14 The Integral-equation Approach


The original paper of Wilson and Cowan (552) can be recommended as the classical reference for population equations. The paper contains the integral equation for neurons with absolute refractoriness as well as, in the appendix, the case of relative refractoriness; Note, however, that the paper is most often cited for a differential equation for an ‘ad hoc’ rate model that does not correctly reflect the dynamics of neurons with absolute refractoriness. It is worth while to also consult the papers of Knight (264) and Amari (15) of the same year that each take a somewhat different approach toward a derivation of population activity equations.

The presentation of the integral equations for time-dependent renewal theory (Eqs. (14.5) and (14.8)) in this chapter follows the general arguments developed in Gerstner (182, 183) emphasizing that the equations do not rely on any specific noise model. The same integral equations can also be found in the appendix of Wilson and Cowan (552) as an approximation to a model with heterogeneity in the firing thresholds and have been derived by integration of the partial differential equations for refractory densities with escape noise in Gerstner and van Hemmen (180). The integral approach for adaptive neurons and the approximation scheme based on the moment-generating function was introduced in (358).

The linearization of the integral equation can be found in Gerstner (183); Gerstner and van Hemmen (181). The model of slow noise in the parameters is taken from Gerstner (183). The escape noise model - which turns out to be particularly convenient for the integral equation approach - is intimately linked to the noise model of Generalized Linear Models, as discussed in Chapter 9 where references to the literature are given.