We study the solutions of a McKean-Vlasov stochastic differential equation (SDE) driven by a Poisson process. In neuroscience, this SDE models the mean field limit of a system of $N$ interacting excitatory neurons with $N$ large. Each neuron spikes randomly with rate depending on its membrane potential. At each spiking time, the neuron potential is reset to the value $\bar v$, its adaptation variable is incremented by $\bar w$ and all other neurons receive an additional amount $J/N$ of potential after some delay where $J$ is the connection strength. Between jumps, the neurons drift according to some two-dimensional ordinary differential equation with explosive behavior. We prove the existence and uniqueness of solutions of a heuristically derived mean-field limit of the system when $N\to\infty$. We then study the existence of stationary distributions and provide several properties (regularity, tail decay, etc.) based on a Doeblin estimate using a Lyapunov function. Numerical simulations are provided to assess the hypotheses underlying the results.