Hoppensteadt 9805544 The investigators introduce a novel methodology for use in mathematical neuroscience: Rather than studying a single model of a particular brain structure, they consider the family of all its models. They use invariant manifold, bifurcation, and singular perturbation theories to prove that under certain natural conditions there is a continuous, noninvertible change of variables that converts every member of the family into a canonical model. They derive canonical models for weakly connected networks of excitable, oscillatory and bursting neurons, arising in studies of the thalamus, hippocampus and neocortex. Exact equations are rarely known to describe biological systems. As a result, various people can use various models for the same biological system, but obtain contradictory results. A reasonable way to resolve this is to study ALL possible models of the same biological system at once. The investigators develop and use state-of-the-art methodologies in system science to show that if a biological system is near a critical regime, then all its models are related to a simpler one, called a canonical model. An advantage of this approach is that the plausibility of the canonical model rests only on the plausibility of the critical regime, but not on the choice of equations describing details of the biological system. This approach enables them to derive simple but exact models for such complex brain structures as the thalamus, hippocampus and neocortex operating near a critical regime, such as the onset of rhythmic activity. These models are used to study attention, memorization and navigation in behaving animals. In addition, these studies may shed light on neurocomputational properties of these brain structures with possible applications to neuroengineering and computer design.
