Subsections

• Literature

6.7 Summary

The dynamics of large populations of homogeneous neurons can efficiently be described by population equations. In the stationary state the population equations can be reduced to standard network equations with sigmoidal gain functions. In the limit of high noise or slowly changing stimulation, the description by the Wilson-Cowan differential equation or field equations is appropriate.

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.

Literature

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).