Dynamical Systems in Neuroscience:

Simple Models 325tional papers can actually suffer from using the model because of its weird properties,such as the logarithmic F-I curve and fixed threshold.The resonate-and-fire model was introduced by Izhikevich (2001), and then byRichardson, Brunel, and Hakim (2003) and Brunel, Hakim, and Richardson (2003).These authors initially called the model “resonate-and-fire”, but then changed its nameto “generalized integrate-and-fire” (GIF), possibly to avoid confusion.A better choice is the quadratic integrate-and-fire neuron in the normal form (8.2)or in the ϑ-form (8.8); see Ex. 7. The ϑ-form was first suggested in the context ofcircle/circle (parabolic) bursting by Ermentrout and Kopell (1986a,b). Later, Ermentrout(1996) used this model to generalize numerical results by Hansel et al. (1995)on synchronization of Class 1 excitable systems, discussed in Chap. 10. Hoppensteadtand Izhikevich (1997) introduced the canonical model approach, provided many examplesof canonical models, and proved that the quadratic integrate-and-fire model wascanonical in the sense that all Class 1 excitable systems could be transformed into thismodel by a piece-wise continuous change of variables. They also suggested to call themodel “Ermentrout-Kopell canonical model”, but most scientists follow Ermentroutand call it “theta-neuron”.The model presented in Sect. 8.1.4 was first suggested by Izhikevich (2000; Eq. (4)and (5) with voltage reset discussed in Sect. 2.3.1) in the ϑ-form. The form presentedhere first appeared in Izhikevich (2003). The representation of the function I + v 2 inthe form (v − v r )(v − v t ) was suggested by Latham et al. (2000).We stress that the simple model is useful only when one wants to simulate largescalenetworks of spiking neurons. He or she still needs to use the Hodgkin-Huxley-typeconductance based models to study the behavior of one neuron or a small network ofneurons. The parameter values that match firing patterns of biological neurons presentedin this chapter are only educated guesses (the same is true for conductance-basedmodels). More experiments are needed to reveal the true spike-generation mechanismof any particular neuron. An additional insights into the question “which model ismore realistic” is in Fig. 1.8.Looking at the simple model, one gets an impression that the spike generationmechanism of RS neurons is the simplest in the neocortex. This is probably true,however, the complexity of the RS neurons, most of which are pyramidal cells, is hiddenin their extensive dendritic trees having voltage- and Ca 2+ -gated currents. Studyingdendritic dynamics is a subject of a 500-page book by itself, and we purposefullyomitted this subject. We recommend reading Dendrites by Stuart et al. (1999), recentreviews by Hausser and Mel (2003) and Williams and Stuart (2003), and the seminalpaper by Arshavsky et al. (1971; Russian language edition - 1969).