Dynamics of a large system of spiking neurons with synaptic delay

We analyze a large system of heterogeneous quadratic integrate-and-fire (QIF) neurons with time delayed, all-to-all synaptic coupling. The model is exactly reduced to a system of firing rate equations that is exploited to investigate the existence, stability, and bifurcations of fully synchronous, partially synchronous, and incoherent states. In conjunction with this analysis we perform extensive numerical simulations of the original network of QIF neurons, and determine the relation between the macroscopic and microscopic states for partially synchronous states. The results are summarized in two phase diagrams, for homogeneous and heterogeneous populations, which are obtained analytically to a large extent. For excitatory coupling, the phase diagram is remarkably similar to that of the Kuramoto model with time delays, although here the stability boundaries extend to regions in parameter space where the neurons are not self-sustained oscillators. In contrast, the structure of the boundaries for inhibitory coupling is different, and already for homogeneous networks unveils the presence of various partially synchronized states not present in the Kuramoto model: Collective chaos, quasiperiodic partial synchronization (QPS), and a novel state which we call modulated-QPS (M-QPS). In the presence of heterogeneity partially synchronized states reminiscent to collective chaos, QPS and M-QPS persist. In addition, the presence of heterogeneity greatly amplifies the differences between the incoherence stability boundaries of excitation and inhibition. Finally, we compare our results with those of a traditional (Wilson Cowan-type) firing rate model with time delays. The oscillatory instabilities of the traditional firing rate model qualitatively agree with our results only for the case of inhibitory coupling with strong heterogeneity.