  We consider a discrete-time version of a Hawkes process defined as a Poisson
auto-regressive process whose parameters depend on the past of the trajectory.
We allow these parameters to take on negative values, modelling inhibition.
More precisely, the model is the stochastic process $(X_n)_{n\ge0}$ with
parameters $a_1,\ldots,a_p \in \mathbb{R}$, $p\in\mathbb{N}$ and $\lambda \ge
0$, such that for all $n\ge p$, conditioned on $X_0,\ldots,X_{n-1}$, $X_n$ is
Poisson distributed with parameter \[ \left(a_1 X_{n-1} + \cdots + a_p X_{n-p}
+ \lambda \right)_+ \] We consider specifically the case $p = 2$, for which we
are able to classify the asymptotic behavior of the process for the whole range
of parameters, except for boundary cases. In particular, we show that the
process remains stochastically bounded whenever the linear recurrence equation
$x_n = a_1x_{n-1} + a_2x_{n-1} + \lambda$ remains bounded, but the converse is
not true. Relatedly, the criterion for stochastic boundedness is not symmetric
in $a_1$ and $a_2$, in contrast to the case of non-negative parameters,
illustrating the complex effects of inhibition.
