Stability of the random neural network model

Abstract

In a recent paper [1] we have introduced a new neural network model, called the Random Network, in which "negative" or "positive" signals circulate, modelling inhibitory and excitatory signals. They are summed at the input of each neuron and constitute its signal potential. The state of each neuron in this model is its signal potential, while the network state is the vector of signal potentials at each neuron. If its potential is positive, a neuron fires, and sends out signals to the other neurons of the network or to the outside world. As it does so its signal potential is depleted. We have shown that in the Markovian case, this model has product form, i.e. the steady-state probability distribution of its potential vector is the product of the marginal probabilities of the potential at each neuron. The signal flow equations of the network, which describe the rate at which positive or negative signals arrive to each neuron, are non-linear, so that their existence and uniqueness is not easily established except for the case of feedforward (or backpropagation) networks [1]. We examine two sub-classes of networks: balanced, and damped networks and obtain stability conditions in each case. A hardware implementation of these networks is also suggested.