Dynamical Systems in Neuroscience:

Bursting 359Spikingu(t)activation of outward currentsinactivation of inward currentsudeactivation of outward currentsdeinactivation of inward currentsRestingFigure 9.17: Hysteresis-loop periodic bursting.SpikingPerturbationRestingFigure 9.18: Burst excitability: A perturbation causes a burst of spikes.in Fig. 9.16, left. Such a hysteresis loop bursting can also occur when u is multidimensional,as we illustrate in Fig. 9.17. The vector-field on the top (spiking) leafpushes u outside the spiking area, whereas the vector-field on the bottom (resting) leafpushes u outside the resting area. As a result, u visits the spiking and resting areasperiodically, and the model exhibits hysteresis-loop bursting.If resting x does not push u into the spiking area, but leaves it in the bistable area,then the neuron exhibits burst excitability: It has quiescent excitable dynamics, but itsresponse to perturbations is not a single spike, but a burst of spikes, as we illustrate inFig. 9.18. Grade III bursters of hippocampus (Fig. 8.34Eb) produce such a response,often called complex spike response, to brief stimuli. In general, many bistable modelsare bistable only because they neglect slow currents and other homeostatic processespresent in real neurons. If the currents are taken into account, then the models becomebistable on a short time scale and burst-excitable on a longer time scale. This justifieswhy many researchers refer to bistable systems as excitable, implicitly assuming thatthe response to superthreshold perturbations is either a single spike or a long train ofspikes.If the fast subsystem does not have a coexistence of resting and spiking states,then the reduced slow subsystem (9.6) must be at least two-dimensional to exhibit