  This article is devoted to the theoretical and numerical analysis of a
network of excitatory and inhibitory neurons of Hodgkin-Huxley (HH) type, for
which the topology is inspired by that of a single local layer of visual cortex
V1. Our model is related to the recent CSY model and therefore differs from
other classical models in the field. It combines a driven stochastic drive --
which may be interpreted as an ambient drive for each neuron -- with recurrent
inputs resulting from the network activity. After a review of the dynamics of a
single HH equation for both the deterministic and the stochastically driven
case, we proceed to an analysis of the network. This analysis reveals emergent
properties of the system such as partial synchronization and synchronization
(defined here as a state of the network for which all the neurons spike within
a short interval of time), correlation between excitatory and inhibitory
conductances, and oscillations in the gamma-band frequency. The collective
behavior enumerated herein is observed when the input-amplitude parameter
$S^{EE}$ measuring excitatory-to-excitatory coupling (recurrent excitation)
increases to within a certain range. Of note, our work indicates a distinct
mechanism for obtaining the emergent properties, some of which have been
classically observed. As a consequence our article contributes to the
understanding of how assemblies of inhibitory and excitatory cells interact
together to produce rhythms in the network. It also aims to bring problems from
neuroscience to the realm of mathematics, where they can be analyzed
rigorously.
