Dynamical Systems in Neuroscience:

Chapter 9BurstingA neuron can fire a single spike or a stereotypical burst of spikes, depending on thenature of stimulation and the intrinsic neuronal properties. Typically, bursting occursdue to the interplay of fast currents responsible for spiking activity, and slow currentsthat modulate the activity. In this chapter we study this interplay in detail.To understand the geometry of bursting, it is customary to assume that the fast andslow currents have drastically different time scales. In this case we can dissect a burster,i.e., freeze its slow currents and use them as parameters that control the fast spikingsubsystem. During bursting, the slow parameters drive the fast subsystem throughbifurcations of equilibria and limit cycles. We provide a topological classification ofbursters based on these bifurcations, and show that different topological types havedifferent neuro-computational properties.9.1 ElectrophysiologyMany spiking neurons can exhibit bursting activity if manipulated, e.g., pharmacologically.In Fig. 9.1 we depict a few well-known examples of neurons that burst undernatural conditions without any manipulation. Some require an injected dc-current tobias the membrane potential, others do not. One can only be amazed by the diversityof bursting patterns and time scales. In this chapter we consider electrophysiologicaland bifurcation mechanisms responsible for the generation of these patterns.Is zebra a black animal with white stripes or a white animal with black stripes? Thisseemingly silly question is pertinent to every bursting pattern: Does bursting activitycorrespond to an infinite period of quiescence interrupted by groups of spikes or doesit correspond to an infinite spike train interrupted by short periods of quiescence?Biologists are mostly concerned with the question of what makes the neuron fire thefirst spike in a burst and what keeps it in the spiking regime afterwards. The questionof why the spiking stops is often forgotten. It turns out that to fully understand theionic mechanism of bursting, we need to concentrate on the second question, i.e., weneed to treat bursting as an infinite spike train that is chopped into short bursts by aslow (resonant) current that builds up during the spiking phase and recovers during the341