Applications of the White Noise Method to Neurobiological Systems

Many of the nonlinear neural systems that can be studied by the white noise method present special problems. The systems described above have assumed that both the white noise input and the NSUS output, y(t), are continuous analog signals. Many neural systems differ in that they can have nerve spike inputs and/or nerve spike outputs. Nerve spike trains are best characterized as point processes; that is, by a train of unit impulses whose occurrence times are the peak times of the recorded spikes, as shown in Equation 8.3-19:

To apply the white noise method effectively, it is expedient to convert yP(t) to a continuous function. One way to do this is to compute the elements of instantaneous frequency, rk, for yP(t). rk is defined at each spike occurrence time, tk as rk = 1/(tk - tk1), k = 2,3,4 ... 8.3-20

rk is also a point process, but it can be converted simply to a stepwise-continuous analog signal by integrating each rk8(t - tk) and holding the resulting step over the interval, {tk, tk+1}. The output of the process is the stepwise waveform, q(t), given by q(t) = Xrk[[- tk)-U(t- tk+1)] 8.3-21

Zero-mean y0(t) response

yo(t)

Cross-

x(t-x)

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