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tially and temporally coarse-scale quantities, e.g. by the number of spikes emitted
by a local population of neurons in a larger time interval (their spike rate) [33].
Nevertheless, there is accumulating evidence [6, 44, 45, 55, 121, 130] that the tim-
ing of spikes may be highly coordinated between neurons and play a role in neural
processing as well. Neurons that spike coincidentally within a few milliseconds,
or with a precise time lag between them have been observed in dierent neuronal
systems [6, 44, 67, 87, 109, 119, 121, 130]. Coincident spiking can occur with high
statistical signicance correlated to internal states of the brain [121, 130] and pat-
terns of spikes may re-occur repeatedly and in a particular order (second order
spike patterns) [44, 45]. Patterns of precisely timed spikes and synchronization
in the millisecond range are therefore discussed to be essential for information
processing in the brain [4{6, 17, 31, 96, 117, 124, 130]. For a number of physi-
ological experiments [6, 67, 87, 130] however, the statistical signicance of some
of the ndings is currently highly debated [14, 99, 109, 119]. It has been argued
that in some experiments [6, 87, 130] the signicance of the occurrences of spike
patterns highly depends on the underlying statistical assumptions about the spike
trains [14, 119]. Further, [109] shows that the occurrence of repeated dynamical
motifs of the membrane potential (which were assumed to indicate and generate
spike patterns) is equally likely in random or randomized sub-threshold dynamics
if the randomly generated membrane potential has similar coarse statistical prop-
erties (such as the power spectrum) as the actually measured one. It is thus still
an open problem whether and how neurons may precisely coordinate their spiking
activity across complex networks, and which role the identity of individual neurons
and their inter-connectivity actually play.
Below we present two classes of hypotheses that may explain the dynamical
origin of patterns of precisely timed spikes and microscopic, inter-neuronal syn-
chronization, i.e. non-random, coincident spiking. One hypothesis states that feed-
forward anatomical structures are embedded in cortical circuits and support the
propagation of synchronous spiking activity of groups of neurons that constitute
the layers of the feed-forward architecture [4, 5, 38, 62]. This kind of dynamics
was termed `synre chain' activity [4]. As in current physiological experiments only
small subsets of neurons are observed, the synre chain hypothesis permits the oc-
currence of spiking activity that is synchronized with millisecond precision as well
as the persistence of spike patterns over longer time periods. A second, alterna-
tive hypothesis states that recurrent networks may collectively organize patterns of
precisely timed spikes without the need of specic feed-forward anatomy. We will
give more emphasis to this latter hypothesis as it is more recent and its theoretical
aspects are only marginally described so far in standard references.
This chapter is organized as follows: In Section 13.2, we briey present the
key ideas underlying synre chain dynamics and state the main results on this
topic. The remainder of this chapter is devoted to recurrent network models. In
Section 13.3 we introduce a class of analytically tractable models of spiking neural
