A very good article. All most important things seem to be in place. The only bit still lacking is a brief, dictionary-like definition which I understand is required of a good Scholarpedia article. The current "Definition" paragraph is a bit too long in my taste, and somewhat vague and informal.
Suggestion: what about adding in the very beginning, before the existing text, something like this:
An excitable medium is a spatially extended, essentially uniform system which
- has a spatially uniform resting state asymptotically stable against small perturbations,
- can support non-decremental, asymptotically stable solitary pulse travelling waves that can be initated by an appropriate over-threshold perturbation of the resting state.
This is just an example illustrating what I understand by dictionary-like definition. Cross-links are obviously less important and could be refined subsequently during the lifetime of the article. NB ability to support periodic waves is a consequence of ability of support solitary waves so probably does not belong to "definition", rather to "properties".
Another, less categorical suggestion. Would it be interesting, among "examples", to give one or two "anti-examples", i.e. systems which resemble excitable and sometimes confused with them, but in fact are not excitable. Say, the KPP-Fisher equation (fronts rather than solitary pulses and more importantly resting state is not stable) or soliton systems (where neither the resting state nor the pulse solitions are asymptotically stable).
A link with dissipative structures would be very appropriate - esp. when that article will eventually come into existence.
I am very thankful to the anonymous reviewer for his suggestion.
In accordance with these suggestions I have moved a short "dictionaly-like" definition to the beginning of the article. I have also extended the section "self-organized patterns" in order to separate more clearly the wave processes in the excitable media from the wave processes in the conservative systems.
I can not accept the short definition suggested by the reviewer since
i. an excitable medium can be nonuniform ii. an excitable medium sometimes is able to support only a local responce, rather than traveling waves. iii. I think it is very important to stress that the interaction between the neighboring segments occurs due to diffusion-like local transport processes. That means, for example, that spiral galaxies and spiral structures during Rayleigh-Benard convection appear due to quite different mechanisms.
Thanks for taking my comments on board and making the changes. Of course, the author should choose the short definition according to his understanding. The question whether a medium that only supports local response should really be called excitable is however debatable. I cannot remember seeing this term used in this sense in literature and would appreciate a reference. I think that in cases where there are varying opinions, a good encyclopedia article should not impose a particular opinion on the reader, but reflect all, e.g. via a comment like
(some authors call excitable the media that only support local responses)
A major deficiency of the presently proposed short definition is that it does not make sense without the definition of excitability, and that article is not written yet so one cannot be sure what will be there. My particular worry is that the concept of the resting state which I thought as fundamental for excitable media, is not reflected in the current definition at all. OK it may not necessarily be spatially uniform, but it should exist, shouldn't it?
An omission which I failed to notice in the first review. As far as I can see, your "most general form" misses an important special case. The dynamical processes in cardiac muscle are more accurately described by a bidomain system of equations, which apart from a parabolic reaction-diffusion part includes an elliptic part describing electrostatic potential extracellular (or intracellular) domain. So your current "most general" definition seems to exclude one of the most important examples. I do not think that full description of the bidomain equations should be given here, but perhaps a comment and a reference should be given. Further refinements: diffusion coefficients could be tensors in the physical space (say to reflect conduction anisotropy of the cardiac muscle) and/or form matrices in the concentrations space (to reflect cross-diffusion processes as appearing in some models). Again, not sure these fine points should be reflected in the equation lest they make it even less readable, but perhaps an explanation in the text would do.
"in an excitable media" -> "in excitable media"?
"to be pass from one" -> "to be passed from one" ?
Thank you very much for careful reading of the article.
I am far away to "call excitable the media that only support local responses". It is written in the first paragraph that "an excitable medium is able to support propagation of undamped solitary excitation waves, as well as wave trains". Of course, it is very important property of an excitable medium. However, in the literature you can find a lot of examples when "asymptotically stable solitary pulse traveling waves" do not exist in an excitable medium.
For example, it is shown in the paper "Tracking waves and vortex nucleation in excitable system with anomalous dispersion" (Manz, Hamik, Steinbock., PRL, 92, 248301, 2004) that in the modified Belousov-Zhabotinsky reaction the first induced wave travels only a short distance before vanishing in the highly inhibited medium. Subsequent pulses, however, travel an increasingly longer distance, since their predecessors have partially "cleared the way" by decreasing the local concentration of inhibitory species.
Another example is described in the paper "Wave instabilities in excitable media with fast inhibitor diffusion" (Zykov, Mikhailov, Mueller, PRL, 81, 2811, 1998). Here the two-dimensional excitable system is analyzed which does not support asymptotically stable propagation of flat waves (or waves in one-dimensional system), but traveling convex wave still exist.
The propagation of chemical waves through narrow capillary tube connecting two thin layers of excitable Belousov-Zhabotinsky mixture has been investigated in the paper "Signal transmition in chemical systems: propagation of chemical waves through capillary tubes" (Toth, Gaspar, Schowalter, J. Phys. Chem., 98, 522, 1994). It was show that when the capillary diameter is greater than a critical value, the excitation propagating through the tube serves to initiate a circular wave in the second compartment; otherwise, the wave propagation failed.
The propagation failure in excitable systems is widely studied problem, which plays very important role in cardiac dynamics, for example. I do not think that cardiac tissue can not be considered as an excitable medium if a morphological or a functional propagation failire takes place. In addition, note, that all biological experiments with excitable media are performed for systems with confined geometries. Usually, there is not enough space and time to distinguish between "asymptotically stable solitary pulse traveling waves" and a local response.
I agree that it will be much more better to refer to a special article "excitability". The second paragraph in the Definition explains this term very shortly. The concept of the resting state is also mentioned here. However, I do not like to overestimate this concept. One can imagine an excitable system with many resting states. In addition, there is such a phenomenon as "adaptation".
I do not think that the bidomain model of cardiac muscle should be considered as "the most important examples" of the excitable media. First of all, the concept "excitable media" has appeared long before the modern bidomain description. Already at this time it was much discussion in the literature regarding possibility to describe cardiac muscle as an excitable medium. Since cardiac cells are connected on a microscopic level discretely, of course, not continuously, the reaction-diffusion description is an approximation simplifying the consideration. An additional simplification is that the extracellular potential assumed to be constant. The bidomain model, of course, produces more accurate description of electrical processes in cardiac muscle. However, the elliptic part included in the bidomain model is absent in many other models of excitable media. In my opinion this part is not generic for common concept of the excitable medium, like a mechanical motion of cells during the Dictyostelium discoideum aggregation.
On the other hand, since there is no separate article "Bidomain Model" it is worth to mention this very important topic. In accordance with the reviewer suggestion, a short description of the bidomain model is inserted in the section Mathematical models and an additional reference to the book of Keener and Sneyd is given.