Estimating the Fractal Exponent of Point Processes in Biological Systems Using Wavelet- and Fourier-Transform Methods
This chapter considers point processes on a line, which model a variety of observed phenomena in the biological sciences. Comparing existing methods of estimating the fractal exponent of point processes with techniques based on wavelets requires a benchmark fractal stochastic point process. Many biological neurons transmit information by means of action potentials, which are localized regions of depolarization traveling down the length of an axon. The Fano factors (FF) and Allan factors (AF) can be expressed as special cases of more general measures based on the statistics of the wavelet- and scaling-coefficient sequences of stochastic point processes. The chapter outlines the relation between the FF and AF on the one hand, and the wavelet- and scaling-coerficient-based measures on the other. As with auditory nerve fibers, some neurons in the visual system transmit information by means of action potentials, and Fractal Stochastic Point Processes provide suitable models for describing the behavior of the neurons.