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We identify and describe the key qualitative rhythmic states in various3-cell network motifs of a multifunctional central pattern generator (CPG).Such CPGs are neural microcircuits of cells whose synergetic interactionsproduce multiple states with distinct phase-locked patterns of burstingactivity. To study biologically plausible CPG models, we develop a suite ofcomputational tools that reduce the problem of stability and existence ofrhythmic patterns in networks to the bifurcation analysis of fixed points andinvariant curves of a Poincar\'e return maps for phase lags between cells.  We explore different functional possibilities for motifs involving symmetrybreaking and heterogeneity. This is achieved by varying coupling properties ofthe synapses between the cells and studying the qualitative changes in thestructure of the corresponding return maps. Our findings provide a systematicbasis for understanding plausible biophysical mechanisms for the regulation ofrhythmic patterns generated by various CPGs in the context of motor controlsuch as gait-switching in locomotion. Our analysis does not require knowledgeof the equations modeling the system and provides a powerful qualitativeapproach to studying detailed models of rhythmic behavior. Thus, our approachis applicable to a wide range of biological phenomena beyond motor control.

Key bifurcations of bursting polyrhythms in 3-cell central pattern generators