Describing function approximation for biomedical engineering applications

Abstract

--This paper focuses on the determination of suitable approximations for sigmoid-type nonlinear characteristics, which are common to physiological systems, particularly cardiovascular regulatory systems. These sigmoid nonlinearities have been implicated in the development of limit cycle oscillations in blood pressure. Approximations of the sigmoid are required since the describing function is not calculable for the all representations of the sigmoid characteristic. In this paper, we present a new approximation, which gives a better overall approximation of the sigmoid and hence, can assist the use of describing functions in the diagnostic analysis of cardiovascular function.

A Simple Soft Limiter Describing Function for Biomedical Applications

“…The DF, as detailed in Section III [see (12)], assumes an input sinusoid of a single frequency. The nonlinear soft limiter characteristic will, however, produce harmonics of this single frequency which may propagate around the feedback loop (Fig.…”

“…This, aside from any other complications, leads to difficulties in analytical evaluation of the Fourier integrals associated with DF calculations for these characteristics and has necessitated the polynomial approximations used in [11] and [12]. One sigmoid description which avoids the use of exponential terms is the Hill function, as used by Abbiw-Jackson and Langford [19] (7)…”

“…Also, the terms can be evaluated over or as desired. The DF, representing the gain between the fundamental of the output and the input, can now be defined as (12) For the sigmoid of (8), with (13) Equation (13) can be recast as (14) and integrated, by parts, to give (15) (16) with the second term of the "integration by parts" disappearing at both upper and lower limits. Equation (16) is easily recast as (17) The remaining integral in (17) may be evaluated [21] as (18) Though the integral appears to be zero at both limits, observation of the symmetry of the integrand in (17) allows the integral to be recast as (19) and, with the final observation that (20) the DF in (17) may be evaluated as …”

“…This has consequences for the stability analysis in Section V in terms of a unique solution for the limit cycle amplitude. The DFs of previous approximations have displayed a variety of properties: the polynomial approximation of Kinnane et al [12] is monotonic, while the Taylor series approximation of Holohan [11] is not.…”

“…0.1 Hz in humans) oscillations in blood pressure [8]. However, achieving 2) has proved somewhat elusive, with highly unwieldy DF expressions resulting from a variety of approximations [11], [12] to the sigmoid characteristics, necessitated due to the inability to get closed-form DF expressions for the original sigmoid descriptions. This paper proposes the ubiquitous arctangent function as a suitable parameterisation for sigmoidal characteristics typical in physiological subsystems.…”