Vasilis Z. Marmarelis

The White-Noise Method in System Identification

Early in the study of the dynamics of a physiological system the bioscientist is faced with the task of recognizing the domains of linearity and of nonlinearity of the stimulus-response transformation that the system performs and how these domains compare with the dominant natural variation of the stimuli in the system’s environment. As we saw in the previous chapter, the analytical advantages of linear systems are many, and, therefore, they justify the search for a linear domain in the system’s operational range (if such exists). However, the bioscientist must resist the temptation of being carried away by a natural desire for beautiful and explicit solutions since they often tend to be unrealistic idealizations. We cannot but bow to the evidence that nonlinear system characteristics are abundant in nature and go far beyond the trite admission that every physical system is in some way nonlinear. In much the same way that nonlinearities optimize the design of artificial systems, nonlinearities seem to be necessary for the optimal functioning of physiological systems from the behavioral point of view. There are many such examples: the logarithmic transformation of sensory input in order to accommodate large stimulus ranges, dynamic asymmetries arising from such physiological necessities as sensing direction, and many others.

The Problem of System Identification in Physiology

Even though the epistemology of the life sciences has a distinctly hierarchical organization — extending from the subcellular level to the behavioral — the main thrust of research up to now has focused on each particular level of this organization, e.g., at the molecular, cellular, or behavioral level. The relationships and interdependences between the various levels have been relatively neglected. This latter endeavor belongs to the realm of systems analysis. In addition, a great part of the methodology employed within each particular level (and being equally applicable to all of them) belongs to systems analysis. Thus, systems analysis, as a methodological tool, has both a “vertical” and a “horizontal” component in the hierarchy of physiological systems.

Peeking into the Black Box

Thus far we have approached the problem of analysis of a physiological system from the point of view of identifying its stimulus-response relationship and describing quantitatively its dynamics. That is, our objective has been the determination of the system functional F, where y(t) = F[x(t)] and x(t), y(t) are the stimulus and response, respectively. The identification problem was posed as follows: Given a system y(t) = F[x(t)], choose a set of stimuli {x i (t)} such that the stimulus-response pairs {x i (t), y i (t)} allow you to determine F as completely and accurately as possible, under given experimental conditions.

Applicability of the White-Noise Method and the Use of Quasiwhite Test Signals

The white-noise method of system identification, i.e., the Wiener formulation of the nonlinear system identification problem in connection with the crosscorrelation technique, appears to be a general, straightforward, and powerful approach to a subject where great mathematical complexity is usually encountered. However, the generality and the elegance of the method is bound to be moderated in actual applications by limitations imposed by reality.

Methods of Computation of System Kernels

The objective of functional identification is the determination of system kernels that furnish a complete description of the stimulus-response functional relation.

Tests and Analyses Preliminary to Identification Experiment

Like most types of analysis, the application of the white-noise method for the characterization of a physiological system can become very difficult and involved if not preceded by considerable preliminary analysis of the system characteristics and the tools available for its study. In addition, the amount of difficulty depends on the nature of the system nonlinearities and the degree of accuracy which we require from the derived characterization (model). In certain cases the application of the method to a physiological system could produce poor results after long experimental procedures and digital computations. Therefore, it is desirable to develop preliminary criteria and simple experiments and tests that give an indication of how complex the system is and how successful the white-noise method will be in each particular case.

Physiological Systems Requiring Special Treatment

In this chapter we discuss the identification procedures to be followed when the system does not fall into any of the categories covered in the previous chapters; for example, systems whose characteristics change with time (nonstationary systems), or whose inputs are functions of more than one independent variable (e.g., spatiotemporal stimuli). Several alternative methods of treating systems with point process (discrete) inputs and outputs (e.g., neural systems) are also presented.

Analysis of Physiological Signals

In this chapter we discuss briefly the nature of signals encountered in physiological systems and some of the analytical tools used to study them. We avail ourselves of this opportunity to present the mathematical preliminaries necessary for the development of the methodology of functional identification of systems (primarily the white-noise approach) that will be discussed in the remainder of this book. In the course of this discussion we present examples of analysis of physiological signals through the use of these tools and concepts of signal analysis.

Applications of the White-Noise Method to Neural Systems

In the previous chapters we covered the theoretical foundation of the white-noise method and the practical considerations of its actual use in physiological system identification and analysis. In this chapter we present some representative applications of the white-noise method to neural system identification and analysis. These applications are chosen because they include rather interesting cases of physiological system identification (e.g., one or two inputs, continuous or discrete input or output, etc.) and illustrate the various techniques and strategies of the white-noise method (e.g., choice of quasiwhite stimulus, representation of discrete data etc.).

Traditional Approaches to Physiological System Identification

Having discussed the analysis of physiological signals, we must now discuss the specific ways in which they propagate through systems and the resulting transformation of a stimulus signal into a response signal.