We study synaptically coupled neuronal networks to identify the role ofcoupling delays in network's synchronized behaviors. We consider a network ofexcitable, relaxation oscillator neurons where two distinct populations, oneexcitatory and one inhibitory, are coupled and interact with each other. Theexcitatory population is uncoupled, while the inhibitory population is tightlycoupled. A geometric singular perturbation analysis yields existence andstability conditions for synchronization states under different firing patternsbetween the two populations, along with formulas for the periods of suchsynchronous solutions. Our results demonstrate that the presence of couplingdelays in the network promotes synchronization. Numerical simulations areconducted to supplement and validate analytical results. We show the resultscarry over to a model for spindle sleep rhythms in thalamocortical networks,one of the biological systems which motivated our study. The analysis helps toexplain how coupling delays in either excitatory or inhibitory synapsescontribute to producing synchronized rhythms.
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