We analyze the dynamics of two coupled identical populations of quadratic integrate-and-fire neurons, which represent the canonical model for class I neurons near the spiking threshold. The populations are heterogeneous; they include both inherently spiking and excitable neurons. The coupling within and between the populations is global via synapses that take into account the finite width of synaptic pulses. Using a recently developed reduction method based on the Lorentzian ansatz, we derive a closed system of equations for the neurons firing rates and the mean membrane potentials in both populations. The reduced equations are exact in the infinite-size limit. The bifurcation analysis of the equations reveals a rich variety of non-symmetric patterns, including a splay state, antiphase periodic oscillations, chimera-like states, also chaotic oscillations as well as bistabilities between various states. The validity of the reduced equations is confirmed by direct numerical simulations of the finite-size networks.
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