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13.4.2. Away from synchrony:
rst hints towards spike patterns in recurrent networks
Systems that exhibit full synchrony, with all units obeying identical dynamics, nec-
essarily are constraint, e.g. they exhibit some invariance. For instance, when con-
sidering networks of identical inhibitory neurons, the total input strength to each
neuron in a complex network needs to be the same for the fully synchronous state
to exist [148, 149]. Such an idealized condition is atypical for biological networks
in which the total synaptic strengths may be roughly the same due to homeosta-
sis [32, 157, 158], but at least weak inhomogeneities are prevalent. Naturally, weak
inhomogeneities and the presence of other less idealized features will induce states
similar to the fully synchronous one [3, 36, 154]. Stronger heterogeneities typi-
cally lead to states that are very distinct from synchrony and sometimes completely
asynchronous [36].
Already in large networks of all-to-all and homogeneously coupled excitatory
neurons with temporally extended synaptic responses, a partially synchronous state
exists for a certain range of temporal extent [160]. In this partially synchronous
state the total network ring rate oscillates periodically whereas the individual
neurons send spikes quasi-periodically. This result by van Vreeswijk provides one
possible mechanism for oscillations in neural circuits and at the same time is of in-
terest mathematically as the local quasi-periodic activity adds up to global periodic
activity. Brunel et al. [24, 25, 48] showed that also suciently strong inhibitory
interactions can lead to high frequency network oscillations where the individual
neurons re irregularly and with low frequency. This type of dynamics has re-
cently been proposed to underlie high frequency oscillations of Purkinje cells in the
cerebellum [34].
Tsodyks, Mitkov and Sompolinsky uncovered a dierent interesting state similar
to synchrony [154]; for globally and excitatorily coupled neurons with temporally
extended synaptic responses arbitrarily weak inhomogeneities in the individual neu-
rons' intrinsic time scales may split the neurons into two sub-populations (Fig. 13.5),
one sub-population consisting of the slower neurons, that stay identically synchro-
nized forever, and a second consisting of the intrinsically faster spiking neurons,
which also have collective frequencies that are dierent from each other and larger
than in the synchronized sub-population. We remark that already in such states,
the timing of spikes of the synchronized sub-population is highly precise, despite
inhomogeneities in the individual neuron features; moreover, the timing of spikes
of neurons in the unlocked sub-population is relatively precise and close to that of
the locked sub-population, for repeated, long stretches of time.
Networks with more complex connectivity may exhibit additional collective dy-
namical features induced by heterogeneities. Recent work [36] has shown that in
networks of inhibitorily coupled IF-like neurons with delayed interactions, weak
inhomogeneities in the coupling strengths (or equivalently, in other system param-
eters) induce a state close to full synchrony that exhibits well-dened patterns of
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