  Similar activity patterns may arise from model neural networks with distinct
coupling properties and individual unit dynamics. These similar patterns may,
however, respond differently to parameter variations and, specifically, to
tuning of inputs that represent control signals. In this work, we analyze the
responses resulting from modulation of a localized input in each of three
classes of model neural networks that have been recognized in the literature
for their capacity to produce robust three-phase rhythms: coupled fast-slow
oscillators, near-heteroclinic oscillators, and threshold-linear networks.
Triphasic rhythms, in which each phase consists of a prolonged activation of a
corresponding subgroup of neurons followed by a fast transition to another
phase, represent a fundamental activity pattern observed across a range of
central pattern generators underlying behaviors critical to survival, including
respiration, locomotion, and feeding. To perform our analysis, we extend the
recently developed local timing response curve (lTRC), which allows us to
characterize the timing effects due to perturbations, and we complement our
lTRC approach with model-specific dynamical systems analysis. Interestingly, we
observe disparate effects of similar perturbations across distinct model
classes. Thus, this work provides an analytical framework for studying control
of oscillations in nonlinear dynamical systems, and may help guide model
selection in future efforts to study systems exhibiting triphasic rhythmic
activity.
