  We study a purely inhibitory neural network model where neurons are
represented by their state of inhibition. The study we present here is
partially based on the work of Cottrell \cite{Cot} and Fricker et al.
\cite{FRST}. The spiking rate of a neuron depends only on its state of
inhibition. When a neuron spikes, its state is replaced by a random new state,
independently of anything else and the inhibition state of the other neurons
increase by a positive value. Using the Perron-Frobenius theorem, we show the
existence of a Lyapunov function for the process. Furthermore, we prove a local
Doeblin condition which implies the existence of an invariant measure for the
process. Finally, we extend our model to the case where the neurons are indexed
by $ \mathbb{Z}. $ We construct a perfect simulation algorithm to show the
recurrence of the process under certain conditions. To do this, we rely on the
classical contour technique used in the study of contact processes, and
assuming that the spiking rate lies on the interval $[ \beta_* , \beta^* ], $
we show that there is a critical threshold for the ratio $ \delta=
\frac{\beta_*}{\beta^* - \beta_*}$ over which the process is ergodic. \\
\textbf{Keywords}: spiking rate, interacting neurons, perfect simulation
algorithm, classical contour technique.
