Balanced Networks of Spiking Neurons with Spatially Dependent Recurrent Connections

Networks of model neurons with balanced recurrent excitation and inhibition capture the irregular and asynchronous spiking activity reported in cortex. While mean-field theories of spatially homogeneous balanced networks are well understood, a mean-field analysis of spatially heterogeneous balanced networks has not been fully developed. We extend the analysis of balanced networks to include a connection probability that depends 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. Notably, this implies that network models with broad recurrent inhibition are inconsistent with the balanced state. For finite size networks, we investigate the pattern-forming dynamics arising when balanced conditions are not satisfied. Our study highlights the new challenges that balanced networks pose for the spatiotemporal dynamics of complex systems.