Chapter 33 dealt with linear systems and stated that there is no general method of describing nonlinearities. Nevertheless, these are so common and important in the visual system that methods of description are important, none more so than kernel analysis. The aims of this chapter are, first, to give a nonrigorous account of this method that will enable the clinical electrophysiologist to use the method and interpret the results and, second, to briefly indicate the rationale for using kernel analysis and the choice of strategy to fit particular situations. Finally, some clinical and experimental results of this method will be given. For a more mathematically rigorous treatment of many of these same points, the reader is referred to other sources.^{15,18,21–23,25,26,32,35,37}

The definitions of a linear system are given in the previous chapter. It will be recalled that if the response to a brief impulse is known, the response of a linear system to any other stimulus can be predicted. This is not true of a nonlinear system. Figure 34.1 provides an example of a non-linear system. Figure 34.1A is a diagram of two stimulus pulses—the stimulus that is readily obtained from Grass stroboscopes, for example; figure 34.1B shows the response. The early part of the response is shown as a solid line. In the absence of a stimulus, the record would continue according to the dotted line, but in the presence of a second flash, the record corresponding to the lower full line is obtained. If the responses to the first and second of the paired flashes were equal (a linear system), the upper of the two lines would be followed. Figure 34.1C shows the difference between the actual response and the larger response that is expected from a linear system. The waveform (for ease, look at the peak times) of the residual “real” second response and also the waveform of the difference bear a complex relationship to the impulse response. Many systems that are nearly linear have thresholds and saturation points, and stimuli of appropriate intensity can evoke nonlinear behavior. Such nonlinearities are typically referred to as nonessential nonlinearities. An example of a nonessential nonlinearity is the clipping observed from the output of a linear filter when the input is too large. Other systems exhibit nonlinearities throughout the full range of their operations. These are referred to as essential nonlinearities. A rectifier is a typical example of a physical system that demonstrates essential nonlinearities. As was indicated with the previous examples, nonlinearities can be modeled by electronic components (e.g., rectifiers, amplifiers, filters). In the case of figure 34.1, which behaves in a way very similar to the electroretinogram (ERG), the nonlinearity occurs at various voltage levels and is time dependent, that is, it demonstrates an essential nonlinearity.

Figure 34.1.  

First- and second-order kernels. The four panels illustrate several of the major points of first- and second-order kernels. A, Two impulses with a fixed separation. B, (1) The response to a single pulse, (2) the linear prediction of the response to two pulses with the delay illustrated in A (the response of two single flashes added together with the appropriate delay), and (3) the obtained response. C, The difference between the predicted response and the response obtained in B. D, One way of presenting the second-order kernel. The second-order kernel has three dimensions. Time from pulse 1 is on the abscissa, and time from pulse 2 is on the ordinate. The difference between linear predictions and obtained results (e.g., C) could be plotted either on the z-axis (not displayed) or as contour lines on the xy-coordinates. The main diagonal represents the response when the two pulses were at the same time and reflects second-order nonlinear ties related to amplitude differences in the pulses. One physical system with second-order amplitude dependent nonlinearities is a rectifier. Off diagonals represent the differences between predicted and obtained responses for a specific difference in time between the two pulses. C would represent an off diagonal, with the time between pulses illustrated in A.

Several strategies exist to characterize a system so that its response to an arbitrary stimulus can be predicted. In the time domain, these strategies are based on computing cross-correlations between the stimulus and the response.^{15,18,20,22,23,25,26,32,37} Stimuli that are used to determine kernels are presented in figure 34.2. Typical stimuli are white noise or pseudorandom sequences (PRS), such as M-sequences. In the frequency domain, the system's responses to a set of sine waves are described by Fourier analysis, and the responses of appropriate order (second order, etc.) are summed.^20,21,26,35

Figure 34.2.  

Stimuli for kernel analysis. The values of +1, 0, and −1 represent arbitrary dimensions. They may be thought of as input voltages, logic levels, or intensity levels. A, An effort to represent band-limited white noise. It should have a flat frequency spectrum and a Gaussian amplitude distribution. White noise may be approximated by using a sum of sinusoids (usually eight or more different frequencies). Binary pseudorandom sequences (B), ternary sequences (C), and sums of two sinusoids (D) have useful properties. Deterministic signals such as B, C, and D may have greater contrast, are computationally easier to analyze, and may be averaged.

In using these input signals, it is possible to calculate a series of integrals that fully characterize the system's response to any arbitrary stimulus. Kernels are the weights of these integrals; as such, they are analogous to the coefficients of a polynomial. The zero-order kernel represents the bias or mean response of a system. The first-order kernel, analogous to the polynomial's first-order coefficient, represents the best linear approximation (in a least mean square error sense) of the response elicited by the stimulus and estimates the impulse response. The second-order kernel is analogous to the second-order coefficient in a polynomial equation; it represents the interactions of two stimulus pulses or variations in stimulus pulse amplitude on the response. It is difficult to record enough response samples to characterize higher-order kernels accurately. Consequently, it is uncommon to calculate kernels beyond the second order. Figure 34.3 illustrates some of the difficulties of linear approximations of nonlinear systems. A linear approximation of a nonlinear system varies with the range of stimulus conditions over which the estimate is made.

Figure 34.3.  

Linear approximations of the response of a non-linear system. The dashed lines extending to the abscissa indicate the limits of the stimulus conditions; those extending to the ordinate indicate the limits of the observed responses. The straight lines were best fit through the indicated regions. Linear estimates of a nonlinear process will be different depending on the input conditions, such as mean luminance or contrast, and are highly dependent on the stimulus values used. Consequently, different experiments can yield very different estimates of the first-order kernel. The presence of a strong, reliable first-order kernel for a particular stimulus range does not mean that the system or the response is linear.