phase-space projections: 2 slow/1 fast in (b1)-(c1) and 1 slow/2 fast in (d1), also showing the critical manifold (fast subsystem’s set of equilibria, in green), the fast subsystem’s limit cycles envelope (blue), as well as relevant bifurcations. In (c1), the trajectory is zoomed near its small oscillations, which follow attracting (red) and repelling (blue) slow manifolds , perturbations of the attracting and repelling parts of the critical manifold, and pass near the folded node (dot). Middle row (b2)-(d2), fast subsystem information: the bifurcation diagram with respect to 1 slow variable (s₁) in (b2), which we can assume persists as such for a small interval of values of the other slow variable (s₂); this allows to superimpose the projection of the full system bursting orbit (c2), as done in the Rinzel/Izhikevich classification, and to compute loci of bifurcation points of this diagram in the 2-parameter plane (s₁, p), as done in the Bertram/Golubitsky and colleagues classification. However, both approaches classify this bursting pattern as fold/homoclinic (square-wave), hence failing to capture the reason for its small oscillations during quiescence, which can only be unraveled by studying the slow subsystem’s information in the bottom row (b3)-(c3) and find the existence of a folded node in the slow singular limit; details on labels in S1 Text.

doi: https://doi.org/10.1371/journal.pcbi.1009752.g007