  A perturbative method is developed for calculating the effects of recurrent
synaptic interactions between neurons embedded in a network. A series expansion
is constructed that converges for networks with noisy membrane potential and
weak synaptic connectivity. The terms of the series can be interpreted as loops
of interactions between neurons, so the technique is called a loop-expansion. A
diagrammatic method is introduced that allows for construction of analytic
expressions for the parameter dependencies of the spike probability function
and correlation functions. An analytic expression is obtained to predict the
effect of the surrounding network on a neuron during an intracellular current
injection. The analytic results are compared with simulations to test the range
of their validity and significant effects of the the recurrent connections in
network are accurately predicted by the loop-expansion.
