The theory of `Balanced Neural Networks’ is a very popular explanation for the high degree of variability and stochasticity in the brain’s

activity. We determine equations for the hydrodynamic limit of a balanced all-to-all network of $2n$ neurons for asymptotically large

$n$. The neurons are divided into two classes (excitatory and inhibitory). Each excitatory neuron excites every other neuron, and each

inhibitory neuron inhibits all of the other neurons. The model is of a stochastic hybrid nature, such that the synaptic response of each

neuron is governed by an ordinary differential equation. The effect of neuron j on neuron k is dictated by a spiking Poisson Process, with

intensity given by a sigmoidal function of the synaptic potentiation of neuron j. The interactions are scaled by O(n^{-1/2}) , which is

much stronger than the O(n^{-1}) scaling of classical interacting particle systems ( the most common scaling used in mathematical

neuroscience). We demonstrate that, under suitable conditions, the system does not blow up as n tends to infinity because the network

activity is balanced between excitatory and inhibitory inputs. If the synaptic dynamics is linear, then the limiting population dynamics is proved to be Gaussian: with the mean

determined by the balanced between excitation and inhibition, and the variance determined by the Central Limit Theorem for

inhomogeneous Poisson Processes. The limiting equations can thus be expressed as autonomous Ordinary Differential Equations for

the means and variances. We search for conditions under which spatially-distributed patterns exist, as well as oscillations.