  We consider a noise driven network of integrate-and-fire neurons. The network
evolves as result of the activities of the neurons following
spike-timing-dependent plasticity rules. We apply a self-consistent mean-field
theory to the system to obtain the mean activity level for the system as a
function of the mean synaptic weight, which predicts a first-order transition
and hysteresis between a noise-dominated regime and a regime of persistent
neural activity. Assuming Poisson firing statistics for the neurons, the
plasticity dynamics of a synapse under the influence of the mean-field
environment can be mapped to the dynamics of an asymmetric random walk in
synaptic-weight space. Using a master-equation for small steps, we predict a
narrow distribution of synaptic weights that scales with the square root of the
plasticity rate for the stationary state of the system given plausible
physiological parameter values describing neural transmission and plasticity.
The dependence of the distribution on the synaptic weight of the mean-field
environment allows us to determine the mean synaptic weight self-consistently.
The effect of fluctuations in the total synaptic conductance and plasticity
step sizes are also considered. Such fluctuations result in a smoothing of the
first-order transition for low number of afferent synapses per neuron and a
broadening of the synaptic weight distribution, respectively.
