## Characterizing Nonlinear Systems

Many of the extensions of linear system theory to the characterization of frankly nonlinear systems presuppose a particular configuration or partitioning of separable linear and nonlinear portions of the system. While such techniques may have some utility in the study of neural systems, it is clear that a more "black box" method that makes no supposition about system configuration could also be useful in the characterization of neuro-sensory systems. The white noise method of characterizing nonlinear systems, as introduced by Wiener (1958), offers in theory this utility as well as concise, quantitative descriptions of the system dynamics and nonlinearity. The Wiener white noise method allows one to build a model that emulates the behavior of the nonlinear system under study (NSUS) that is optimum in a minimum MSE sense, but sheds little light on the biology of the system; it provides an optimum black box model.

The Wiener white noise approach is related to the Volterra (1959) expansion of the input/output characteristics of the NSUS in terms of a power series with func-tionals as terms. (A functional is a term whose argument is a function, and whose value is a number. A definite integral is a functional; a real convolution is a functional.) Wiener showed that a nonlinear dynamic system excited by Gaussian white noise could be described in terms of an infinite series of orthogonal functionals in an elegant derivation. The output of the NSUS can be written:

where {G} is a complete set of orthogonal functionals with respect to the input Gaussian white noise, x(t), and {h} is the set of kernels that characterizes the impulse responses of the NSUS. The first four Wiener functionals are k=0

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