  We study the dynamical stability of pulse coupled networks of leaky
integrate-and-fire neurons against infinitesimal and finite perturbations. In
particular, we compare current versus fluctuations driven networks, the former
(latter) is realized by considering purely excitatory (inhibitory) sparse
neural circuits. In the excitatory case the instabilities of the system can be
completely captured by an usual linear stability (Lyapunov) analysis, on the
other hand the inhibitory networks can display the coexistence of linear and
nonlinear instabilities. The nonlinear effects are associated to finite
amplitude instabilities, which have been characterized in terms of suitable
indicators. For inhibitory coupling one observes a transition from chaotic to
non chaotic dynamics by decreasing the pulse width. For sufficiently fast
synapses the system, despite showing an erratic evolution, is linearly stable,
thus representing a prototypical example of Stable Chaos.
