Bursting, the regular alternation between active spiking and quiescence, is typical of many classes of neurons. Many basic nervous system behaviors, such as heartbeat, digestion, breathing, and walking, are controlled by networks of bursting neurons called central pattern generators (CPGs). Establishing the appropriate phasing of bursts within CPGs is essential for producing the correct rhythmic patterns for these behaviors, and understanding how CPGs do this is a major research area in neuroscience.
I will talk about some computational and mathematical approaches to this subject in the context of modeling two CPGs, one responsible for coordinating hindlimb movement in mice and one involved in the digestive rhythm of lobsters. When we perturb bursting neural models with realistic synaptic inputs, we discover that the phase responses of the bursts are very sensitive to the structure of the spikes within the bursts. Fast-slow analysis gives us insight into the mechanisms underlying the phase response sensitivity. I will show how we can reduce the phase response of single bursting neural models to low-dimensional maps of spike trains. Iteration of these maps predict the full system dynamics, and the phasing of bursts in networks can be well-approximated by iteration of appropriately coupled maps.