Dynamical Systems in Neuroscience:

Bursting 345interburst periodquiescentperiodactivephaseinterspike(intraburst)periodduty cycle =active phaseinterburst periodV(t)Figure 9.5: Basic characteristics of bursting dynamics.jected. From a mathematical point of view, every burster is conditional, since it existsfor some values of the parameters but not others.9.1.2 Fast-Slow DynamicsIn general, every bursting pattern consists of oscillations with two time scales: fastspiking oscillation within a single burst (intraburst oscillation, or spiking), modulatedby a slow oscillation between the bursts (interburst oscillation); see Fig. 9.5. Typically,though not necessarily (see exercises at the end of this chapter), two time scales resultfrom two interacting processes involving fast and slow currents. For example, thespiking in Fig. 9.4 is generated by the fast I Na,p +I K -subsystem and modulated by theslow I K(M) -subsystem.There are two questions associated with each bursting pattern:• What initiates sustained spiking during the burst?• What terminates sustained spiking temporarily and ends the burst?The answer to the first question is relatively simple: Repetitive spiking is initiated andsustained by the positive injected current I, or some other source of persistent inwardcurrent that causes the neuron to fire (most biologists are interested in identifyingthis source). Surprisingly, the second question is the most important for building amodel of bursting. While the neuron fires, relatively slow processes somehow make itnon-excitable and eventually terminate the firing. Such slow processes result in a slowbuildup of an outward current or in a slow decrease of an inward current needed tosustain the spiking. During the quiescent phase, the neuron slowly recovers and regainsthe ability to generate action potentials again.Let us discuss possible ionic mechanisms responsible for the termination of spikingwithin a burst. Suppose we are given a neuronal model that is capable of sustainedspiking activity, at least when a positive I is injected. To transform an infinite spiketrain into a finite burst of spikes, it suffices to add a slow resonant current or gating