title: Multi-timescale systems and fast-slow analysis.
abstract: Mathematical models of biological systems often have components that vary on different timescales. This multi-timescale character can lead to problems when doing computer simulations, which can require a great deal of computer time so that the components that change on the fastest time scale can be resolved. Mathematical analysis of these multi-timescale systems can be greatly simplified by partitioning them into subsystems that evolve on different time scales. The subsystems are then analyzed semi-independently, using a technique called fast-slow analysis. In this review we describe the fast-slow analysis technique and apply it to relaxation oscillations, neuronal bursting oscillations, canard oscillations, and mixed-mode oscillations. Although these examples all involve neural systems, the technique can and has been applied to other biological, chemical, and physical systems. It is a powerful analysis method that will become even more useful in the future as new experimental techniques push forward the complexity of biological models.
authors.affiliation: Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University, Florida State University, Tallahassee, FL, United States. Electronic address: bertram@math.fsu.edu.
authors.forename: Richard
authors.initials: R
authors.lastname: Bertram
authors.affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, United States.
authors.forename: Jonathan E
authors.initials: JE
authors.lastname: Rubin
mesh_terms: D001683:Biological Clocks; D008954:Models, Biological
keywords: Bursting; Canards; Fast-slow analysis; Mixed-mode oscillations; Multiscale analysis; Relaxation oscillations