  Finite-sized populations of spiking elements are fundamental to brain
function, but also used in many areas of physics. Here we present a theory of
the dynamics of finite-sized populations of spiking units, based on a
quasi-renewal description of neurons with adaptation. We derive an integral
equation with colored noise that governs the stochastic dynamics of the
population activity in response to time-dependent stimulation and calculate the
spectral density in the asynchronous state. We show that systems of coupled
populations with adaptation can generate a frequency band in which sensory
information is preferentially encoded. The theory is applicable to fully as
well as randomly connected networks, and to leaky integrate-and-fire as well as
to generalized spiking neurons with adaptation on multiple time scales.
