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Fractal intensity point processes--doubly stochastic point processes with a fractal waveform intensity process--are required to describe the discharge patterns recorded from the auditory and visual systems. The Fano factor--the ratio of the variance of the number of events in an interval to the mean of this number--captures the self-similar characteristics of the intensity via two quantities: fractal dimension and fractal time. The fractal dimension is the exponent of the asymptotic power law behavior of the Fano factor with interval duration. The fractal time delineates long-term fractal behavior from short-term characteristics of the data. The average rate and self-similarity parameter of the intensity process, absolute and relative refractory effects, and serial dependence all modify the fractal time. To generate fractal intensity point processes, stochastic fractal processes are derived by applying memoryless, nonlinear transformations to fractional Gaussian noise. The intensity's amplitude distribution in combination with the Fano factor form criteria to choose the transformation that best describes data.

The temporal pattern of the action potential discharges of auditory nerve fibers have been presumed to be random in nature and well described as a renewal process.Recently, we observed that the discharges were characterized by excess variability and low frequency spectra. Simple stationary point process models, point process models with a chaotic intensity, and doubly stochastic point process models with a strongly mixing intensity process do not describe the data. But a fractal point process, defined to be a doubly stochastic point process with a fractal waveform intensity process, described the the data more generally. The sample paths of the counting process of a fractal point process are not self-similar; it displays self-similar characteristics only over large time scales.The Fano factor, rather than the power spectrum, captures these self-similar characteristics. The fractal dimension and fractal time characterize the Fano factor. The fractal dimension is characteristic of the fractal intensity process and is measured from the asymptotic slope of the Fano factor plot. The fractal time is the time before the onset of the fractal behavior and delineates the short-term characteristics of the 11 data. Absolute and relative refractory effects, serial dependence, and average rate modify the fractal time.To generate a fractal point process, fractal intensity processes are derived through memoryless, nonlinear transformations of fractional Gaussian noise. All the transformations considered preserve the self-similarity property of the fractional Gaussian noise. A theorem due to Snyder [32) relates the estimate of the asymptotic pulse number distribution (PND) to the asymptotic distribution of the integrated intensity process of a doubly stochastic Poisson process. The shape of the probability density function of the intensity process is greatly influenced by the choice of the transformation. The intensity density function, in combination with the Fano factor form a criterion to choose the transformation that best describes the data.

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The spontaneous activity and the responses to single, high-frequency tones of single auditory-nerve fibers are often described as a stationary point processes. Using a doubly stochastic Poisson process model as the analysis framework, it is shown that the instantaneous rate of discharge varies more than is consistent with a stationary model. Variability indices (which equal unity for a stationary Poisson process model) were found to be no smaller than 1.5 and at least as large as 6.9 for the data analyzed. Assuming this variability to be due to the presence of a noise process modulating the instantaneous rate, the spectra of the noise processes were measured. These spectra were low pass, having upper frequencies no greater than about 0.1 Hz. Some spectra may contain spectral lines, indicating the presence of nearly periodic components. The presumed noise process, whether containing periodic components or not, could not be readily related to ambient acoustic noise. [Work supported by NINCDS.]