Spatially Localized Synchronous Oscillations in Synaptically Coupled Neuronal Networks: Conductance-based Models and Discrete Maps

Abstract

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.