  Correlations are employed in modern physics to explain microscopic and
macroscopic phenomena, like the fractional quantum Hall effect and the Mott
insulator state in high temperature superconductors and ultracold atoms.
Simultaneously probed neurons in the intact brain reveal correlations between
their activity, an important measure to study information processing in the
brain that also influences macroscopic signals of neural activity, like the
electro encephalogram (EEG). Networks of spiking neurons differ from most
physical systems: The interaction between elements is directed, time delayed,
mediated by short pulses, and each neuron receives events from thousands of
neurons. Even the stationary state of the network cannot be described by
equilibrium statistical mechanics. Here we develop a quantitative theory of
pairwise correlations in finite sized random networks of spiking neurons. We
derive explicit analytic expressions for the population averaged cross
correlation functions. Our theory explains why the intuitive mean field
description fails, how the echo of single action potentials causes an apparent
lag of inhibition with respect to excitation, and how the size of the network
can be scaled while maintaining its dynamical state. Finally, we derive a new
criterion for the emergence of collective oscillations from the spectrum of the
time-evolution propagator.
