Fractal character of the electrocardiogram: distinguishing heart-failure and normal patients.

Statistical analysis of the sequence of heartbeats can provide information about the state of health of the heart. We used a variety of statistical measures to identify the form of the point process that describes the human heartbeat. These measures are based on both interevent intervals and counts, and include the interevent-interval histogram, interval-based periodogram, rescaled range analysis, the event-number histogram, Fano-factor, Allan Factor, and generalized-rate-based periodogram. All of these measures have been applied to data from both normal and heart-failure patients, and various surrogate versions thereof. The results show that almost all of the interevent-interval and the long-term counting statistics differ in statistically significant ways for the two classes of data. Several measures reveal 1/f-type fluctuations (long-duration power-law correlation). The analysis that we have conducted suggests the use of a conveniently calculated, quantitative index, based on the Allan factor, that indicates whether a particular patient does or does not suffer from heart failure. The Allan factor turns out to be particularly useful because it is easily calculated and is jointly responsive to both short-term and long-term characteristics of the heartbeat time series. A phase-space reconstruction based on the generalized heart rate is used to obtain a putative attractor's capacity dimension. Though the dependence of this dimension on the embedding dimension is consistent with that of a low-dimensional dynamical system (with a larger apparent dimension for normal subjects), surrogate-data analysis shows that identical behavior emerges from temporal correlation in a stochastic process. We present simulated results for a purely stochastic integrate-and-fire model, comprising a fractal-Gaussian-noise kernel, in which the sequence of heartbeats is determined by level crossings of fractional Brownian motion. This model characterizes the statistical behavior of the human electrocardiogram remarkably well, properly accounting for the behavior of all of the measures studied, over all time scales.

A nonstationary Poisson point process describes the sequence of action potentials over long time scales in lateral-superior-olive auditory neurons.

The behavior of lateral-superior-olive (LSO) auditory neurons over large time scales was investigated. Of particular interest was the determination as to whether LSO neurons exhibit the same type of fractal behavior as that observed in primary VIII-nerve auditory neurons. It has been suggested that this fractal behavior, apparent on long time scales, may play a role in optimally coding natural sounds. We found that a nonfractal model, the nonstationary dead-time-modified Poisson point process (DTMP), describes the LSO firing patterns well for time scales greater than a few tens of milliseconds, a region where the specific details of refractoriness are unimportant. The rate is given by the sum of two decaying exponential functions. The process is completely specified by the initial values and time constants of the two exponentials and by the dead-time relation. Specific measures of the firing patterns investigated were the interspike-interval (ISI) histogram, the Fano-factor time curve (FFC), and the serial count correlation coefficient (SCC) with the number of action potentials in successive counting times serving as the random variable. For all the data sets we examined, the latter portion of the recording was well approximated by a single exponential rate function since the initial exponential portion rapidly decreases to a negligible value. Analytical expressions available for the statistics of a DTMP with a single exponential rate function can therefore be used for this portion of the data. Good agreement was obtained among the analytical results, the computer simulation, and the experimental data on time scales where the details of refractoriness are insignificant.(ABSTRACT TRUNCATED AT 250 WORDS)

Rate fluctuations and fractional power-law noise recorded from cells in the lower auditory pathway of the cat.

The noise properties of the sequence of action potentials recorded from adult-cat auditory nerve fibers and lateral superior olivary units have been investigated under various stimulus conditions. Large fluctuations exhibited by the spike rate, and spike clusters evident in the pulse-number distribution, both indicate an unusual underlying sequence of neural events. We present results demonstrating that (i) the firing rate calculated with different averaging times can exhibit self-similar behavior; (ii) the pulse-number distribution remains irregular even for large numbers of samples; (iii) the spike-number variance-to-mean ratio increases with the counting time T in fractional power-law fashion for sufficiently large T; and (iv) the exponent in the power law generally depends on the stimulus level. The results obtained in our laboratories support the notion that all auditory-nerve and LSO units exhibit fractal neural firing patterns, as indicated earlier by Teich (IEEE Trans. Biomed. Eng. 36, 150-160, 1989).

Multinomial pulse-number distributions for neural spikes in primary auditory fibers: theory.

We previously reported experimental short- and long-counting-time pulse-number distributions (PND's) for the neural spike train in cat primary auditory nerve fibers. Data were obtained for spontaneous activity, pure-tone stimuli with a wide range of frequencies and intensity levels, and Gaussian noise. The irregular shapes of the PND's are an indication of the presence of spike clusters of various sizes in the neural impulse train. We develop a family of theoretical cluster counting distributions and examine their suitability for describing the experimental PND's. The reduced-quintinomial distribution provides theoretical results that describe the characteristics of the PND's quite well, accounting for the smooth or scalloped behavior of short-counting-time data, the jagged nature of long-counting-time data, and the Poisson-like character of very-short-counting-time data. This family of distributions admits values for the spike-number mean-to-variance ratio that are independent of stimulus level, in agreement with experimental observation. A number of procedures for fitting the theoretical distributions to the experimental PND's are studied. These include the use of a minimum mean-square error criterion, the factorial moments of the data, and the discrete Fourier transform of the PND. The first of these techniques appears to be the most useful.