Dynamical Systems in Neuroscience:

Bursting 351SpikingBistabilityu(t)RestinguFigure 9.11: Parameter u can control spikingbehavior of the fast subsystem in (9.1).When u changes slowly, the model exhibitsbursting behavior.9.1.4 Central pattern generators and half-center oscillatorsBursting can also appear in small circuits of coupled spiking neurons, such as the twomutually inhibitory oscillators in Fig. 9.10, called half-center oscillators. While onecell fires, the other is inhibited, then they switch roles, and so on. Such small circuits,suggested by Brown (1911), are the building blocks of central pattern generators inpyloric network of the lobster stomatogastric ganglion, medicine leech heartbeat, fictivemotor patterns and swimming patterns of many vertebrates and invertebrates (Marderand Bucher 2001).What makes the oscillators in Fig. 9.10 alternate? Wang and Rinzel (1992) suggestedtwo mechanisms, release and escape, which were later refined to intrinsic orsynaptic by Skinner et al. (1994):• intrinsic release: The active cell stops spiking, terminates inhibition andallows inhibited cell to fire.• intrinsic escape: Inhibited cell recovers, starts to fire and shuts off the activecell.• synaptic release: The inhibition weakens, e.g., due to spike frequency adaptationor short-term synaptic depression, and allows the inhibited cell to fire.• synaptic escape: Inhibited cell depolarizes above certain threshold and startsto inhibit the active cell.All four mechanisms assume that in addition to fast variables responsible for spiking,there are also slow adaptation variables responsible for slowing down or termination ofspiking, recovery, or synaptic depression. Thus, similarly to the minimal models above,the circuit has at least two time scales, i.e., it is a fast-slow system.9.2 GeometryTo understand the neuro-computational properties of bursters, we need to study thegeometry of their phase portraits. In general, it is quite a difficult task. However, itcan be accomplished in the special case of fast-slow dynamics.