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Fig. 6 | The Journal of Mathematical Neuroscience

Fig. 6

From: Fast–Slow Bursters in the Unfolding of a High Codimension Singularity and the Ultra-slow Transitions of Classes

Fig. 6

Bifurcation diagrams along bursting paths. For each bursting class found in the unfolding we produced the bifurcation diagram of the fast subsystem using z as bifurcation parameter and plotting the fast variable x as ordinate. The simulated bursting trajectory, in blue, is superimposed to the bifurcation diagram. Bifurcations are marked with a colored star (same color code as in Fig. 4) and vertical dotted line of the same color separate regions of the unfolding. The Roman numeral of the relative region is written in gray in the lower part of the plots. Onset and offset bifurcations are written in red. In all the nine classes (eight point-cycle and one point-point bursters) the upper branch of the z-shaped curve of fixed points acts as silent state. When the fast subsystem is in the silent state, the slow variable z is instructed to increase. At SN r (or at the subcritical Hopf point in classes c14b and c16b) the resting-state destabilizes, the system moves towards another attractor, and z start decreasing until the system goes back to the silent state and a new bursting cycle is started. The active phase takes place on the limit cycle surrounding the lower branch of the equilibrium manifold. The top panel shows bifurcation diagrams for bursters in the LCs region, the bottom panel for those in the LCb region. In the LCs region we can have a saddle-node onset if the limit cycle exists already when the silent state destabilizes at SN r (first row) or supercritical Hopf onset when the limit cycle is created for smaller values of z (second row). Oscillations can end because the limit cycle coalesces with the saddle (middle column) or because of another supercritical Hopf bifurcation (right column). In the LCb region panel, in the first row oscillations are started through saddle-node bifurcation, while the resting-state destabilizes earlier in the second row via subcritical Hopf bifurcation. The limit cycle coalesces with the saddle in the left column and with an unstable limit cycle in the right column. Bursting paths and entire arcs of great circles used to produce these bifurcation diagrams are shown in Sect. 5.3

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