The article gives a nice overview.
Considering the references to work by Sherman and others, I would suggest to mention in the introduction that \(\beta\)-cells show bursting.
"Square wave burster" is never defined. It would be helpful to refer to e.g. Fig. 2.
Also in the section on Arbitrary coupling, "codimension-one and -two Z2-equivariant bifurcations" are not well-known. This part should either be explained better or left out.
Concerning noise and emergent bursting, I would suggest to refer to DeVries & Sherman, J. Theor. Biol. (2000), and Pedersen, J. Theor. Biol. (2005). Pedersen showed that the effect of noise is effectively that it introduces heterogeneity. This could be done, e.g., in the section on heterogenous networks.
A phenomenom related to burst synchronization is the "near-synchronization" giving wave propagation for example across pancreatic as found by Aslanidi et al., Biophys. J. (2001), and others. A comment on this might be appropriate in the section on large networks.
This is a well written paper on an interesting topic. I do have some suggestions for the author to consider.
I would suggest that the author be more careful to emphasize that the many results he describes are examples of what may happen when bursters are coupled together, but these results may depend on the specific choice of model or parameters. For example, in the diffusive coupling section, the author states that "Weak diffusive coupling of calcium has been shown to enhance burst synchronization, but stronger coupling was found to yield a death of oscillations... ". I'm wondering if this is always true, or depends on specific details of the model under consideration. For another example, the paper gives the impression that diffusive and fast excitatory synaptic coupling always leads to burst synchronization. This is probably true (most of the time) but I wouldn't be surprised if this type of coupling may also lead to stable antiphase bursting under certain conditions. For relaxation oscillators, this has been shown in the case of excitatory coupling in a paper by Kopell (and LaFarro, I believe). A paper by Bem and Rinzel considers electrical coupling.
I'm worried that many potential readers may find much of the article hard to follow unless they are already familiar with the material. Of course, more detailed explanations will add to the length of the paper. The author may wish to consider eliminating or shortening some sections towards the end of the paper in order to expand on the earlier ones. The material on time delays, arbitrary coupling, common inputs and maps are interesting, but especially hard to follow.
The three forms of synaptic coupling on page four do not depend on the postsynapic voltage. However, the coupling term c(V_pre,V) given on the bottom of that page does. This is confusing.
Perhaps the author should give a reference for half-center oscillation (Brown).
I didn't follow comments in the first paragraph of the synaptic coupling section. The author states that in order for the half-center oscillation to be robust, some condition such as the deinactivation of an inward current needs to hold. However, I believe that a condition of this type needs to hold in order for the individual cell to exhibit bursting, not that the coupling leads to half-centered oscillations.
COMMENT FROM AUTHOR TO REVIEWERS: Thank you for the many helpful suggestions. I have implemented most of these. In the interest of space, I did not insert a comment on wave propagation based on the Aslanidi et al. paper. While this is an interesting result, it was in the setting of pancreatic beta cells (although I obviously did include other material from that setting) and, importantly, I did not want to open the door to a whole range of other results on waves in neuronal models. Also, if you would like to see it included, please give more details on the Brown reference on half-center oscillations, as I am not familiar with this and have not turned it up. Indeed, feel free to insert this reference on your own, which is easy with this Wikipedia format.