  Large sparse circuits of spiking neurons exhibit a balanced state of highly
irregular activity under a wide range of conditions. It occurs likewise in
sparsely connected random networks that receive excitatory external inputs and
recurrent inhibition as well as in networks with mixed recurrent inhibition and
excitation. Here we analytically investigate this irregular dynamics in finite
networks keeping track of all individual spike times and the identities of
individual neurons. For delayed, purely inhibitory interactions we show that
the irregular dynamics is not chaotic but in fact stable. Moreover, we
demonstrate that after long transients the dynamics converges towards periodic
orbits and that every generic periodic orbit of these dynamical systems is
stable. We investigate the collective irregular dynamics upon increasing the
time scale of synaptic responses and upon iteratively replacing inhibitory by
excitatory interactions. Whereas for small and moderate time scales as well as
for few excitatory interactions, the dynamics stays stable, there is a smooth
transition to chaos if the synaptic response becomes sufficiently slow (even in
purely inhibitory networks) or the number of excitatory interactions becomes
too large. These results indicate that chaotic and stable dynamics are equally
capable of generating the irregular neuronal activity. More generally, chaos
apparently is not essential for generating high irregularity of balanced
activity, and we suggest that a mechanism different from chaos and
stochasticity significantly contributes to irregular activity in cortical
circuits.
