  We examine analytically the existence and stability of phase-locked states in
a weakly heterogeneous neuronal network. We consider a model of N neurons with
all-to-all synaptic coupling where the heterogeneity is in the firing frequency
or intrinsic drive of the neurons. We consider both inhibitory and excitatory
coupling. We derive the conditions under which stable phase-locking is
possible. In homogeneous networks, many different periodic phase-locked states
are possible. Their stability depends on the dynamics of the neuron and the
coupling. For weak heterogeneity, the phase-locked states are perturbed from
the homogeneous states and can remain stable if their homogeneous conterparts
are stable. For enough heterogeneity, phase-locked solutions either lose
stability or are destroyed completely. We analyze the possible states the
network can take when phase-locking is broken.
