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A Balance Equation Determines a Switch in Neuronal Excitability

Figure 11

Three bistable phase portraits of model (1) and cartoon of the associated hysteretic bifurcation diagrams.

In the phase portraits, a solid curve denotes the -nullcline, whereas a dashed curve denotes the -nullcline. Stable fixed points are depicted as filled circles, whereas unstable as circles and saddle points as cross. Stable limit cycles are drawn as solid oriented blue curves, whereas unstable as red dashed curves. The stable manifolds of saddle points are depicted as green oriented curves. In bifurcation diagrams, a solid curve denotes branches of stable fixed points, whereas a dashed curve denotes branches of unstable or saddle points. Branches of stable limit cycles are depicted as blue curves, whereas branches of unstable limit cycles as red dashed curves. sub.HB denotes a subcritical Hopf bifurcation, SNLC a saddle-node limit cycles bifurcation, SN a saddle-node bifurcation, and SH a saddle-homoclinic bifurcation. A–B. Restorative bistability. A. Subcritical Hopf bifurcation. Hysteresis vanishes exponentially fast as timescale separation increases. B. Restorative saddle-homoclinic bifurcation. Not physiological because it violates the time scale separation between and . C. Regenerative bistability ruled by a regenerative saddle-homoclinic bifurcation. Hysteresis is barely affected by time-scale separation.

doi: https://doi.org/10.1371/journal.pcbi.1003040.g011