Reduced Order Modeling of Nonlinear Dynamical Systems Using Slow Manifolds

TL;DR

This paper explores a method for reducing complex nonlinear dynamical systems by identifying slow manifolds through intersections of stable and unstable manifolds, demonstrated on neural and circadian models.

Contribution

It introduces a novel approach to model reduction using slow manifolds defined by manifold intersections, with detailed biological examples and control applications.

Findings

Reduced models accurately capture system dynamics

Method applicable to highly nonlinear biological systems

Effective for control objectives in neuroscience and circadian rhythms

Abstract

Model order reduction in high-dimensional, nonlinear dynamical systems if often enabled through fast-slow timescale separation. One such approach involves identifying a low-dimensional slow manifold to which the state rapidly converges and subsequently studying the behavior on the slow manifold. This work investigates slow manifolds defined by the intersection of an unstable manifold of an unstable fixed point or periodic orbit and the stable manifold of a stable attractor. When the decay rates of perturbations transverse to the unstable manifold are sufficiently large, the resulting slow manifold can be used for reduced modeling purposes by leveraging the isostable coordinate framework. Detailed examples are provided for two different highly nonlinear dynamical systems, the first being a coupled system of Hodgkin-Huxley neurons and the second being a biophysically detailed model of circadian oscillations. The resulting reduced order models are illustrated in two different biologically motivated control objectives.