  We study the emergent dynamics of a network of synaptically coupled slow-fast
oscillators. Synaptic coupling provides a network-level positive feedback
mechanism that cooperates with cellular-level positive feedback to ignite
in-phase network oscillations. Using analytical bifurcation analysis, we prove
that the Perron-Frobenius eigenvector of the network adjacency matrix fully
controls the oscillation pattern locally in a neighborhood of a Hopf
bifurcation. Besides shifting the focus from the spectral properties of the
network Laplacian matrix to the network adjacency matrix, we discuss other key
differences between synaptic and diffusive coupling.
