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tic dynamics of large recurrent networks. Of course in all the above-mentioned
frameworks, also additional features may be studied, including synaptic plasticity
on short [2, 88, 93, 153] and long time-scales [63, 68, 90, 112], ion channel co-
operativity [115], compartmental or spatially extended structure [20, 82, 135] and
non-additive features of the interactions [9, 88, 93, 106, 115, 153].
Moreover, various models of abstract rate-coded often binary-state or discrete-
time neurons [16, 66, 91, 92, 131, 165] exist that are valuable for studying conceptual
problems of computation or information processing in neural systems, but by their
very nature generically do not capture the precise timing of spikes. Very recently the
link between continuous-time and discrete-time models has been reconsidered with
the interesting resulting suggestion [29, 30] that under certain conditions specic
discrete time models may actually more appropriately describe the spiking dynamics
of recurrent networks.
13.4. Basic Collective States of Recurrent Networks
Here we provide a brief overview on basic collective states of deterministic recurrent
networks. This should pave the way to a better understanding of the concepts and
the complex spatio-temporal dynamics presented in the next section.
13.4.1. Quiescence and synchrony
Obviously, the simplest dynamical state of a spiking neural network is global quies-
cence, where no neuron is emitting any spike. This is a trivial network state because
its dynamics is just the collection of all individual neuron dynamics, even in the
presence of driving signals and uctuations. Still it is sometimes valuable to know
under which conditions the quiescent state exists and is stable, for instance if the
emergence of a complex, persistent state from the quiescent one (or from an almost
quiescent one) is to be understood [57, 59].
In fact, single neurons are intrinsically excitable systems and typically quiescent
if not driven by synaptic inputs, external currents, uctuations or by other means.
Therefore, the modeling of single spiking neurons ranges essentially between two
extreme limits (see, e.g., [155, 156]). One limit is stochastic: neurons receive inde-
pendent stochastic sequences of spikes (e.g. Poisson spike trains) and therefore also
generate stochastic spiking dynamics theirselves (e.g. [27, 28, 156]). The second
limit is deterministic: neurons receive suciently strong temporally uniform input
currents such that their dynamics becomes tonic periodic spiking. We here focus
on deterministic models of spiking neural networks. As we will see below, how-
ever, these neurons collectively still often exhibit dynamics that resembles random
processes [58, 74, 106, 162, 163].
Arguably the simplest non-trivial and truly collective state is the fully syn-
chronous state, a periodic orbit in which every neuron emits spikes periodically and
at the same times as all the other neurons in the network. It is dominant in globally
