Dynamical Systems in Neuroscience:

Simple Models 283which we considered in Sect. 3.3.8. Here v peak is not a threshold, but the peak (cut off)of a spike, as we explain below. It is useful to use v peak = +∞ in analytical studies. Insimulations, the peak value is assumed to be large but finite, so it can be normalizedto v peak = 1.Notice that ˙v = b + v 2 is a topological normal form for the saddle-node bifurcation.That is, it describes dynamics of any Hodgkin-Huxley-type system near that bifurcation,as we discuss in Chap. 3 and 6. There we derived the normal form (6.3) for theI Na,p +I K -model and showed that the two systems agree quantitatively in a reasonablybroad voltage range. By resetting v to v reset , the quadratic integrate-and-fire modelcaptures the essence of recurrence when the saddle-node bifurcation is on an invariantcircle.When b > 0, the right-hand side of the model is strictly positive, and the neuronfires a periodic train of action potentials. Indeed, v increases, reaches the peak, resetsto v reset , and then increases again, as we show in Fig. 3.35, top. In Ex. 3 we prove thatthe period of such spiking activity isT = √ 1 (atan v peak√ − atan v )reset√ < √ π ,b b b bso that the frequency scales as √ b, as in Class 1 excitable systems.When b < 0, the parabola b + v 2 has two zeroes, ± √ |b|. One corresponds to thestable node equilibrium (resting state), the other corresponds to the unstable node(threshold state); see Ex. 2. Subthreshold perturbations are those that keep v belowthe unstable node. Superthreshold perturbations are those that push v beyond theunstable node, resulting in the initiation of an action potential, reaching the peakvalue v peak , and then resetting to v reset . If, in addition, v reset > √ |b|, then there is aco-existence of resting and periodic spiking states, as in Fig. 3.35, bottom. The periodof the spiking state is provided in Ex. 4. A two-parameter bifurcation diagram of (8.2)is depicted in Fig. 8.3.Unlike its linear predecessor, the quadratic integrate-and-fire neuron is a genuineintegrator. It exhibits saddle-node bifurcation, it has a soft threshold, and it generatesspikes with latencies, like many mammalian cells do. Besides, the model is canonicalin the sense that the entire class of neuronal models near saddle-node on invariant circlebifurcation can be transformed into this model by a piece-wise continuous changeof variables (see Sect. 8.1.5 and the Ermentrout-Kopell theorem in Hoppensteadt andIzhikevich 1997). In conclusion, the quadratic and not the leaky integrate-and-fireneuron should be used in simulations of large-scale networks of integrators. A generalizationof this model is discussed next.8.1.4 Simple model of choiceA striking similarity among many spiking models, discussed in Chap. 5, is that theycan be reduced to two-dimensional systems having a fast voltage variable and a slower“recovery” variable, which may describe activation of K + current or inactivation of Na +