The formulation of the population dynamics
on a *microscopic* level leads to
partial differential equations
for densities in the internal variables,
e.g., refractory or membrane potential density.
A description of the dynamics on the *macroscopic*
level leads to an integral equation
that is based on the
input-dependent interval distributions.
The relation between the
macroscopic activity equations
and the microscopic density equations
can be most easily demonstrated for
a population of Spike Response Model neurons
with escape noise.
Finally, we have seen that in
a sparsely connected network
of excitatory and inhibitory neurons
noise-like fluctuations
may arise even with deterministic dynamics.

The original paper of Wilson and Cowan (1972) can be recommended as the classical reference for population equations. It is worth while to also consult the papers of Knight (1972a) and Amari (1972) of the same year that each take a somewhat different approach towards a derivation of population activity equations. Some standard references for field equations are Wilson and Cowan (1973), Ellias and Grossberg (1975), and Amari (1977a).

The study of randomly connected networks has a long tradition in the mathematical sciences. Random networks of formal neurons have been studied by numerous researchers, e.g., (Sompolinsky et al., 1988; van Vreeswijk and Sompolinsky, 1996; Amari, 1977b; van Vreeswijk and Sompolinsky, 1998; Amari, 1972,1974; Cessac et al., 1994). The theory for integrate-and-fire neurons (Brunel, 2000; Brunel and Hakim, 1999; Amit and Brunel, 1997a) builds upon this earlier work. Finally, as an introduction to the density equation formalism for neurons, we recommend, apart from Abbott and van Vreeswijk (1993) and Brunel and Hakim (1999), the recent paper by Nykamp and Tranchina (2000). For the general theory of Fokker-Planck equations see Risken (1984).

Cambridge University Press, 2002

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