We study synaptically coupled neuronal networks to identify the role of
coupling delays in network's synchronized behaviors. We consider a network of
excitable, relaxation oscillator neurons where two distinct populations, one
excitatory and one inhibitory, are coupled and interact with each other. The
excitatory population is uncoupled, while the inhibitory population is tightly
coupled. A geometric singular perturbation analysis yields existence and
stability conditions for synchronization states under different firing patterns
between the two populations, along with formulas for the periods of such
synchronous solutions. Our results demonstrate that the presence of coupling
delays in the network promotes synchronization. Numerical simulations are
conducted to supplement and validate analytical results. We show the results
carry over to a model for spindle sleep rhythms in thalamocortical networks,
one of the biological systems which motivated our study. The analysis helps to
explain how coupling delays in either excitatory or inhibitory synapses
contribute to producing synchronized rhythms.Comment: 43 pages, 12 figure
