HOUSE_OVERSIGHT_013709.jpg

along the x axis and the logarithm of the ratio of the range to the standard deviation on the y axis. An independent random system has a Hurst of 0.5. If a sequential increase or decrease in an amplitude or inter-event time tends to be followed by a change in the same direction, the Hurst > 0.5. If an increase in the measure tends to be followed by a decrease, then Hurst < 0.5. Computation of the Fano factor (power law exponent) exploits the same general strategy using the variance/mean in place of the range/variance and counting the number of events (such as single neuron discharges or heartbeats) in time windows of increasing length, generating a similar log-log graph. There is a relatively long history of the use of spike-number variance-to mean ratio in studies of response variability in visual cortical neurons (see Teich et al, 1996 for a review). The Allen factor (power law exponent) tends to reduce the influence of local trends by a computation of the variance of the difference between the number of events in two successive time windows divided by twice the mean number of events in the window.

Each system’s invariant logarithmic slope across sample segment sizes takes the place of its missing finite variance in characterizing experimental data in which the distributional tails do not converge (or do so very slowly) to the x axis. Recent approaches to these measures in the context of stochastic analysis of DNA sequences, but also applied to normal and pathological cardiac inter-beat intervals and gait interval sequences, have dealt with the influence of non-stationarity due to apparent trends in the data on α-equivalent indices by local mean-normalization of the fluctuations at each window size (Peng et al, 1993; Peng et al, 1995; Hausdorff et al, 1995).

The rate of decay of the densities in the tails of the probability distribution as they approach extreme values along the x axis, called the Levy exponent when represented in Fourier space (technically, as a “characteristic function” of the probability distribution) (Shlesinger, 1988; Shlesinger et al, 1995), can also be computed directly on the distribution by fitting the tails with a two parameter curve quantifying their “fatness” and rates of decay (Mantegna, 1991). We can speak of a Gaussian tail as having an exponential decay rate representable by α = 2 implying

209

HOUSE_OVERSIGHT_013709