Collective oscillations in disordered neural networks

We investigate the onset of collective oscillations in a excitatory pulse-coupled network of leaky integrate-and-fire neurons in the presence of quenched and annealed disorder. We find that the disorder induces a weak form of chaos that is analogous to that arising in the Kuramoto model for a finite number N of oscillators [O. V. Popovych , Phys. Rev. E 71 065201(R) (2005)]. In fact, the maximum Lyapunov exponent turns out to scale to zero for N ->infinity, with an exponent that is different for the two types of disorder. In the thermodynamic limit, the random-network dynamics reduces to that of a fully homogeneous system with a suitably scaled coupling strength. Moreover, we show that the Lyapunov spectrum of the periodically collective state scales to zero as 1/N-2, analogously to the scaling found for the "splay state.".