Rapid synchronization under weak synaptic coupling

Biophysical coupling in neuronal networks is either through gap junctions or synapses. This paper explores how models of synaptic coupling relate to the standard diffusive coupling model studied in the control theory of networks. In particular, we investigate how weak synaptic coupling can locally realize strong diffusive coupling in the vicinity of nonlinear oscillations of the relaxation type. Although often perceived as a source of complexity, the combination of nodal excitability with synaptic coupling is shown to be a source of tractability, allowing for robust mathematical predictions on the basis of linear analysis.