  The purpose of this paper is trifold -- to serve as an instructive resource
and a reference catalog for biologically plausible modeling with i)
conductance-based models, coupled with ii) strength-varying slow synapse
models, culminating in iii) two canonical pair-wise rhythm-generating networks.
We document the properties of basic network components: cell models and
synaptic models, which are prerequisites for proper network assembly. Using the
slow-fast decomposition we present a detailed analysis of the cellular dynamics
including a discussion of the most relevant bifurcations. Several approaches to
model synaptic coupling are also discussed, and a new logistic model of slow
synapses is introduced. Finally, we describe and examine two types of
bicellular rhythm-generating networks: i) half-center oscillators ii)
excitatory-inhibitory pairs and elucidate a key principle -- the network
hysteresis underlying the stable onset of emergent slow bursting in these
neural building blocks. These two cell networks are a basis for more
complicated neural circuits of rhythmogensis and feature in our models of swim
central pattern generators.

Neurons and Cognition;Dynamical Systems;Adaptation and Self-Organizing Systems;Chaotic Dynamics