We introduce a new class of random “neural” networks in which signals are either negative or positive. A positive signal arriving at a neuron increases its total signal count or potential by one; a negative signal reduces it by one if the potential is positive, and has no effect if it is zero. When its potential is positive, a neuron “fires,” sending positive or negative signals at random intervals to neurons or to the outside. Positive signals represent excitatory signals and negative signals represent inhibition. We show that this model, with exponential signal emission intervals, Poisson external signal arrivals, and Markovian signal movements between neurons, has a product form leading to simple analytical expressions for the system state.