Dynamical Systems in Neuroscience:

292 Simple Modelsgeneral algorithm for doing this is not known. However, much success has been achievedin some important cases. The canonical model for a system near an equilibrium is thetopological normal form at the equilibrium (Kuznetsov 1995). Such canonical model islocal, but it can be extended to describe global dynamics. For example, the quadraticintegrate-and-fire model with a fixed v reset < 0 is a global canonical model for all Class1 excitable systems, i.e., systems near saddle-node on invariant circle bifurcation. Thesame model with variable v reset is a global canonical model for all systems near saddlenodehomoclinic orbit bifurcation considered in Sect. 6.3.6. The phase model ˙ϑ = 1derived in Chap. 10 is a global canonical model for the family of nonlinear oscillatorshaving exponentially stable limit cycle attractors. Other examples of canonical modelsfor spiking and bursting can be found in subsequent chapters of this book.The vector-field of excitable conductance-based models in the subthreshold regionand in the region corresponding to the up-stroke of the spike can be converted into thesimple form (8.3, 8.4), possibly with u being a vector. Therefore, the simple model(8.3, 8.4) is a local canonical model for the spike-generation mechanism and the spikeup-stroke of the Hodgkin-Huxley-type neuronal models. It is not a global canonicalmodel because it ignores the spike downstroke. Yet, it describes remarkably well thespiking and bursting dynamics of many biological neurons, as we demonstrate next.8.2 CortexIn this section we consider the six most fundamental classes of firing patterns observedin the mammalian neocortex and depicted in Fig. 8.11 (Connors and Gutnick 1990,Gray and McCormick 1996, Gibson et al. 1999). Though most biologists agree withthe classification in the figure, many would point out that it is greatly oversimplified(Markram et al. 2004), that the distinction between the classes is not sharp, that thereare subclasses within each class (Nowak et al. 2003, Toledo-Rodriguez et al. 2004),and that neurons can change their firing classes depending on the state of the brain(Steriade 2004).• (RS) Regular spiking neurons fire tonic spikes with adapting (decreasing) frequencyin response to injected pulses of dc-current. Most of them have Class 1excitability in the sense that the interspike frequency vanishes when the amplitudeof the injected current decreases. Morphologically, these neurons are spinystellate cells in layer 4 and pyramidal cells in layers 2,3,5, and 6.• (IB) Intrinsically bursting neurons generate a burst of spikes at the beginning ofa strong depolarizing pulse of current, and then switch to tonic spiking mode.Those are excitatory pyramidal neurons found in all cortical layers, but mostabundant in layer 5.• (CH) Chattering neurons fire high-frequency bursts of spikes with relatively shortinterburst periods, hence another name — FRB or fast rhythmic bursting. Outputof such a cell fed to the loudspeaker “sounds a lot like a helicopter — cha,