FIGURE 8.3-1 Block diagram describing how the zeroth, first, and second-order kernels are calculated for a SISO nonlinear system, using the white noise method developed by Marmarelis (1972). Note that the second-order kernel, h2(ij, t2), is presented as a contour plot.

where U(t - tk) is a unit step that occurs at t = tk, and U(t - tk+1 ) is a unit step at tk+1. Figure 8.3-3 illustrates the generation of the instantaneous frequency (IF) signal, q(t). This function is physically realizable because rk is not defined until the kth pulse in the sequence occurs, triggering a step at tk of height rk and lasting until the next pulse at tk+1.

An application of the IF description of a spike train used in the white noise method was given by Poliakov (Poliakov et al., 1997; Poliakov, 1999). Poliakov injected a dc plus broadband noise current into a motoneuron soma with a micro-electrode and recorded its spike activity on its axon. A mild, second-order nonlin-earity was assumed in which the input noise first acted on the linear kernel, h1(t). The output of h1(t), u, was assumed to be the input to a nonlinearity of the form: y(t) = h0 + u(t) + au2(t). y(t) is the output of the NSUS, in this case, the IF of the motoneuron's spike train. Figure 8.3-4 illustrates the typical form of h1(t), h2(x1, t1)

FIGURE 8.3-2 Block diagram showing how the Fourier transforms of the first- and second-order kernels can be calculated.

and h2(xj, t2) found for the motoneurons he studied. Note that the second-order kernel is symmetrical around the line Tj = t2.

Another example of the application of the white noise method to a neuro-sensory system was described by Marmarelis and Naka (1973a). In this case the system was the single-input/single output catfish retina photoreceptors to a spiking ganglion cell (GC). The input was noise-intensity-modulated light in the form of the general RF, a spot at the center of the RF, or an annulus around the center of the RF. The first system studied by Marmarelis and Naka (1973a) using white noise analysis was the analog input/analog output light (whole RF) to horizontal cell (HC) pathway. (For a description of the anatomy of the vertebrate retina, see Section 6.1.) Because of the spatiotemporal filtering properties of the retina, and the sigmoidal (log intensity)

FIGURE 8.3-3 A spike train, showing how the spikes' instantaneous frequency elements can be converted to a stepwise, analog output, ry(t), by use of a holding process. The height of each step of ry(t) is proportional to the IF of the preceding pulse interval.

response characteristic of photoreceptors, the calculated kernels for light input/HC response are different for different mean light intensities and stimulus shapes (spot, annulus, whole RF). Figure 8.3-5 illustrates the gross cellular features of the retina. Figure 8.3-6 show the linear kernels for a horizontal cell membrane potential when its whole RF was stimulated by "white" noise at two different mean intensities. (Note that the RF of the HC recorded from contains many photoreceptors and other HCs.) Figure 8.3-7 shows two second-order kernels calculated for the same two mean intensities. The higher mean intensity gives more pronounced peaks and valleys in the h2(xj, t2) plots. Finally, Figure 8.3-8 shows hj(t) plots, each calculated for three increasing mean intensities (C highest) for a spot stimulus (A) and for an annulus stimulus (B). Note that the annulus impulse response becomes more under-damped with increasing mean intensity.

When Marmarelis and Naka (1973a, b) used white noise analysis to characterize the light to GC responses of the catfish retina, they faced the problem of converting the point process describing the GC spikes to a continuous signal proportional to frequency. In this case, they did not use IF, but instead used the following procedure. Five to ten runs were made using identical noise records. A pooled, PST histogram was made from the resulting GC spikes for each run. (The pooled PST diagram had the dimensions of spikes/s.) The pooled PST diagram was smoothed with an "appropriate smoothing window," to form y(t) used in kernel calculations. Figure 8.3-9 shows three h1(t) kernels, all from the same GC system calculated for uniform RF stimulation, and spot and annulus stimuli. Note the strong biphasic response for the annulus and uniform RF stimulation, denoting temporal differentiation of the input. The first and second-order kernel contour plots for the GC system with entire RF stimulation are shown in Figure 8.3-10. Note the sharp peaks and valleys in h2(x1, t2). To interpret these features, note that a positive impulse of light intensity (a short flash) given 100 ms in the past followed by a second flash at 80 ms in the past will

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