Neuronal populations in biology are neither completely homogeneous nor infinitely large. In order to treat heterogeneity in local neuronal parameters, the variability of a parameter between one neuron and the next is often replaced by slow noise in the parameters. For example, a population of integrate-and-fire neurons where the reset value $u_{r}$ is different for each neuron is replaced by a population where the reset values are randomly chosen after each firing (and not only once at the beginning). Such a model of slow noise in the parameters has been discussed in the example of Section 14.3. The replacement of heterogeneity by slow noise neglects, however, correlations that would be present in a truly heterogeneous model. To replace a heterogeneous model by a noisy version of a homogeneous model is somewhat ad hoc, but common practice in the literature.

The second question is whether we relax the condition of a large network. For $N\to\infty$ the population activity shows no fluctuations and this fact has been used for the derivation of the population equation. For systems of finite size fluctuations are important since they limit the amount of information that can be transmitted by the population activity. For a population without internal coupling ($J_{0}=0$), fluctuations can be calculated directly from the interval distribution $P_{I}(t\,|\hat{t})$ if the population consists of neurons that can be described by renewal theory; cf. Chapter 9. For networks with recurrent connections, several attempts toward a description of the fluctuations have been made (491; 341; 299). Here we present a different approach.

If we consider a network with a finite number $N$ of neurons, the integral equation (14.5) which describes the evolution of the population activity $A(t)$ in terms of the input-dependent interval distribution $P_{I}(t|\hat{t})$ should be written more carefully with expectations signs,

so as to emphasize that the left-hand side is the expected population activity at time $t$, given the observed population activity at earlier times $\hat{t}$. In other words $N\,\langle A(t)\rangle\,\Delta t=N\langle m_{0}(t)\rangle$ is the expected number of spikes to occur in a a short interval $\Delta t$. Here we have defined $m_{0}(t)$ as the fraction of neurons that fire in a time step $\Delta t$, just as in the previous section. Given the past input for $t^{\prime}<t$ (which is the same for all the $N$ neurons in the group), the firing of the neurons is independent in the next time step (’conditional independence’). Therefore in the limit of $N\to\infty$ the observed variable $m_{0}(t)$ approaches $\langle m_{0}(t)\rangle$ and we can drop the expectation signs.

For finite $N$, the variable $m_{0}(t)$ fluctuates around $\langle m_{0}(t)\rangle$. In order to determine these fluctuations, we assume that $N$ is large, but finite. For finite $N$ the population activity $A(t)$ can be written in the form of a ‘noisy’ integral equation

where $\xi(t)$ is a Gaussian white noise, $A(\hat{t})$ is the observed activity in the past, $S_{I}^{\rm noise}(t|\hat{t})$ is the fraction of neurons that have survived up to time $t$ after a last spike at time $\hat{t}$, and $\rho(t|\hat{t})^{\rm noise}$ is the stochastic intensity of that group of neurons. Starting from discrete time steps, and then taking the continuum limit, it is possible to determine the amplitude of the fluctuations as $\sigma(t)=\sqrt{\langle A(t)\rangle/N}$. Eq. 14.98 can be used to evaluate the correlations $\langle A(t)\,A(t^{\prime})\rangle$, in coupled networks of finite size (122).