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collective dynamics of recurrent neural networks. Compared to patterns generated
by feed-forward networks, this possibility so far is much less explored. Below, we
present some recent developments where in part a detailed understanding of the
collective phenomena is possible.
13.5.1. Spike patterns as attractors of recurrent networks
In certain networks that are dominated by inhibitory non-delayed interactions, the
collective network dynamics converges to periodic spike patterns [75, 98]. These
spike patterns are typically of low period and reached quickly such that they domi-
nate the network dynamics on the relevant time scales. As shown before already for
globally coupled networks [40, 41], interaction delays may have a drastic inuence
on the collective network dynamics. This is even more so if the network topology
is complex and local dissipation becomes relevant, compare, e.g., [169] vs. [50, 108].
For instance, in inhibitorily coupled units delays strongly enhance the transient
times towards periodic spike patterns [73, 74] such that stable irregular transients
dominate the dynamics. Moreover, periodic orbits in these systems typically are
long. In another example, very long delays induce switching between sequences [54]
that recur several times and afterwards are non-recurrent. The results of Ref. [74]
strongly suggest that the dynamics is nevertheless stable. An explanation for the
switching phenomenon is detailed in [102].
13.5.2. Realizing spike patterns in complex networks {
An inverse problem
Nearly all the above studies considered certain pre-specied networks of spiking
neurons and studied what kinds of dynamics they may exhibit. In recent years, an
inverse perspective was introduced [36, 97, 98, 103, 104, 123, 147, 166] where now the
central question becomes \What kind of networks exhibit a given dynamics?". Such
questions have been conceptually addressed two decades ago in abstract networks
of non-spiking neurons [37, 66, 91, 92].
Prinz and coworkers [123] presented an extensive numerical analysis of three-
neuron circuits and identied broadly distinct networks that exhibit nearly the
same spiking dynamics. Makarov et al. [97] used stochastic optimization to nd
networks of given model neurons that most closely match observed spiking data. An
analytical deterministic framework of network design was introduced recently [103,
104] to nd the set of all possible networks that exhibit a predened, e.g. periodic,
spike pattern (Fig. 13.7). If a network solution exists at all (which it does under mild
constraints), there typically is a high-dimensional set of networks that exhibit the
same spike pattern as a possible dynamics. This set is parametrized, for instance, by
the coupling strengths. The set is restricted by spike timing conditions, equations
that impose the constraint that a given spike occurs where predened, and by silence
conditions, inequalities that ensure that a neuron does not emit a spike when none
