Canonical models for mathematical neuroscience

A major drawback to most mathematical models in neuroscience is that they are either far away from reality or the results depend on the specific model. A promising alternative approach takes advantage of the fact that many complicated systems behave similarly when they operate near critical regimes, such as bifurcations. Using nonlinear dynamical system theory it is possible to prove that all systems near certain critical regimes are governed by the same model, namely a canonical model. Briefly, a model is canonical if there is a continuous change of variables that transforms any other model that is near the same critical regime to this one. Thus, the question of plausibility of a mathematical model is replaced by the question of plausibility of the critical regime. Another advantage of the canonical model approach to neuroscience is that rigorous derivation of the models is possible even when only partial information is known about anatomy and physiology of brain structures. Then, studying canonical models can reveal some general laws and restrictions. In particular, one can determine what certain brain structures cannot accomplish regardless of their mathematical model. Since the existence of such canonical models might sound too good to be true, we present a list of some of them for weakly connected neural networks. Studying such canonical models provides information about all weakly connected neural networks, even those that have not been discovered yet