Dynamical Systems in Neuroscience:

Bursting 379stable equilibriumBautinbifurcationunstalimitcyclestable limit cycle|V|unstable equilibriumusupercritical Andronov-HopfbifurcationHopf/fold cycleburstingHopfHopf/HopfburstingsubHopf/Hopfburstingfold limit cyclebifurcationsubcriticalAndronov-HopfbifurcationsubHopf/fold cycleburstingfold cyclesubHopfFigure 9.42: A neural system near co-dimension-2 Bautin bifurcation (central dot) canexhibit 4 different types of fast-slow bursting, depending on the trajectory of the slowvariable u ∈ R 2 in the parameter space. The “subHopf/fold cycle” bursting occurs viaa hysteresis loop and requites only one slow variable. Solid (dotted) lines correspondto spiking (resting) regimes (modified from Izhikevich 2000).A prominent feature of “subHopf/fold cycle” bursting, as well as any other typeof fast-slow bursting involving Andronov-Hopf bifurcation (“subHopf/*” or “Hopf/*”,where the wildcard “*” means any bifurcation) is that the transition from resting tospiking does not occur at the moment the resting state becomes unstable. The fastsubsystem continues to dwell at the unstable equilibrium for quite some time before itjumps rather abruptly to a spiking state, as we can clearly see in Fig. 9.41. This delayedtransition is due to the slow passage through Andronov-Hopf bifurcation, discussed inSect. 6.1.4. Delayed transitions through Andronov-Hopf bifurcation are ubiquitous inneuronal models, but they have never been seen in real neurons. Conductance noise,always present at physiological temperatures, constantly kicks the membrane potentialaway from the stable equilibrium, as one can see in the inset in Fig. 9.38, so transitionto spiking in real neurons is never delayed. Instead, it can occur even before theequilibrium becomes unstable, as we show in Sect. 6.1.4.Suppose that the hysteresis loop oscillation of the slow variable has a small amplitude.That is, the subcritical Andronov-Hopf bifurcation and the fold limit cycle