Dynamical Systems in Neuroscience:

236 Excitabilitysaddle-node bifurcationsubcritical Andronov-Hopf bifurcationinhibitorypulseattractiondomainn-nullclinespiking limit cycleexcitatorypulsenV-nullclineVV-nullclineexc.pulsespiking limit cycleunstablen-nullclineinhibitorypulseattractiondomainof spikinglimit cycleFigure 7.11: Co-existence of stable equilibrium and spiking limit cycle attractor in theI Na,p +I K -model. Left: The rest state is about to disappear via saddle-node bifurcation.Right: The rest state is about to lose stability via subcritical Andronov-Hopfbifurcation. Right (left) arrows denote the location and the direction of an excitatory(inhibitory) pulse that switches spiking behavior to resting.Fig. 7.10d and e, where we consider the model’s phase portraits. Notice the co-existenceof the resting state and a limit cycle attractor. The resting state loses stability via subcriticalAndronov-Hopf bifurcation at I = 5.25, so the emerging spiking has non-zerofrequency at I ≈ 5.25. However, injecting steps of current results in transitions to thelimit cycle even before the resting state loses its stability. The limit cycle in the modelappears via saddle homoclinic orbit bifurcation at I ≈ 3.8866, its period is quite largeresulting in the Class 1 response to steps of current. The F-I curves for homoclinicbifurcations have logarithmic scaling, so small frequency oscillations are difficult tocatch numerically let alone experimentally.The surprising discrepancy in Fig. 7.10a occurs because the resting state of theI Na,p +I K -model is near the Bogdanov-Takens bifurcation, i.e., the model is near atransition from resonator to integrator. Such a bifurcation was recorded, though indirectly,in some neocortical pyramidal neurons, as we will show later in this chapter andin Chap. 8. Another surprising example of Andronov-Hopf bifurcation with Class 1excitability is presented in Ex. 6. To avoid such surprises, we adopt the ramp definitionof excitability throughout the book.7.1.6 BistabilityWhen transition from resting to spiking states occurs via saddle-node (off invariantcircle) or subcritical Andronov-Hopf bifurcation, there is a co-existence of a stableequilibrium and a stable limit cycle attractor just before the bifurcation, as we illustratein Fig. 7.11. We refer to such systems as bistable. They have a remarkable