  Networks of model neurons with balanced recurrent excitation and inhibition
produce irregular and asynchronous spiking activity. We extend the analysis of
balanced networks to include the known dependence of connection probability on
the spatial separation between neurons. In the continuum limit we derive that
stable, balanced firing rate solutions require that the spatial spread of
external inputs be broader than that of recurrent excitation, which in turn
must be broader than or equal to that of recurrent inhibition. For finite size
networks we investigate the pattern forming dynamics arising when balanced
conditions are not satisfied. The spatiotemporal dynamics of balanced networks
offer new challenges in the statistical mechanics of complex systems.
