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 (Xn) n≥0 with parameters a1,. .. , ap ∈ R, p ∈ N and λ ≥ 0, such that for all n ≥ p, conditioned on X0,. .. , Xn−1, Xn is Poisson distributed with parameter (a1Xn−1 + • • • + apXn−p + λ) + 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 xn = a1xn−1 + a2xn−1 + λ remains bounded, but the converse is not true. Relatedly, the criterion for stochastic boundedness is not symmetric in a1 and a2, in contrast to the case of non-negative parameters, illustrating the complex effects of inhibition.