  We present a simple Markov model of spiking neural dynamics that can be
analytically solved to characterize the stochastic dynamics of a finite-size
spiking neural network. We give closed-form estimates for the equilibrium
distribution, mean rate, variance and autocorrelation function of the network
activity. The model is applicable to any network where the probability of
firing of a neuron in the network only depends on the number of neurons that
fired in a previous temporal epoch. Networks with statistically homogeneous
connectivity and membrane and synaptic time constants that are not excessively
long could satisfy these conditions. Our model completely accounts for the size
of the network and correlations in the firing activity. It also allows us to
examine how the network dynamics can deviate from mean-field theory. We show
that the model and solutions are applicable to spiking neural networks in
biophysically plausible parameter regimes.
