  We identify and describe the key qualitative rhythmic states in various
3-cell network motifs of a multifunctional central pattern generator (CPG).
Such CPGs are neural microcircuits of cells whose synergetic interactions
produce multiple states with distinct phase-locked patterns of bursting
activity. To study biologically plausible CPG models, we develop a suite of
computational tools that reduce the problem of stability and existence of
rhythmic patterns in networks to the bifurcation analysis of fixed points and
invariant curves of a Poincar\'e return maps for phase lags between cells.
  We explore different functional possibilities for motifs involving symmetry
breaking and heterogeneity. This is achieved by varying coupling properties of
the synapses between the cells and studying the qualitative changes in the
structure of the corresponding return maps. Our findings provide a systematic
basis for understanding plausible biophysical mechanisms for the regulation of
rhythmic patterns generated by various CPGs in the context of motor control
such as gait-switching in locomotion. Our analysis does not require knowledge
of the equations modeling the system and provides a powerful qualitative
approach to studying detailed models of rhythmic behavior. Thus, our approach
is applicable to a wide range of biological phenomena beyond motor control.
