Dynamical Systems in Neuroscience:

238 Excitability(a) resting spiking (b) spiking resting (c)Class 2 Class 1800F-I curveI=4.51spikingI=3.08frequency, Hz(b)Class 1Class 2(a)restingspikingresting02 4 6current, IFigure 7.13: (a) The frequency of emerging oscillations at the transition “resting →spiking” defines the class of excitability. (b) The frequency of disappearing oscillationsat the transition “spiking → resting defines the class of spiking. (c) The I Na,p +I K -model with high-threshold K + current exhibits class 2 excitability but class 1 spiking.Its F-I curve has a hysteresis.used when determining experimentally the class of excitability; only spike trains withregular interspike periods should be accepted to measure the F-I relations.7.1.7 Class 1 and 2 spikingThe class of excitability is determined by the frequency of emerging oscillations atthe transition “resting → spiking”, as in Fig. 7.13a. Let us look at the frequencyof disappearing oscillations at the transition “spiking → resting”. To induce such atransition, we inject a strong pulse of dc-current of slowly decreasing amplitude, as inFig. 7.13b. Similarly to the Hodgkin’s classification of excitability, we say that a neuronhas Class 1 spiking if the frequency-current (F-I) curve at the transition “spiking →resting” decreases to zero, as in Fig. 7.13c, and Class 2 spiking if it stops at a certainnon-zero value.The class of excitability coincides with the class of spiking when the transitions“resting ↔ spiking” occur via saddle-node on invariant circle bifurcation or supercriticalAndronov-Hopf bifurcation. Indeed, if the current ramps are sufficiently slow, theneuron as a dynamical system goes through the same bifurcation, just in the oppositedirections. The classes may differ when the bifurcation is of the saddle-node (off invariantcircle) type or subcritical Andronov-Hopf type because of the bistability of theresting and spiking states. Such a bistability results in the hysteresis behavior of thesystem when the injected current I increases and decreases slowly, which may resultin the hysteresis of the F-I curve. For example, the transition “resting → spiking” inFig. 7.13a occurs via saddle-node bifurcation at I = 4.51, and the frequency of spikingequals the frequency of the limit cycle attractor, which is non-zero at this value of I.Decreasing I results in the transition “spiking → resting” via the saddle homoclinic