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a
b
5690
40
5680
30
20
5670
10
5660
0
−1
−0.5
0
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x 10
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d
Time
Fig. 13.5. Network activity and rates of individual neurons in a network of excitatory neurons
with weak inhomogeneities as studied in [154]. The neurons split into two populations, neurons that
are constantly phase-locked with zero lag and neurons which are not phase-locked. The network
activity (a) shows large peaks of well synchronized ring together with some non-synchronous
activity. The rate prole (b), displaying the rate of the neurons versus the perturbation of their
driving, shows that there is a driving I
c
separating phase locked neurons (driving I < I
c
) and
non-phase locked neurons (driving I > I
c
).
spiking activity coordinated between the neurons. The exact analysis of spike times
revealed also the transition point at which inhomogeneities become too strong such
that states close to full synchrony (short patterns) cease to exist. Furthermore,
the same work considered the occurrence of patterns of precisely timed spikes as
an inverse problem (see sec. 13.5.2): For any given, predened pattern of spikes
that spreads over a suciently short time interval, Denker et al. [36] showed how
to nd the set of networks that exhibit that pattern as an invariant solution of its
collective dynamics. Interestingly, in homogeneous sparsely connected random net-
works (or in those with suciently weak inhomogeneities) a synchronous (or almost
synchronous) state may coexist [148, 149] with highly irregular asynchronous states
(see Figs. 13.3 and 13.4 and also the next subsection).
Numerical investigations of inhomogeneous networks of inhibitory and excitatory
sub-populations with delayed, temporally extended interactions [18] have shown
that the two sub-populations may send spikes phase-locked but out-of-phase with
each other, with all neurons in the separate sub-population close to synchronous
with each other. As a sideline, that work suggests that patterns of locked spikes may
occur also in neural circuits with a mixture of excitatory and inhibitory neurons;
the mechanism underlying this phenomenon is similar to that described in Ref. [36]
for purely inhibitory recurrent interactions.
13.4.3. Asynchrony: Irregular, chaotic, and balanced activity
Besides simple synchronous states, asynchronous states provide a second type of
basic activity in spiking neural networks. Depending on the features of the network
considered, asynchronous states may predominantly emerge (i) as states in which
neurons emit spikes individually and periodically and phase-locked to all other neu-
