Mean-Field Theory of Irregularly Spiking Neuronal Populations and Working Memory in Recurrent Cortical Networks - Computational Neuroscience: A Comprehensive Approach - page 472

15.4 Summary and future directions
In this chapter, we have presented analytical mean-field techniques that can be used
to study the collective properties of large networks of spiking neurons. In analyz-
ing the self-consistent steady-states of these networks, we observed that the self-
consistency equations have sometimes multiple stable states. This leads quite natu-
rally to the the interpretation of these networks as models of working memory sys-
tems. The methods discussed here help to understand in detail in which conditions
multistability can be achieved in large networks of spiking neurons. The results that
have been discussed are of course only the current status of a rapidly growing field.
Extensions of both the mean-field techniques and of network architectures for work-
ing memory are either already done, under way, or should be done in the near future.
We discuss here several of these possible extensions.

•

More realistic single neuron models. The LIF lacks several features of real
neurons. First, it lacks any sub-threshold resonance phenomena [65]. General-
izations of LIFs with several variables have been introduced that possess such
sub-threshold resonance properties and can be studied analytically in stochas-
tic contexts along the lines of Section 15.2.3 [98]. Second, it lacks an intrinsic
spiking mechanism. The firing rate of neurons with intrinsic spike genera-
tion mechanism can be studied in the context of the 'quadratic integrate-and-
fire' neuron [22], and even more realistic neurons can be studied analytically
(Fourcaud et al. SFN 2002 abstract). Furthermore, mean-field theory can be
extended to a recurrent network of Hodgkin-Huxley-type conductance-based
single neurons [109]. This generalization may be important, e.g., the network
stability may be different depending on whether single neurons are described
by Hogkin-Huxley-type models or leaky integrate-and-fire models [26, 50].

• More realistic synaptic dynamics. The mean-field description of realistic

synaptic interactions can be improved in at least two ways. First, synaptic
fluctuations act through conductance changes, which are multiplied with the
driving force

to yield synaptic current. Therefore the noise is mul-
tiplicative. We have sidepassed this difficulty by replacing V with its average,
so that the noise term becomes additive to the voltage equation. It would be
desirable to be able to deal analytically with multiplicative noise. Second,
synapses display short-term depression and facilitation [113, 131]. Mean-field
models that incorporate synaptic depression have been investigated [115, 120],
but the implications of short-term plasticity to recurrent networks, especially
to working memory models, still await to be fully explored.

( V − E syn )

• Extension to correlations between neurons. In this chapter we have al-

ways assumed that the spiking activity of different cells was independent.
Although the experimentally observed cross-correlations are relatively weak
[14, 31, 74, 130], they might have a large impact on the input-output relation-