A computational tool for the reduction of nonlinear ODE systems possessing multiple scales

Robert Clewley, Horacio G. Rotstein, Nancy Kopell

Research output: Contribution to journalReview articlepeer-review

21 Scopus citations


Near an orbit of interest in a dynamical system, it is typical to ask which variables dominate its structure at what times. What are its principal local degrees of freedom? What local bifurcation structure is most appropriate? We introduce a combined numerical and analytical technique that aids the identification of structure in a class of systems of nonlinear ordinary differential equations (ODEs) that are commonly applied in dynamical models of physical processes. This "dominant scale" technique prioritizes consideration of the influence that distinguished "inputs" to an ODE have on its dynamics. On this basis a sequence of reduced models is derived, where each model is valid for a duration that is determined self-consistently as the system's state variables evolve. The characteristic time scales of all sufficiently dominant variables are also taken into account to further reduce the model. The result is a hybrid dynamical system of reduced differential-algebraic models that are switched at discrete event times. The technique does not rely on explicit small parameters in the ODEs and automatically detects changing scale separation both in time and in "dominance strength" (a quantity we derive to measure an i nput's influence). Reduced regimes describing the full system have quantified domains of validity in time and with respect to variation in state variables. This enables the qualitative analysis of the system near known orbits (e.g., to study bifurcations) without sole reliance on numerical shooting methods. These methods have been incorporated into a new software tool named Dssrt, which we demonstrate on a limit cycle of a synaptically driven Hodgkin-Huxley neuron model.

Original languageEnglish (US)
Pages (from-to)732-759
Number of pages28
JournalMultiscale Modeling and Simulation
Issue number3
StatePublished - 2005
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Chemistry
  • Modeling and Simulation
  • Ecological Modeling
  • General Physics and Astronomy
  • Computer Science Applications


  • Analytic approximation of solutions
  • Bifurcation analysis
  • Biophysical neural networks
  • Computational methods
  • Methods for differential-algebraic equations
  • Multiple scale methods


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