Peeking into the Black Box

Introduction

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.

We have argued—and, we hope, shown—that a quasiwhite stimulus is indeed the most desirable stimulus with regard to both completeness and accuracy of the identification, for the kind of nonlinear systems that appear functionally to us as "black boxes," as most physiological systems do.

Following the determination of F, the next question concerns the structure of the system. That is, how is F realized by the different interacting components of the system? How do changes in the characteristics of a particular component affect the system function F? Paradoxically, subanalysis of the system structure often becomes possible if the system is nonlinear, while it is nearly impossible if the system is linear. This is due to the fact that nonlinear operators, unlike linear ones, do not commute, in general. For example, as shown in Fig. 9.1, if systems U and/or W are nonlinear, it may be possible to determine whether U precedes or follows W, while this is not possible if U and W are linear. We show how below.