In this chapter we have studied an
integral equation for the population dynamics.
and discussed its relation to density methods.
The validity of the population equations
relies on three assumptions:
(i) a *homogeneous* population of
(ii) an *infinite* number of neurons
which show (iii) *no adaptation*.

It is clear that there are no large and completely *homogeneous*
populations in biology. The population equations may nevertheless be a useful
starting point for a theory of heterogeneous populations
(Brunel and Hakim, 1999; Chow, 1998; Senn et al., 1996; Tsodyks et al., 1993; Pham et al., 1998). We may distinguish between
heterogeneity in the coupling weights *w*_{ij} and heterogeneity in the local
parameters of the neurons, e.g., the threshold or reset value. The case of
randomly chosen weights has been discussed in Section 6.4.3.
In the stationary case, the population activity equations can be discussed by
solving simultaneously for the mean activity *A*_{0} and the noise amplitude
. The form of the population activity is similar to that of a
homogeneous network. 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 noise model
has been termed `noisy reset' in Chapter 5.4 and discussed
as an example of slow noise in parameters. 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 condition is the limit of a *large network*. For
*N*
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* |); cf. Chapter 5. For networks
with internal coupling, an exact treatment of finite size effects is
difficult. For escape noise first attempts towards a description of the
fluctuations have been made (Spiridon et al., 1998; Meyer and van Vreeswijk, 2001). For diffusive noise,
finite size effects in the low-connectivity limit have been treated by
Brunel and Hakim (1999).

The limit of *no adaptation* seems to be valid for fast-spiking neurons
(Connors and Gutnick, 1990). Most cortical neurons, however, show adaptation. From the
modeling point of view, all integrate-and-fire neurons that have been
discussed in Chapter 4 are in the class of non-adaptive neurons,
since the membrane potential is reset (and the past forgotten) after each
output spike. The condition of short memory (= no adaptation) leads to the
class of renewal models (Perkel et al., 1967a; Cox, 1962; Stein, 1967b) and this is where the
population equation applies; cf. (Gerstner, 1995,2000b). A
generalization of the population equation to neuron models with adaptation is
not straightforward since the formalism assumes that only the last spike
suffices. On the other hand, adaptation could be included phenomenologically
by introducing a slow variable that integrates over the population activity in
the past. A full treatment of adaptation would involve a density description
in the high-dimensional space of the microscopic neuronal variables
(Knight, 2000).

Cambridge University Press, 2002

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