Chaos and stability in a model of inhibitory neuronal network

We analyze the dynamics of a deterministic model of inhibitory neuronal networks proving that the discontinuities of the Poincare map produce a never empty chaotic set, while its continuity pieces produce stable orbits. We classify the systems in three types: the almost everywhere (a.e.) chaotic, the a.e. stable, and the combined systems. The a.e. stable are periodic and chaos appears as bifurcations. We prove that a.e. stable systems exhibit limit cycles, attracting a.e. the orbits.