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together, current theoretical knowledge supports the synre hypothesis in systems
with specic constraints on synaptic dynamics, single neuron features, neural inter-
actions, and inter-connectivity between groups.
The current stage of experimental research is inconclusive. Despite some inter-
esting studies which might support the synre chain hypothesis [94, 113, 125, 139],
there is no key experiment that directly proves { or disproves { the existence of
synre chain anatomy or dynamics. Such an experiment would require either a
large-scale structural investigation of local cortical anatomy, proving or excluding
the necessary non-random feed-forward connectivity; or a large-scale dynamical
study, recording spikes of a large number of neurons simultaneously and repeatedly
under controlled conditions, such as to explicitly show (or exclude) the existence of
synchronous activity propagating along xed paths.
13.3. Recurrent Neural Networks
Alternatively, in recurrent networks without specically embedded feed-forward
structures, mechanisms other than synchronous excitation along feed-forward
anatomy might generate spikes that are precisely coordinated in time and among
dierent neurons. For recurrent networks, however, theoretical investigations that
take into account individual neurons' spike times and thus go beyond mean-eld de-
scriptions are often highly non-standard. Thus up to date precise timing of spikes
in recurrent networks is far less understood than synre chain dynamics. Con-
ceptual challenges include the nonlinear features of individual neurons and their
interactions, the complex recurrent connectivity of the networks, the existence of
transmission delays that make the dynamical systems formally innite-dimensional,
and strong heterogeneities that might be present among the neurons and their in-
teractions.
To cope with these challenges, many studies have focused on networks of ide-
alized model neurons, e.g. of integrate-and-re type [33, 51, 71, 86, 120]. In the
following, we introduce a class of spiking neural network models for which a wide
range of dynamical phenomena becomes analytically accessible. We briey list re-
lated model classes and biophysically more detailed models at the end of this section
and describe some basic and more involved dynamical states of spiking activity in
the subsequent sections.
13.3.1. An analytically accessible class of models
Consider a network of N2N neurons that interact by sending and receiving spikes
(see, e.g., [40, 60, 108, 149]). The state of each neuron j at time s is specied by a
single real variable, the membrane potential V (s) that evolves according to
X
N
X
d
ds
V
j
(s) = g(V
j
(s)) +
"
ji
K(s(s
i
+
v
))
(13.1)
i=1
m2Z
