Triggered Correlation Applied To The Auditory System

Triggered correlation (TC), first described by de Boer (1967) and deBoer and Kuyper (1968), is a conditional expectation statistic, which under certain conditions allows an estimate to be made of an equivalent linear system weighting function associated with frequency discrimination in auditory systems. Broadband Gaussian noise is generated and used as a stimulus to the animal's "ear." The sound pressure level at the ear is x(t). A single eighth nerve or cochlear nucleus neuron responding to the auditory noise stimulus is isolated and recorded from electrophysiologically. A parsimonious TC system model is assumed, shown in Figure 8.2-1. The linear portion of the system is assumed to precede the spike generation process. Below is shown that the TC process provides a biased statistical estimate of the impulse response, h(t), of the equivalent linear filter. The impulse response of the filter is the primary, time-domain descriptor of the linear filter. It allows estimation of the set of ODEs that describe the filter, and calculation of the filter frequency response function by Fourier transform.

Narrow BPF

Narrow BPF

FIGURE 8.2-1 A parsimonious model for a frequency-selective auditory neuron. A linear. narrow bandpass filter processes the auditory information. Its output is the input to a spike generator.

The TC algorithm allows estimation of the cross-correlation function, Rxy(x), between the linear filter input, x(t), and its unobservable output, y(t). It is well known that in the limiting case where x is Gaussian white noise with a two-sided power density spectrum described by ®xx(f) = n/2 msV/Hz (or ®xx(ro) = n/(4n) msV/r/s), the cross-correlation function Rxy(x) can be shown to be related to the impulse response by (Lee, 1960):

The spike generation process is assumed to occur when the filter output, y(t), crosses a positive threshold, b, with positive slope. That is, w(t) = 8(t - tk) when y(tk) = b and y(tk) > 0. The spike generation process is illustrated in Figure 8.2-2. As will be seen below, a conditional expectation, x+(x) = E{x(t - t) | y = b, y > 0} is estimated each time an output nerve spike occurs, and x+(t) is averaged NT times for NT spikes. x(t - t) are past values of x relative to tk that led to w(t) = S(t - tk).

FIGURE 8.2-2 Block diagram of the spike generator model assumed to operate in the TC model architecture. The AND gate output m goes high (to 1) when the input derivative is > 0, and the input exceeds a positive threshold, b. The event m ^ 1 generates a positive output spike, w.

The sample mean of x+(t) is The underlying assumptions for the probability calculations that follow is that x, y, and y = z are joint Gaussian variables, characterized by a mean matrix n in which nx = E{x(t)}, ny = E{y(t)}, nz = 0, and a 3 x 3 covariance matrix, p: