What causes a group of disembodied neurons in a dish to fire after a silence? If there are no neurons providing a stimulus for more firing (as in a dish of disembodied rat neurons), then why don't they simply cease firing after the synchronized burst completes?
There are a number of different wiring scenarios that could lead to this type of behavior, even in a dissociated neuronal cell culture. The two major cases are neurons that are truly isolated from each other, and collections of neurons that connect together. Since your question refers to firing and synchronized bursts, I'll restrict my discussion to the generation of spontaneous action potentials - but keep in mind that not all synaptic communication requires an action potential (the vast majority of internal connections within the retina, for example)
In the case of ensembles neurons in vitro, spontaneous activity is fairly typical. One simple way to get this activity is for a subset of cells in the network to be inherently spontaneously active. In the case of single, isolated neurons, this kind of individual cell spontaneous activity could also result in a single neuron firing spikes in a periodic fashion. So for example purposes, let's explore how a single cell could produce this kind of activity.
Since action potentials are triggered by a sufficiently strong depolarization, we need to identify potential membrane polarizing sources. When dealing with single cells, the major controllers of cell membrane voltage potential are the voltage gated ion channels. When open these channels can cause, depending on their type and cellular/extracellular conditions, either a depolarization or hyperpolarization in the cell. And, most interestingly, whether they are open or not depends on the cell membrane's polarization state.
That makes voltage gated channels excellent at feedback:
Physiologists have thoroughly characterized many of these cells, and they have many more states beyond an "open" and "closed", such as different levels of "inactivated". But we don't need to know about those to answer your question.
The last piece we do need is brownian motion which is the movement of particles due to their bombardment by other particles. This motion can with some frequency cause voltage gated channels to spontaneously open or close, which introduces noise into the system.
As you can imagine, if you have a positive feedback channel that is gated spontaneously, it will make it briefly more likely for other depolarization gated channels to be activated. Positive feedback can quickly lead to a rapid depolarization of the cell, which could be observed as a burst of spikes. If you include a set of slower reacting depolarization gated hyperpolarizing channels, then it's easy to construct a scenario where the cell would depolarize spontaneously with no apparent input, then have a quiescent period where a prolonged hyperpolarization makes it less likely to fire spontaneously.
Voila, bursting in an isolated neuron. Place a set of those neurons in a network and you can get generate a lot of spontaneous activity.