We present a detailed theoretical framework for statistical descriptions of
neuronal networks and derive $(1+1)$-dimensional kinetic equations, without
introducing any new parameters, directly from conductance-based
integrate-and-fire neuronal networks. We describe the details of derivation
of our kinetic equation, proceeding from the simplest case of one excitatory
neuron, to coupled networks of purely excitatory neurons, to coupled
networks consisting of both excitatory and inhibitory neurons. The dimension
reduction in our theory is achieved via novel moment closures. We also
describe the limiting forms of our kinetic theory in various limits, such as
the limit of mean-driven dynamics and the limit of infinitely fast
conductances. We establish accuracy of our kinetic theory by comparing its
prediction with the full simulations of the original point-neuron networks.
We emphasize that our kinetic theory is dynamically accurate, i.e., it
captures very well the instantaneous statistical properties of neuronal
networks under time-inhomogeneous inputs.