  We introduce an exactly integrable version of the well-known leaky
integrate-and-fire (LIF) model, with continuous membrane potential at the
spiking event, the c-LIF. We investigate the dynamical regimes of a fully
connected network of excitatory c-LIF neurons in the presence of short-term
synaptic plasticity. By varying the coupling strength among neurons, we show
that a complex chaotic dynamics arises, characterized by scale free avalanches.
The origin of this phenomenon in the c-LIF can be related to the order symmetry
breaking in neurons spike-times, which corresponds to the onset of a broad
activity distribution. Our analysis uncovers a general mechanism through which
networks of simple neurons can be attracted to a complex basin in the phase
space.
