Qualitative parameter estimation for a class of relaxation oscillators

Abstract

Motivated by neuroscience applications, we introduce the concept of qualitative estimation as an adaptation of classical parameter estimation to nonlinear systems characterized by i) the presence of possibly many redundant parameters, ii) a small number of possible qualitatively different behaviors, iii) the presence of sharply different characteristic timescales and, consequently, iv) the generic impossibility of quantitatively modeling and fitting experimental data. As a first application, we illustrate these ideas on a class of nonlinear systems with a single unknown sigmoidal nonlinearity and two sharply separated timescales. This class of systems is shown to exhibit either global asymptotic stability or relaxation oscillations depending on a single ruling parameter and independently of the exact shape of the nonlinearity. We design and analyze a qualitative estimator that estimates the distance of the ruling parameter from the unknown critical value at which the transition between the two behaviors happens without using any quantitative fitting of the measured data.