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 neu
rons 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.
Neural firing is often subject to negative feedback by adaptation currents. These currents can induce strong correlations among the time intervals between spikes. Here we study analytically the interval correlations of a broad class of noisy neural o
scillators with spike-triggered adaptation of arbitrary strength and time scale. Our weak-noise theory provides a general relation between the correlations and the phase-response curve (PRC) of the oscillator, proves anti-correlations between neighboring intervals for adapting neurons with type I PRC and identifies a single order parameter that determines the qualitative pattern of correlations. Monotonically decaying or oscillating correlation structures can be related to qualitatively different voltage traces after spiking, which can be explained by the phase plane geometry. At high firing rates, the long-term variability of the spike train associated with the cumulative interval correlations becomes small, independent of model details. Our results are verified by comparison with stochastic simulations of the exponential, leaky, and generalized integrate-and-fire models with adaptation.
