Multi-timescale systems and fast-slow analysis
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Updated: Mar 17, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
Published on: December 4, 2017
Richard Bertram1, Jonathan E Rubin2
1Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University, Florida State University, Tallahassee, FL, United States.
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.
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