"Dominant-scale analysis for automatic reduction of high-dimensional ODE systems"
Systems of ordinary differential equations arise commonly as models in the natural sciences, often with multiple time-scales. State-dependent coupling can add to the complexity, introducing new time-scales. These time-scales may not be explicit in the equations. We have developed a computational technique that can be used to reduce the study of such networks evolving near a known trajectory, to a set of low-dimensional approximate models. The dominant-scale technique adds rigor to intuitive reduction techniques that are ubiquitous in modeling high-dimensional coupled systems, and is different to a center manifold reduction. In particular, it quantifies the robustness and parametric dependence of coherent temporal activity along the entire length of a known trajectory. It also provides a quantitative basis for rigorously defining intuitive concepts such as "emergent structure", "evolving sub-systems", etc., in spatially-extended physical models. We demonstrate our analysis software on an example network of Hodgkin-Huxley equations for biological membrane excitability.