# Chapter 14 Quasi-Renewal Theory and the Integral-equation Approach

In the previous Chapter it was shown that an approach based on membrane potential densities can be used to analyze the dynamics of networks of integrate-and-fire neurons. For neuron models that include biophysical phenomena such as refractoriness and adaptation on multiple time-scales, however, the resulting system of partial differential equations is situated in more than two dimensions and therefore difficult to solve analytically; even the numerical integration of partial differential equations in high dimensions is slow. To cope with these difficulties, we now indicate an alternative approach to describing the population activity in networks of model neurons. The central concept is expressed as an integral equation of the population activity.

The advantage of the integral equation approach is four-fold. First, the approach works for a broad spectrum of neuron models, such as the Spike Response Model with escape noise and other Generalized Linear Models (cf. Chapter 9) for which parameters can be directly extracted from experiments (cf. Chapter 11). Second, it is easy to assign an intuitive interpretation to the quantities that show up in the integral equation. For example the interspike interval distribution plays a central role. Third, an approximative mathematical treatment of adaptation is possible not only for the stationary population activity, but also for the case of arbitrary time-dependent solutions. Fourth, the integral equations provide a natural basis for the transition to classical ‘rate equations’ which will be discussed in Chapter 15.

In Section 14.1, we derive, starting from a small set of assumptions, an integral equation for the population activity. The essential idea of the mathematical formulation is to remain at the macroscopic level as much as possible, without reference to a specific model of neuronal dynamics. Knowledge of the interval distribution $P_{I}(t|\hat{t})$ for arbitrary input $I(t)$ is enough to formulate the population equations.

For didactic purposes, we begin by treating neurons without adaptation. In this case, the internal state of the neurons depends solely on the input and on the time since the last spike. The formulation of the macroscopic integral equation exploits the concepts of a time-dependent version of renewal theory that we have already encountered in Chapter 7. In the presence of adaptation, however, the state of the neuron depends not only on the last spike, but also on all the previous spike times. But since the refractoriness caused by the last spike dominates over the effects of earlier spikes we can approximate the interval distribution for adaptive neurons by a ‘quasi-renewal’ theory.

A theory for networks consisting of several interacting populations of spiking neurons is formulated in Section 14.2. In order to analyze the stability of stationary solutions of asynchronous firing with population activity $A_{0}$ in connected networks of integrate-and-fire neurons, we need to know the linear response filter. The linearization of the integral equations under the assumption of a small perturbation is presented in Section 14.3.

The integral equation of Section 14.1, is exact in the limit of a large number of neurons and can be interpreted as a solution to partial differential equations analogous to those of the previous chapter. Section 14.4, which is slightly more technical, presents the relation of the integral equation to an approach by membrane potential density equations.

Section 14.5 describes in detail, how we can formulate integral equations for adaptive neurons. Finally, Section 14.6 contains variants of the theory that are applicable to populations of a finite number of neurons.