Dynamical Systems in Neuroscience:

276 ExcitabilityBibliographical NotesThere is no universally accepted definition of excitability. Our definition is consistentwith the one involving ε-pseudo-orbits (Izhikevich 2000). R. FitzHugh (1955, 1960,1976) pioneered geometrical analyses of phase portraits of neuronal models with theview to understand their neuro-computational properties. It is amazing that such importantneuro-computational properties as all-or-none action potentials, firing thresholds,and integration of EPSPs are still introduced and illustrated using the Hodgkin-Huxley model, which according to FitzHugh, cannot have these properties. Throughoutthis chapter we follow Izhikevich (2000) to compare and contrast neuro-computationalproperties of integrators and resonators.The frozen noise experiment in Fig. 7.24 was pioneered by Bryant and Segundo in1976, but due to an interesting quirk of history, it is better known at present as theMainen-Sejnowski (1995) experiment (despite the fact that the latter paper refers tothe former). Post-inhibitory facilitation was pointed out by Luk and Aihara (2000),Izhikevich (2001). John Rinzel suggested to call it “post-inhibitory exaltation” (ina similar vain, the phenomenon in Fig. 7.49b may be called “post-excitatory hesitation”).Richardson et al. (2003) pointed out that frequency preference and resonanceoccurs without subthreshold oscillations when the system is near the transition froman integrator to a resonator.The Hodgkin’s classification of neuronal excitability can be applied to classify anyrhythmic system, e.g., contractions of uterus during labor. Typically, the contractionsstart with low frequency that gradually increases — Class 1 excitability. The author’swife had to be induced pharmacologically to evoke labor contractions, which is a typicalmedical intervention when the baby is overdue. The contraction monitor showed asinusoidal signal with constant period, around 2 minutes, but slowly growing amplitude— Class 2 excitability via supercritical Andronov-Hopf bifurcation! Since the motherhad an advance degree in applied mathematics, the author waited for a 1-minute periodof quiescence between the contractions and managed to explain to the delivering motherthe basic relationship between bifurcations and excitability. Five years later, induceddelivery of the author’s second daughter resulted in the same supercritical Andronov-Hopf bifurcation. The author reminded this concept to the mother and explained it tothe obstetrician minutes after the delivery.Exercises1. When can the FitzHugh-Nagumo model (4.11, 4.12) exhibit inhibition-inducedspiking, such as the one in Fig. 7.32?2. (French ducks) Numerically investigate the quasi-threshold in the FitzHugh-Nagumo model (4.11, 4.12). How is it related to the French duck (canard; seeEckhaus 1983) limit cycles discussed in Sect. 6.3.4?