Figure 4

Multifractal analysis of numerically-generated time series and experimentally obtained residence-time series of active and inactive states in a representative C. elegans animal. (A-D) MF-DFA of numerically-generated time series with monofractal-like (A) and multifractal (B) properties. The absolute values of white and pink noise time series were shown in (A). Time series with small/middle/great multifractality were generated by the multiplicative cascading processes using log-normal functions whose variances were determined by random noises with small/middle/large variances (^41,42; Supplemental Information) (B). Experimentally-obtained active/inactive state residence-time series from high-motility period of Long-activity wild-type (C) and egl-4 mutant (D) animals. Time series [blue, in (A) and (B)] and cumulative sum of the deviations from the average of values in the time series [red, in (A), (C), and (D)] are shown in upper three (A, B) and two (C, D) graphs. \(F(q,\;s)\) vs. scale s plot data (dots) and their fit functions (lines) are shown at lower q values (cold colors) and higher q values (hotter colors) in a range of − 10 < q < 10 [left column in the lower three (A, B) and two (C, D) graphs]; corresponding multifractal spectra are shown in right columns of each graph. Original white, pink, and Brown noise time series are shown, respectively, after 10, 10², and 10⁴-time magnifications to adjust the amplitude of their cumulative sums in (A). Note that cumulative sums of noise (A, B) and residence-time series (C, D) can take negative values because they are calculated from deviations from the average of the values in the time series. Also note that there is no cumulative sum of the multifractal time series in panel B, because cumulative sums of multifractal time series in log-scale result in a profound loss of information about the smoothness as discussed in cumulative sums shown in a linear sale (C, D). Instead the intermittent activity bursts in numerically-generated multifractal time series in (B) can be compared with experimentally-obtained residence-time series in Fig. 3B,C (both were shown in log scale). The smoothness of the cumulative sums of numerically-generated monofractal-like time series in A can be compared with those of experimentally-obtained cumulative sums of residence-time series in (C) and (D) (both were shown in linear scale).