Dynamical Systems in Neuroscience:

214 BifurcationsIn contrast, if the equilibrium loses stability via subcritical Andronov-Hopf bifurcation,the state of the system diverges from it, which results in an immediate spike orsome kind of a large-amplitude jump. Such a loss of stability is called hard: neither theequilibrium nor its neighborhood are attractive. The hard loss of stability usually leadsto noticeable or catastrophic changes in systems behavior, and the stability boundaryis called dangerous (Bautin 1949). Changing the bifurcation parameter in the oppositedirection will make the equilibrium stable again, but may not bring the state of thesystem back to it. Saddle-node bifurcation is hard unless it is on an invariant circle.In this case, the loss of stability is catastrophic, i.e., leading to noticeable spikes, butreversible. Saddle homoclinic orbit bifurcation is hard, regardless of whether it is subcriticalor supercritical. In general, most bifurcations in neurons or at least in neuronalmodels are hard.Review of Important Concepts• Stable equilibrium (resting state) in a typical neuronal model caneither– disappear via saddle-node bifurcation, which can be off or oninvariant circle, or– lose stability via Andronov-Hopf bifurcation, which can be supercriticalor subcritical.These four cases are summarized in Fig. 6.46.• Stable limit cycle (periodic spiking state) in a typical twodimensionalneuronal model can either– be cut by saddle-node on invariant circle bifurcation,– shrink to a point via supercritical Andronov-Hopf bifurcation,– disappear via fold limit cycle bifurcation, or– disappear via saddle homoclinic orbit bifurcation.These four cases are summarized in Fig. 6.47.• Some atypical (codimension-2) bifurcations may play importantroles in neuronal dynamics.• Bogdanov-Takens bifurcation separates integrators from resonators.Bibliographical NotesThough bifurcation theory can be traced back to Poincare and Andronov, it is a relativelynew branch of mathematics. The first attempt to apply it to neuroscience was as