  The quadratic adaptive integrate-and-fire model (Izhikecih 2003, 2007) is
recognized as very interesting for its computational efficiency and its ability
to reproduce many behaviors observed in cortical neurons. For this reason it is
currently widely used, in particular for large scale simulations of neural
networks. This model emulates the dynamics of the membrane potential of a
neuron together with an adaptation variable. The subthreshold dynamics is
governed by a two-parameter differential equation, and a spike is emitted when
the membrane potential variable reaches a given cutoff value. Subsequently the
membrane potential is reset, and the adaptation variable is added a fixed value
called the spike-triggered adaptation parameter. We show in this note that when
the system does not converge to an equilibrium point, both variables of the
subthreshold dynamical system blow up in finite time whatever the parameters of
the dynamics. The cutoff is therefore essential for the model to be well
defined and simulated. The divergence of the adaptation variable makes the
system very sensitive to the cutoff: changing this parameter dramatically
changes the spike patterns produced. Furthermore from a computational
viewpoint, the fact that the adaptation variable blows up and the very sharp
slope it has when the spike is emitted implies that the time step of the
numerical simulation needs to be very small (or adaptive) in order to catch an
accurate value of the adaptation at the time of the spike. It is not the case
for the similar quartic (Touboul 2008) and exponential (Brette and Gerstner
2005) models whose adaptation variable does not blow up in finite time, and
which are therefore very robust to changes in the cutoff value.
