Here's an analogy to help answer this question. Suppose that you have two guys, Frick and Frack. If left alone, Frick and Frack will just sit there, but if they are hit on the head, then they will hit whoever hit them. Alone, neither Frick nor Frack will spend any time hitting, but if you put them in a room together and add a little misunderstanding to get things going, then Frick and Frack will continually hit one another (at least until they get tired or resolve their misunderstanding).
Frick and Frack are an example of a network oscillator with emergent hitting. Once it's started, there's no assigning blame to one element or the other for their collective behavior, they're both responsible. You might feel a little cheated by the example since either Frick or Frack could hit on their own, if they desired to do so. But if Frick and Frack were much simpler, say, just one neuron each, then their ability to spontaneously generate hitting movements could be greatly diminished - they might need to be hit before they could hit back. Or, if there was some good reason why they shouldn't both hit at the same time, then there would be a good reason why they should cooperate in some way.
The Frick and Frack example allows for some intuition into how networks can
oscillate, but is that the only way non-oscillating entities can collectively
oscillate? Well, you could have a chain of Frick and Frack types, where instead
of hitting the person that hit them, they just hit the person in front of them.
So if they are arranged in a circle, they'll produce a network oscillation.
What are other ways they could be arranged?
To get some insight into this question, we will look into some bifurcation theory.