In Section 14.1 we discussed a single homogeneous population of $N$ neurons. The formalism of integral equations introduced there can readily be adapted to several populations with recurrent interactions within and across populations.

We consider a coupled network of spiking neurons of the renewal type, such as nonlinear (or leaky) integrate-and-fire neurons with escape noise or a Spike Response Model SRM${}_{0}$. Neurons within a population have the same set of parameters whereas neurons of different populations can have different parameters. The activity of population $k$ is described by Eq. (14.5)

where $P_{I_{k}}(t|\hat{t})$ is the input-dependent interval distribution of population $k$. The input to population $k$ is

Here $N_{n}$ is the number of neurons in the presynaptic population $n$, $w_{kn}$ is the strength of a synaptic connection of a neuron in population $n$ to a neuron in population $k$ and $\alpha_{kn}$ the time course of the postsynaptic current into a neuron in population $k$, caused by spike firing of a neuron in population $n$. Conductance-based synapses are treated in the current-based approximation (cf. Section 13.6.3 in Ch. 13). Connections can be excitatory or inhibitory, depending on the choice of $w_{kn}$. Because the overall strength of the connection is incorporated in $w_{kn}$, we can, without loss of generality, assume a normalization $\int_{0}^{\infty}\alpha_{kn}(s){\text{d}}s=1$.

The noise level of each neuron in population $k$ is fixed to a value of $\sigma_{k}$. The choice of noise is arbitrary. If our aim is to mimic stochastic spike arrivals in randomly connected networks by an escape rate, a suitable choice of escape function has been given in Chapter 9. In practice this implies that noise in the input to a neuron is effectively described and replaced by escape noise in its output.

In the following we assume a fully connected network with interaction strength $w_{kn}=J_{kn}/N_{n}$; cf. Chapter 12. If the theory is applied to a random network where each neuron in population $k$ has a fixed number of presynaptic partners $C_{kn}$ in population $n$, we can use the same theory, except that (i) we use $w_{kn}=J_{kn}/C_{kn}$; and (ii) we increase the noise level $\sigma$ in the escape rate function so as to mimic the additional noise caused by stochastic spike arrival; see Section 9.4.

We are interested in the value $A_{k}(t)=A_{k,0}$ of the population activity in the stationary state of asynchronous firing. We recall from Ch. 12 that stationary activity means that the expected value of the population activity is constant whereas in a simulation the actual value $A(t)$ always fluctuates (Fig. 14.5). To lighten the notation, we consider a single population $k$ with self-interaction $w_{kk}=J_{0}/N_{k}$ and drop the index $k$ in the following. The input from other populations is summarized as a constant external input $I^{\rm ext}$. According the our assumptions, all neurons in the population have the same parameters and can be described by time-dependent renewal theory. The level of noise is indicated by an index $\sigma$.

A stationary state of asynchronous firing requires that the total input $I_{0}$ is constant (or at least stationary). In Chapter 12, we have seen that the population activity $A_{0}$ in the state of asynchronous firing is given by the single-neuron firing rate $\nu=g_{\sigma}(I_{0})$. We thus have

Given constant activity $A_{0}$ of the population and constant external input $I_{0}^{\rm ext}$, the total input $I_{0}$ to each neuron is constant. From Eq. (14.18) we find the total input to a neuron in population $k$

Eq. (14.19) and (14.20) together yield the following equation for the population activity $A_{0}$

This result agrees with the general result found earlier in Chapter 12 for the stationary state in a network with self-coupling. Solutions can be found graphically (Fig. 14.6) using the same method as in Chapter 12.

The advantage of the integral equation approach is two-fold. First, for the integral-equations we have transparent mathematical tools to analyze the stability of the stationary solution, as shown in Section 14.2.3.

Second, we can write down the gain function $g_{\sigma}(I)$ and an expression for $A_{0}$ in a compact form, as will be shown now. Because the input is constant, the state of each neuron depends only on the time since the last spike $t-\hat{t}$. We are thus in the situation of stationary renewal theory. Therefore, the survivor function and the interval distribution cannot depend explicitly upon the absolute time, but only on the time difference $s=t-\hat{t}$. Hence we set

The value of the stationary activity $A_{0}$ follows now directly from the normalization Eq. (14.8),

We use $\frac{d}{ds}S_{0}(s)=-P_{0}(s)$ and integrate by parts

where the last equality follows from the definition of the mean interspike interval (Chapter 9). Hence

The result has an intuitively appealing interpretation: if everything is constant, then averaging over time (for a single neuron) is the same as averaging over a population of identical neurons; cf. the discussion in Chapter 12.

In the previous subsection we have assumed that the network is in a state of asynchronous firing. In this section, we study whether asynchronous firing can indeed be a stable state of a fully connected population of spiking neurons – or whether the connectivity drives the network toward oscillations. For the sake of simplicity, we restrict the analysis to SRM${}_{0}$ neurons; the same methods can, however, be applied to integrate-and-fire neurons or spiking neurons formulated in the framework of Generalized Linear Models.

For SRM${}_{0}$ neurons (cf. Chapter 9), the membrane potential is given by

where $\eta(t-\hat{t}_{i})$ is the effect of the last firing of neuron $i$ (i.e., the spike afterpotential) and $h(t)$ is the total postsynaptic potential caused by presynaptic firing. If all presynaptic spikes are generated within the homogeneous population under consideration, we have

Here $\epsilon_{0}(t-t_{j}^{(f)})$ is the time course of the postsynaptic potential generated by a spike of neuron $j$ at time $t_{j}^{(f)}$ and $w_{ij}=J_{0}/N$ is the strength of lateral coupling within the population. The second equality sign follows from the definition of the population activity, i.e., $A(t)=N^{-1}\sum_{j}\sum_{f}\delta(t-t_{j}^{(f)})$; cf. Chapter 12. For the sake of simplicity, we have assumed in Eq. (14.33) that there is no external input.

The state of asynchronous firing corresponds to a fixed point $A(t)=A_{0}$ of the population activity. We have already seen in the previous subsection as well as in Chapter 12 how the fixed point $A_{0}$ can be determined either numerically or graphically. To analyze its stability we assume that for $t>0$ the activity is subject to a small perturbation,

with $A_{1}\ll A_{0}$. This perturbation in the activity induces a perturbation in the input potential,

The perturbation of the potential causes some neurons to fire earlier (when the change in $h$ is positive), and others to fire later (whenever the change is negative). The perturbation may therefore build up ($\lambda>0$, the asynchronous state is unstable) or decay back to zero ($\lambda<0$, the asynchronous state is stable). At the transition between the region of stability and instability the amplitude of the perturbation remains constant ($\lambda=0$, marginal stability of the asynchronous state). These transition points, defined by $\lambda=0$, are determined now.

We start from the population integral equation $A(t)=\int_{-\infty}^{t}P_{I}(t|\hat{t})\,A(\hat{t})\,{\text{d}}\hat{t}$ that has been introduced in Section 14.1. Here $P_{I}(t|\hat{t})$ is the input-dependent interval distribution, i.e., the probability density of emitting a spike at time $t$ given that the last spike occurred at time $\hat{t}$. The linearized response of the population activity to a small change $\Delta I$ in the input can, under general smoothness assumptions, always be written in the form

where $G(s)$ is the linear response filter in the time domain. The Fourier transform $\hat{G}(\omega)$ is the frequency dependent gain function. The explicit form of the filter will be derived in the framework of the integral equations in Section 14.3.

Instead of thinking of a stimulation by an input current $\Delta I(t)$, it is more convenient to work with the input potential $h(t)=\int_{0}^{\infty}\kappa(s)I(t-s)\,{\text{d}}s$, because the neuron model equations have been defined on the level of the potential; cf. Eq. (14.32). We use $\Delta A(t)=A_{1}\,e^{i\omega t+\lambda t}$ and $\Delta h(t)=h_{1}\,e^{i\omega t+\lambda t}$ in Eq. (14.36) and search for the critical value $\lambda=0$ where the stable solution turns into an unstable one. After cancellation of a common factor $A_{1}\exp(i\omega t)$ the result can be written in the form

Here, $\hat{\kappa}(\omega)$ and $\hat{\epsilon}$ are the Fourier transform of the membrane kernel $\kappa(s)$, and the time course of the postsynaptic potential $\epsilon_{0}(s)$ caused by an input spike, respectively. Typically, $\kappa(s)=(R/\tau_{m})\,\exp(-s/\tau_{m})$ where $\tau_{m}$ is the membrane time constant. If the synaptic input is a short current pulse of unit charge, $\epsilon_{0}$ and $\kappa$ are identical, but we would also like to include the case of synaptic input currents with arbitrary time dependence and therefore keep separate symbols for $\epsilon_{0}$ and $\kappa$. The second equality sign defines the real-valued functions $S_{f}(\omega)$ and $\Phi(\omega)$.

Equation (14.37) is thus equivalent to

Solutions of Eq. (14.38) yield bifurcation points where the asynchronous firing state looses its stability toward an oscillation with frequency $\omega$.

We have written Eq. (14.38) as a combination of two requirements, i.e., an amplitude condition $S_{f}(\omega)=1$ and a phase condition $\Phi(\omega)\;{\rm mod}\;2\pi=0$. Let us discuss the general structure of the two conditions. First, if $S_{f}(\omega)\leq 1$ for all frequencies $\omega$, an oscillatory perturbation cannot build up. All oscillations decay and the state of asynchronous firing is stable. We conclude from Eq. (14.37) that by increasing the absolute value $|J_{0}|$ of the coupling constant, it is always possible to increase $S_{f}(\omega)$. The amplitude condition can thus be met if the excitatory or inhibitory feedback from other neurons in the population is sufficiently strong. Second, for a bifurcation to occur we need in addition that the phase condition is met. Loosely speaking, the phase condition implies that the feedback from other neurons in the network must arrive just in time to keep the oscillation going. Thus the axonal signal transmission time and the rise time of the postsynaptic potential play a critical role during oscillatory activity (4; 181; 519; 523; 182; 78; 79; 183).