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be largely (though not completely) averaged out [105, 106], and the estimate of the
average activity of the pre-synaptic pool can be quite accurate even with a small D
t
.
In other words, the population firing rate can be defined (almost) instantaneously
in real time. Moreover, such a rate code can be readily decoded by a post-synaptic
cell: the summation of thousands of synaptic inputs provides a means to readout the
population firing rate at any time.
Population firing rate models were introduced in the early 1970s and have since
become widely popular in theoretical neuroscience. These models are described as
non-linear differential equations, to which tools of mathematical analysis are appli-
cable. Thus, concepts like attractor dynamics, pattern formation, synchronous net-
work oscillations, etc, have been introduced in the field of neurobiology (See [37]
for a review and references). Early models, such as associative memory models,
were formulated in terms of firing-rates [27, 64]. Broadly speaking, two different
approaches can be used to construct a firing-rate model. A rate model can be built
heuristically
: for example, a unit is assumed to have a threshold-linear or sigmoid
input-output relation [5, 125, 126]. This class of rate models is valuable for its sim-
plicity; important insights can be gained by detailed analysis of such models. The
drawback is that these models tend to be not detailed enough to be directly related to
electrophysiology. For example, the baseline and range of firing rates are arbitrarily
defined so they cannot be compared with those of real neurons. It is therefore dif-
ficult to use the available data to constrain the form of such models. On the other
hand, a firing-rate model can also be
derived
, either rigorously or approximately,
from a spiking neuron model. To do that, the dynamics of spiking neurons must
be well understood. The analytical study of the dynamics of spiking neuron mod-
els was pioneered by [68], and has witnessed an exponential growth in recent years.
Up to date, most of the work was done with the leaky-integrate-and-fire (LIF) neu-
ron model [1, 8, 11, 18, 19, 21, 24, 51, 68, 77, 88, 114, 120]. The LIF model is
a simple spiking model that incorporates basic electrophysiological properties of a
neuron: a stable resting potential, sub-threshold integration, and spikes. A network
model can be constructed with LIF neurons coupled by realistic synaptic interac-
tions. Such models have been developed and studied for many problems, such as
synchronization dynamics, sensory information processing, or working memory. In
some instances, firing-rate dynamics can be derived from the underlying spiking
neuron models [26, 36, 40, 109]. These firing rate models provide a more compact
description that can be studied in a systematical way.
Analytical studies of networks of neurons are usually performed in the context of
'mean-field' theories. In such theories, the synaptic input of a neuron in the network
is traditionally only described by its average: the 'mean-field'. This first class of
models is applicable to networks in which neurons are weakly coupled and fire in a
regular fashion. More recently, mean-field theories have been introduced in which
the synaptic inputs are described not only by their mean, but also by the fluctuations
of their synaptic inputs, which come potentially both from outside the network, and
from the recurrent inputs. This second class of models is applicable to strongly
coupled networks in which neurons fire irregularly [11, 118, 119].
The objective of this chapter is to provide a pedagogical summary of this latter
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