Project3NewThought

There has been a lot of papers developing and applying techniques of testing whether a system is chaotic. I don't know what the current state of the field is but the problem has been that it is difficult to tell between a chaotic system and a noisy system. Nevertheless it has been a problem that has captivated the imagination of many researchers.

The ghostburster model cell exhibits chaos for some values of the parameters and not for others. Models that exhibit qualitatively different behavior for different parameters have a bifurcation structure that can be visualized with a bifurcation diagram (shown in the paper). Usually, when you make small changes to the parameters, the qualitative properties of the cell (e.g. quiesient, bistable, tonically active, chaotic) don't change (the resulting models are equivalent). Only at discrete values of the parameters do "bifurcations" (qualitative changes in the model) occur (e.g. quiesent --> tonically active).

Let's say you fix all but one (or two) of the parameters and try to fit the one (or two) that remain. In these situations you can actually draw both the bifurcation diagram and the likelihood function. It would interesting to see what the relationship between the other figures are as you vary the other ("fixed") parameters. When will the maximum likelihood estimates of the parameters tell you in what region of the bifurcation diagram you lie? When will they tell you that the cell exhibits backpropagation? When will they distiguish between a nonchaotic cell perturbed by noise and a chaotic cell perturbed by noise?

I said in class that this project was the most risky in terms of being publishable. This is only because the story we would tell is not yet clear. Sometimes the most risk involves the most potential payoff. That is not yet clear, but it is clear that this project would be doable and interesting.