Data-Driven Inference of High-Accuracy Isostable-Based Dynamical Models in Response to External Inputs

TL;DR

This paper introduces a data-driven method to accurately infer isostable-based reduced models for nonlinear dynamical systems with fixed points, especially effective for large inputs where traditional linear models fail.

Contribution

It develops a novel inference strategy that estimates isostable response functions and output relationships directly from data, enabling high-accuracy modeling without known equations.

Findings

Effective for large magnitude inputs

Accurate inference of isostable response functions

Validated on neuronal and fluid dynamics models

Abstract

Isostable reduction is a powerful technique that can be used to characterize behaviors of nonlinear dynamical systems in a basis of slowly decaying eigenfunctions of the Koopman operator. When the underlying dynamical equations are known, previously developed numerical techniques allow for high-order accuracy computation of isostable reduced models. However, in situations where the dynamical equations are unknown, few general techniques are available that provide reliable estimates of the isostable reduced equations, especially in applications where large magnitude inputs are considered. In this work, a purely data-driven inference strategy yielding high-accuracy isostable reduced models is developed for dynamical systems with a fixed point attractor. By analyzing steady state outputs of nonlinear systems in response to sinusoidal forcing, both isostable response functions and isostable-to-output relationships can be estimated to arbitrary accuracy in an expansion performed in the isostable coordinates. Detailed examples are considered for a population of synaptically coupled neurons and for the one-dimensional Burgers' equation. While linear estimates of the isostable response functions are sufficient to characterize the dynamical behavior when small magnitude inputs are considered, the high-accuracy reduced order model inference strategy proposed here is essential when considering large magnitude inputs.