  Bidimensional spiking models currently gather a lot of attention for their
simplicity and their ability to reproduce various spiking patterns of cortical
neurons, and are particularly used for large network simulations. These models
describe the dynamics of the membrane potential by a nonlinear differential
equation that blows up in finite time, coupled to a second equation for
adaptation. Spikes are emitted when the membrane potential blows up or reaches
a cutoff value. The precise simulation of the spike times and of the adaptation
variable is critical for it governs the spike pattern produced, and is hard to
compute accurately because of the exploding nature of the system at the spike
times. We thoroughly study the precision of fixed time-step integration schemes
for this type of models and demonstrate that these methods produce systematic
errors that are unbounded, as the cutoff value is increased, in the evaluation
of the two crucial quantities: the spike time and the value of the adaptation
variable at this time. Precise evaluation of these quantities therefore involve
very small time steps and long simulation times. In order to achieve a fixed
absolute precision in a reasonable computational time, we propose here a new
algorithm to simulate these systems based on a variable integration step method
that either integrates the original ordinary differential equation or the
equation of the orbits in the phase plane, and compare this algorithm with
fixed time-step Euler scheme and other more accurate simulation algorithms.
