  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
neuron's 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.
