Mathematical analysis of the model will be published in the monograph
by Izhikevich [8]. The derivation of the first (1) is based on bifurcation
theory and normal form reduction [2], [5], and the part v0 =
v2 + I is sometimes referred to as being a quadratic integrate-and-fire
neuron. The full model was first published in [10, eqns. (4) and (5)
with voltage reset discussed in Sect. 2.3.1 ] in a trigonometric form
more suitable for mathematical analysis. The form presented here is
more suitable for large-scale simulations.


[7] E. M. Izhikevich, N. S. Desai, E. C. Walcott, and F. C. Hoppensteadt,
gBursts as a unit of neural information: selective communication via
resonance,h Trends in Neurosci., vol. 26, pp. 161.167, 2003.
[8] E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry
of Excitability and Bursting, to be published.
[9] , gResonate-and-fire neurons,h Neural Networks, vol. 14, pp.
883.894, 2001.
[10] , gNeural excitability, spiking, and bursting,h Int. J. Bifurc. Chaos,
vol. 10, pp. 1171.1266, 2000.


Our gone-fits-allh choice of the function 0:04v2+5v+140 in (1) is
justified when large-scale networks of spiking neurons are simulated,
as we discuss below. However, if one is interested in the behavior of
a single neuron, then other choices of the function are available, and
sometimes more preferable. For example, the function 0:04v2+4:1v+
108 with b = ..0:1 is a better choice for the RS neuron, since it leads to
the saddle-node on invariant circle bifurcation and Class 1 excitability
[10].