The dynamical behavior of a weakly diluted fully inhibitory network of pulse-coupled spiking neurons is investigated. Upon increasing the coupling strength, a transition from regular to stochasticlike regime is observed. In the weak-coupling phase, a periodic dynamics is rapidly approached, with all neurons firing with the same rate and mutually phase locked. The strong-coupling phase is characterized by an irregular pattern, even though the maximum Lyapunov exponent is negative. The paradox is solved by drawing an analogy with the phenomenon of "stable chaos," i.e., by observing that the stochasticlike behavior is "limited" to an exponentially long (with the system size) transient. Remarkably, the transient dynamics turns out to be stationary.

The stability of the dynamical states characterized by a uniform firing rate (splay states) is analyzed in a network of globally coupled leaky integrate-and-fire neurons. This is done by reducing the set of differential equations to a map that is investigated in the limit of large network size. We show that the stability of the splay state depends crucially on the ratio between the pulse width and the interspike interval. More precisely, the spectrum of Floquet exponents turns out to consist of three components: (i) one that coincides with the predictions of the mean-field analysis [Abbott and van Vreesvijk, Phys. Rev. E 48, 1483 (1993)], (ii) a component measuring the instability of "finite-frequency" modes, (iii) a number of "isolated" eigenvalues that are connected to the characteristics of the single pulse and may give rise to strong instabilities (the Floquet exponent being proportional to the network size). Finally, as a side result, we find that the splay state can be stable even for inhibitory coupling.