Modeling and analyzing fractal point processes

Abstract

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 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 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.