Dynamical Systems in Neuroscience:

232 ExcitabilityClass 3 excitable neuron0 mV10 mV50 ms4,000 pA-60 mV0 pAFigure 7.7: A Class 3 excitable brainstem mesV neuron does not fire in response to aramp current, even though the injected current is stronger than the one in Fig. 7.4.Among all four codimension-1 bifurcations of equilibrium, discussed in the previouschapter and mentioned in Fig. 7.2, only saddle-node on invariant circle bifurcationresults in a limit cycle attractor with arbitrary small frequency and continuous F-Icurve. The other three bifurcations result in limit cycle attractors with relatively largefrequencies and discontinuous F-I curves. Therefore,• Class 1 neural excitability corresponds to the rest state disappearing viasaddle-node on invariant circle bifurcation.• Class 2 neural excitability corresponds to the rest state disappearing viasaddle-node (off invariant circle) bifurcation or losing stability via subcritical orsupercritical Andronov-Hopf bifurcations.Of course, the rest state can lose stability or disappear via other bifurcations havinghigher codimension, sometimes leading to counter-intuitive results (e.g., Class 1 excitabilitynear Andronov-Hopf bifurcation; see Ex. 6 and Sect. 7.2.11). In this chapterwe concentrate on the four bifurcations above because they have the lowest codimensionand hence are the most likely to be seen experimentally.7.1.4 Class 3In Fig. 7.7 we inject a slow ramp current into the Class 3 excitable system. In contrastto Fig. 7.6, no spiking and no bifurcation occurs in this experiment despite the factthat the membrane potential goes all the way to 0 mV. Therefore,• Class 3 neural excitability occurs when the resting state remains stablefor any fixed I in a biophysically relevant range.Then, why are there single spikes in Fig. 7.4? Their existence in the figure and theirabsence in the ramp experiment is related to the phenomenon of accommodation thatwe show now describe.