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.
Faced, therefore, with nonlinear physiological systems, for which the principle of superposition does not hold, the question naturally arises as to which types of test stimuli should be applied, so that from the resulting responses the bioscientist is able to deduce (or learn as much as possible about) the functional relationship between the stimulus and response of the system. The application of sinusoidal stimuli is certainly unpromising since the response of the system to two such stimuli of different frequencies is different, in general, from the sum of the responses due to each stimulus separately—so no additional information is gained besides the sinusoidal behavior. In general, the same will hold true for other types of stimulus such as steps, pulses, etc. Figure 4.1 illustrates the situation. If a pulse stimulus is applied, we learn the response of the system to this pulse and have little notion of the response of the system to any other type of stimulus. If the stimulus consists of two triangular pulses as shown in the figure, then we know the response of the system to these two triangular pulses and little else. The same applies for any other specific waveform, such as the one shown in part C of the same figure. Therefore, faced with a nonlinear system it appears that we can do nothing less than test it with a great variety of stimuli and catalog the responses to them. This is exactly what a white-noise stimulus accomplishes in an efficient manner.
KeywordsPhysiological System Volterra Series Volterra Kernel System Kernel Wiener Kernel
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