{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:32:00Z","timestamp":1787232720520,"version":"build-2736575974"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>We study the qualitative behavior of localized synchronous oscillations organized by synaptic inhibition in two types of spatially extended neuronal network models driven by a time-independent, localized excitatory input. Each network is formulated as a one-dimensional network of conductance-based models constituting a high-dimensional dynamical system of nonlocal differential equations. Although such equations readily generate highly complex dynamic behavior, in the case of strong inhibitory coupling the response of the network to a localized Gaussian input is a solution in which a single, continuous band of cells fire nearly synchronous action potentials, in an approximately periodic fashion in time. Tracking the cycle-to-cycle evolution of the width of the band of synchronous action potentials reveals the characteristic behavior of low-dimensional, discrete dynamical systems. Based upon a continuum formulation of the conductance-based model, we heuristically develop and analyze one- and two-dimensional implicit discrete maps for both a purely inhibitory and an excitatory-inhibitory network of neurons. Although the discrete maps do not predict the band widths precisely, they generally reflect the qualitative behavior of the conductance-based model. The most salient features of the bifurcations of fixed points to period 2 orbits and resonances indicate that in some cases these high-dimensional continuous dynamical systems exhibit behavior which can be captured in related low-dimensional discrete maps. Finally, we describe a global bifurcation in the discrete map for the excitatory-inhibitory network in which a strong (1:2) resonance bifurcation occurs on a period 2 orbit, giving rise to a pair of double homoclinic tangles that generate nontrivial dynamics.<\/jats:p>","DOI":"10.1137\/090780092","type":"journal-article","created":{"date-parts":[[2010,9,7]],"date-time":"2010-09-07T18:10:25Z","timestamp":1283883025000},"page":"1019-1060","source":"Crossref","is-referenced-by-count":3,"title":["Spatially Localized Synchronous Oscillations in Synaptically Coupled Neuronal Networks: Conductance-based Models and Discrete Maps"],"prefix":"10.1137","volume":"9","author":[{"given":"Stefanos E.","family":"Folias","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"G. 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