Volterra-Wiener Analysis of a Class of Nonlinear Feedback Systems and Application to Sensory Biosystems

Abstract

This article addresses an issue that sits at the confluence of two important, yet marginally explored, problems: the Volterra-Wiener expansions of nonlinear differential equations and the Wiener (white noise) analysis of nonlinear feedback systems. These two problems can be addressed in the same methodological context, as explained below. The importance of this issue derives from the desire to relate parametric (i.e., differential equations) models with nonparametric (i.e., Volterra-Wiener functional expansions) models for nonlinear dynamic systems whose internal structure and functional organization are inadequately known to the investigator to allow the development of an explicit mathematical model on the basis of physical/chemical principles („black box“ formulation). This problem has been addressed extensively in the linear case (linear realization theory) and practical methods have been developed for this purpose. However, this problem has received little attention in the nonlinear case, not for lack of importance but because of its considerable complexity. Notable efforts have been made for development of a „nonlinear realization theory“ (e.g., Rugh, 1981) but no practical methods are currently available that can be employed by investigators who study nonlinear physiological „black box“ systems in order to arrive at parametric (nonlinear differential equation) models from inputoutput experimental data. On the other hand, many investigators have followed the Wiener approach to obtain nonparametric models (in the form of Volterra-Wiener expansions) of nonlinear physiological systems in recent years.