Is there a user-friendly building unit to replicate rhythmic patterns of CPG systems? Synchrony transition and application of the delayed bursting-HCO model

The CPG neural system is an important local circuit to control rhythmic movement of vertebrates and invertebrates. The half-central oscillator (HCO), i.e. the building unit of the CPG neural circuit, should present the adjustable in-phase and anti-phase synchronous patterns. In this study, we introduce time delay into a pair of Hindmash-Rose (HR) neuronal model to construct a delayed bursting-HCO (DB-HCO) system. Employing numerical methods including phase-lag and its probability, we determine how the pattern of rhythmic activity of the DB-HCO evolves with time delay. The DB-HCO translates rhythmic activity from anti-phase to in-phase synchrony with adjusting time delay. Further, in the synchrony transition between in-phase and anti-phase states, there exist multiple types of stability coexistence including bi-stable and tri-stable stages. The DB-HCO model switches synchrony states among in-phase, anti-phase, and out-of-phase of the spiking and bursting activities. At last, based on the regulatory mechanism of coupling delay, we propose two models as application examples to replicate rhythmic patterns of CPG systems. The rhythm signals of neuronal cells having the fixed phase difference are reproduced in the pyloric and gastric mill CPGs. The DB-HCO model is regarded as a user-friendly building unit to obtain the expected rhythmic patterns of the whole CPG neural circuit.