  Synchronization of coupled dynamical systems is a widespread phenomenon in
both biological and engineered networks, and understanding this behavior is
crucial for controlling such systems. Considerable research has been dedicated
to identifying the conditions that promote synchronization in diffusively
coupled systems, where coupling relies on the difference between the states of
neighboring systems and vanishes on the synchronization manifold. In
particular, contraction theory provides an elegant method for analyzing
synchronization patterns in diffusively coupled networks. However, these
approaches do not fully explain the emergence of synchronization behavior in
non-diffusively coupled networks where the coupling does not vanish on the
synchronization manifold and hence the dynamics on the synchronization manifold
differ from the uncoupled systems. Inspired by neuronal networks connected via
non-diffusive chemical synapses, we extend contraction theory to establish
sufficient conditions for global synchronization in general non-diffusively
coupled nonlinear networks. We demonstrate the theoretical results on a network
of Hindmarsh-Rose oscillators connected via chemical synapses and networks of
FitzHugh-Nagumo oscillators connected via chemical synapses and additive
coupling.
