Neurons do not work in isolation, but are embedded in networks of neurons with similar properties. Such networks of similar neurons can be organized as distributed assemblies or as local pools of neurons. Groups of neurons with similar properties can be approximated as homogeneous or weakly heterogeneous populations of neurons. In mathematical models, the connectivity within the population is typically all-to-all or random.

The population activity is defined as the number of spikes fired in a short instant of time, averaged across the population. Since each neuron in a population receives input from many others (either from the same and/or from other populations) its total input at each moment in time depends on the activity of the presynaptic population(s). Hence the population activity $A(t)$ controls the mean drive of a postsynaptic neuron.

If the population in a self-connected network is in a state of stationary activity, the expected value $\langle A_{0}\rangle$ of the population activity can be determined self-consistently. To do so, we approximate the mean drive of a neuron by $\langle A_{0}\rangle$ and exploit that the firing rate of the population must be equal to that of a single neuron. In the stationary state of asynchronous activity the population activity is therefore fully determined by the gain function of a single neuron (i.e., its frequency-current curve) and the strength of feedback connections. This result, which is an example of a (stationary) mean-field theory, is independent of any neuron model. The mean-field solution is exact for a fully connected network in the limit of a large number of neurons ($N\to\infty$), and a good approximation for large randomly connected networks.

The assumption of a stationary state is, of course, a strong limitation. In reality, the activity of populations in the brain responds to external input and may also show non-trivial intrinsic activity changes. In other words, the population activity is in most situations time-dependent. The mathematical description of the dynamics of the population activity is the topic of the next three chapters.

The development of population equations, also called ‘neural mass’ equations, had a first boom around 1972 with several papers by different researchers (552; 264; 15). Equations very similar to the population equations have sometimes also been used as effective rate model neurons (199). The transition from stationary activity to dynamics of population in the early papers is often ad hoc (552).

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., (15; 16; 17; 487; 91; 532; 533), and a mathematically precise formulation of mean-field theories for random nets is possible (146).

The theory for randomly connected integrate-and-fire neurons (21; 78; 79; 426) builds on earlier studies of formal random networks. The Siegert-formula for the gain function of a leaky integrate-and-fire model with diffusive noise appears in several classic papers (476; 21; 78; 79). In arbitrarily connected integrate-and-fire networks, the dynamics is highly complex (92).

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