Irregularity and asynchrony in biologic network signals.

The principal focus of this chapter has been the description of both ApEn, a quantification of serial irregularity, and of cross-ApEn, a thematically similar measure of two-variable asynchrony (conditional irregularity). Several properties of ApEn facilitate its utility for biological time series analysis: (1) ApEn is nearly unaffected by noise of magnitude below a de facto specified filter level; (2) ApEn is robust to outliers; (3) ApEn can be applied to time series of 50 or more points, with good reproducibility; (4) ApEn is finite for stochastic, noisy deterministic, and composite (mixed) processes, these last of which are likely models for complicated biological systems; (5) increasing ApEn corresponds to intuitively increasing process complexity in the settings of (4); and (6) changes in ApEn have been shown mathematically to correspond to mechanistic inferences concerning subsystem autonomy, feedback, and coupling, in diverse model settings. The applicability to medium-sized data sets and general stochastic processes is in marked contrast to capabilities of "chaos" algorithms such as the correlation dimension, which are properly applied to low-dimensional iterated deterministic dynamical systems. The potential uses of ApEn to provide new insights in biological settings are thus myriad, from a complementary perspective to that given by classical statistical methods. ApEn is typically calculated by a computer program, with a FORTRAN listing for a "basic" code referenced above. It is imperative to view ApEn as a family of statistics, each of which is a relative measure of process regularity. For proper implementation, the two input parameters m (window length) and r (tolerance width, de facto filter) must remain fixed in all calculations, as must N, the data length, to ensure meaningful comparisons. Guidelines for m and r selection are indicated above. We have found normalized regularity to be especially useful, as in the growth hormone studies discussed above; "r" is chosen as a fixed percentage (often 20%) of the subject's SD. This version of ApEn has the property that it is decorrelated from process SD--it remains unchanged under uniform process magnification, reduction, and translation (shift by a constant). Cross-ApEn is generally applied to compare sequences from two distinct yet interwined variables in a network. Thus we can directly assess network, and not just nodal, evolution, under different settings--e.g., to directly evaluate uncoupling and/or changes in feedback and control. Hence, cross-ApEn facilitates analyses of output from myriad complicated networks, avoiding the requirement to fully model the underlying system. This is especially important, since accurate modeling of (biological) networks is often nearly impossible. Algorithmically and insofar as implementation and reproducibility properties are concerned, cross-ApEn is thematically similar to ApEn. Furthermore, cross-ApEn is shown to be complementary to the two most prominent statistical means of assessing multivariate series, correlation and power spectral methodologies. In particular, we highlight, both theoretically and by case study examples, the many physiological feedback and/or control systems and models for which cross-ApEn can detect significant changes in bivariate asynchrony, yet for which cross-correlation and cross-spectral methods fail to clearly highlight markedly changing features of the data sets under consideration. Finally, we introduce spatial ApEn, which appears to have considerable potential, both theoretically and empirically, in evaluating multidimensional lattice structures, to discern and quantify the extent of changing patterns, and for the emergence and dissolution of traveling waves, throughout multiple contexts within biology and chemistry.