Steady State of an Inhibitory Neural Network

We investigate the dynamics of a neural network where each neuron evolves according to the combined effects of deterministic integrate-and-fire dynamics and purely inhibitory coupling with K randomly-chosen "neighbors". The inhibition reduces the voltage of a given neuron by an amount Delta when one of its neighbors fires. The interplay between the integration and inhibition leads to a steady state which is determined by solving the rate equations for the neuronal voltage distribution. We also study the evolution of a single neuron and find that the mean lifetime between firing events equals 1+K*Delta and that the probability that a neuron has not yet fired decays exponentially with time.